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Phase transitions in the MSSM

M.N.Dubinin (INP MSU)
Work with M.V.Dolgopolov, E.N.Rykova (Samara State Univ.)


Outline


One-dimensional picture of T evolution Two-dimensional picture ( v1 (T), v2(T) ) Bifurcation sets in the catastrophe theory Results for the MSSM parameter space






One-dimensional picture. The effective high temperature MSSM potential 2=v21+v22, tg = v2/v1

A.Brignole, J.Espinosa, M.Quiros, F.Zwirner, PL B324 (1994) 181


From D.Gorbunov, V.Rubakov, solid lines -- numerical calculation, dashed -- high T expansion, for different T and mH



Two-dimensional effective potential in the v1, v2 plane (in red, U=0 surface in green colour)




Evolution parameters

Control parameters

Ueff ( v1(T) , v2(T) | 1(T), 2(T), 3(T), 4(T), 5(T), 6(T), 7(T) )
Implicitly
1...7

( T | mU, mD, mQ, At, Ab, )


First order phase transition with Shaposhnikov criteria vc / Tc > 1

Where is does take place?



Higgs boson masses mh(T), mH(T), mA(T) which are always positively defined and large enough at low T Mixing angles , which respect some phenomenological constraints at low T


Transformation of SU(2) eigenstates to mass eigenstates




lead to the nonlinear equations for effective parameters lambda

M.D., A Semenov, Eur.J.Phys. C28 (2003) 223 M.Dolgopolov, M.D.,E.Rykova, Phys.Rev. D71 (2005) 075008

and the minimization conditions for dimension 2 parameters mu


Calculation of the one-loop threshold corrections in the framework of the finite temperature field theory (imaginary time formalism, Matsubara series) gives the result for efeective parameters lambda


Hurwitz zeta-function


General formalism is known as the theory of catastrophes

Isolated (nondegenerate) critical points Nonisolated (degenerate) critical points. Defines «bifurcation sets» as zero det of the equilibrium matrix (Hessian)

V.I. Arnold, Critical points of smooth functions and their canonical forms, Uspekhi Math. Nauk (USSR), 30 (1975) 3 R. Thom, Structural stability and morphogenesis, Reading, Benjamin, 1975 M. Morse, The critical points of a function of n variables, Trans. Am. Math. Soc., 33 (1931) 72



The four bifurcation sets for U(v1,v2 | 1,2,3,4,5)

Set (1) is an elementary Sylvester's criteria Set (4) also elementary, sets (2) and (3) give


Where is the light stop?


The regions of light stop and light sbottom in the (A, ) plane


Parameter set (A) everywhere (the light stop)
(A, ) contours for bifurcation set (1) (A, ) contours for bifurcation set (2)

( v, tg ) contours for set (4)


Temperature evolution of masses


Summary


High temperature MSSM one-loop effective Higgs potential with threshold corrections is explicitly constructed. Temperature evolution of masses and mixings from high T down to zero is explicitly obtained. The regions of MSSM parameter space where the mass(T) eigenstates exist are separated. Four bifurcation sets are found in the general THDM then projected onto the MSSM parameter space. MSSM EWPT favors the case of light stop.