Theoretical and Mathematical Physics, 132(2): 1144–1147 (2002) QUASIPOTENTIAL APPROACH TO QUANTUM CHROMODYNAMICS Nguyen Van Hieu∗ We use the Logunov–Tavkhelidze quasipotential approach to study some processes of quantum chromodynamics at finite densities and temperatures The dynamic equation for the color superconductivity is derived We discuss the spontaneous breaking of the chiral, flavor, and color symmetries and also the possibility of constructing a field theory of the composite three-quark systems Keywords: quantum chromodynamics, chiral symmetry, flavor symmetry, color symmetry, three-quark systems About forty years ago, Logunov and Tavkhelidze proposed the quasipotential approach in quantum field theory [1]–[4] This approach is an universal method for studying the bound states of relativistic two-particle systems and the asymptotic behavior of scattering amplitudes at high energies It is based on introducing an energy-dependent quasipotential and using a relativistic Schră odinger-type equation with the quasipotential It was then my fortune to work in the collaboration of scientists headed by A A Logunov at the Joint Institute for Nuclear Research The knowledge and experience in scientific work acquired then proved of substantial value In this paper, we propose using the Logunov–Tavkhelidze quasipotential method to study two relevant problems in quantum chromodynamics (QCD) at finite densities and temperatures: color superconductivity and spontaneous symmetry breaking We also discuss the possibility of constructing a field theory of the three-quark systems Color superconductivity The superconducting pairing of quarks in QCD can occur because of two mechanisms: the gluon exchange and the instanton-induced quark–quark interactions The individual contributions of each of these mechanisms to color superconductivity were considered in many papers [5]–[15] In the general case, it is necessary to take both of these mechanisms into account simultaneously The quasipotential method using functional integrals is adequate for this purpose We use the formalism with imaginary time Our condensed notation is β x = (x, τ ), dx = dτ dx, β= , kT where k is the Boltzmann constant and T is the absolute temperature We let ψA and ψ¯A denote the respective quark field and its conjugate, where the index A = (α, a, i) describes the set of the spinor index α = 1, 2, 3, 4, the color symmetry index a = 1, 2, , Nc , and the flavor symmetry index i = 1, 2, , Nf ∗ National Centre for Natural Sciences and Technology, Hanoi, Vietnam; Faculty of Technology, Vietnam National University - Hanoi, Hanoi, Vietnam, e-mail: nvhieu@ims.ncst.ac.vn Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol 132, No 2, pp 295–299, August, 2002 Original article submitted October 15, 2001 1144 0040-5779/02/1322-1144$27.00 c 2002 Plenum Publishing Corporation The starting point in constructing the theory of color superconductivity is the partition function for the system of interacting quarks, Z= ¯ exp − [Dψ] [Dψ] × exp dx dx ψ¯A (x)LB A ψB (x) × BD dy ψ¯A (x)ψ¯C (y)VCA (x − y)ψD (y)ψB (x) , (1) where ∂ −µ +γ∇+M ∂τ b j LB A = δa δi γ4 β , (2) α BD µ and M are the respective chemical potential and the mass of the bare quark, and VCA (x − y) is the quasipotential For simplicity, we consider the approximation in which the quasipotential depends only on the difference x − y of the two coordinates Our considerations are easily applicable to the general case where the complex quasipotential depends on the differences of the various coordinates of all four quark fields AC We introduce the bilocal bispinor composite fields ΦAC (x, y) and their conjugates Φ (x, y) satisfying the Fermi–Dirac statistics, ΦCA (y, x) = −ΦAC (x, y), Φ CA (y, x) = −Φ AC (x, y), and apply the Hubbard–Stratonovich transformation Then the partition function Z can be represented as the functional integral in the composite fields, Z= [DΦ] [DΦ] exp Seff [Φ, Φ] , (3) with some effective action Seff [Φ, Φ] The equations for the composite fields δSeff [Φ, Φ] δΦ AC =0 (4) (x, y) following from the principle of extreme action are the generalized BCS equations for the order parameter of the flavor superconductivity These equations were considered for particular cases of specific mechanisms of quark–quark pairing in [5]–[14] The quasipotential approach using the functional integrals [15] is the unique method for studying the general case where both mechanisms of quark–quark pairing operate simultaneously Spontaneous symmetry breaking The presence of the nonzero order parameters ΦAB results in the spontaneous breaking of the corresponding symmetry properties of the interacting quark systems with finite density and temperature Because these order parameters are the second-rank spinors (with two indices a, b = 1, 2, 3) of the color symmetry group SU (3)c , they contribute to the matrix elements of the various processes breaking the color symmetry including the processes of transmutation of a photon into a gluon and vice versa (photon–gluon mixing) [16] In addition, breaking the gauge color symmetry SU (3)c gives rise to a nonzero mass of the gluon with the appropriate quantum numbers of the group SU (3)c Because of the photon–gluon mixing, this massive gluon of the quantum chromodynamics of matter at finite densities and temperatures decays 1145 into electron–positron pairs This process can serve as experimental evidence for the quark–gluon plasma formation Quantum chromodynamics with the zero mass of the bare quark has the chiral symmetry The presence of the nonzero order parameters ΦAB gives rise to a nonzero mass of the quarks and thus spontaneously breaks the chiral symmetry Another mechanism for the spontaneous chiral symmetry breaking is the quark–antiquark pairing To study the quark–antiquark pairing [17], [18], we use partition function (1) in a somewhat different form: ¯ exp − [Dψ] [Dψ] Z= × exp dx dx ψ¯A (x)LB A ψB (x) × BD dy ψ¯A (x)ψB (y)UAC (x − y)ψ¯C (y)ψD (x) , (5) BD where the quasipotential is denoted by UAC (x − y) We introduce the composite bilocal meson fields A ΦB (x, y) with the transformation properties of the product ψ¯A (x)ψB (y) Using the Hubbard–Stratonovich transformation, we represent partition function (5) as the functional integral in the meson fields, Z= [DΦ] exp Seff [Φ] (6) with some effective action Seff [Φ] The equations for the composite meson fields δSeff [Φ] =0 δΦA B (x, y) (7) following from the principle of extreme action are the dynamic equations for defining the order parameters of the interacting quark system in the presence of quark–antiquark pairing If these order parameters are the singlets of the color and flavor symmetry groups, they give rise only to the spontaneous chiral symmetry breaking It was shown that in the case of the flavor symmetry group SU (2), there exist nonzero order parameters forming a triplet of this group and the flavor symmetry is therefore spontaneously broken [18] But there is a degeneration between the phases with the triplet and the singlet order parameters if the quark–quark interactions are induced by the instantons Formation of the triquark The quasipotential approach proves a rather convenient method for studying the formation of the triquark, the bound state of the three-quark system in QCD For this purpose, we use the functional integral for the interacting quark system in the form Z= ¯ exp −i [Dψ] [Dψ] × exp i d4 x d4 y ˆ B + mδ B ]ψB (x) × d4 x ψ¯A (x)[(∂) A A d4 z d4 u d4 v CAB (u, v, w; z, y, x)ψA (x)ψB (y)ψC (z) × VDEF d4 w ψ¯F (w)ψ¯E (v)ψ¯D (u) × (8) CBA with some quasipotential VDEF (u, v, w; z, y, x) of the interaction between three quarks Introducing the composite trilocal trispinor baryon fields ΨABC (x, y, z) and their conjugates ΨABC (x, y, z) and performing 1146 the corresponding Hubbard–Stratonovich transformation, we obtain functional integral (8) in the form Z= [DΨ] [DΨ] exp iSeff [Ψ, Ψ] (9) with some effective action Seff [Ψ, Ψ] The equations for the composite fields δSeff [Ψ, Ψ] ABC δΨ =0 (10) (x, y, z) following from the principle of extreme action are the system of dynamic integral equations for the bound states of the three-quark system These equations were obtained in [19] The further study of their solutions is an interesting problem in hadron physics REFERENCES 10 11 12 13 14 15 16 17 18 19 A A Logunov and A N Tavkhelidze, Nuovo Cimento, 29, 380 (1963) A A Logunov, A N Tavkhelidze, I T Todorov, and O A Khrustalev, Nuovo Cimento, 30, 134 (1963) A A Logunov, Nguyen Van Hieu, and O A Khrustalev, Nucl Phys., 50, 295 (1964) Nguyen Van Hieu and R N Faustov, Nucl Phys., 53, 337 (1964) M Alford, K Rajagopal, and F Wilczek, Phys Lett B, 422, 247 (1998); 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