Proc Natl Conf Theor Phys 36 (2011), pp 80-88 PHASE TRANSITION IN THE ELECTROWEAK THEORY PHAN HONG LIEN Military Technical Academy DO THI HONG HAI Hanoi University of Mining and Geology Abstract Spontaneous symmetry breaking in the electroweak theory is investigated at finite temperature and non-zero chemical potential We consider two sceneries µ2 > m2 > and m2 < ≤ µ2 The dispersion relations and critical temperature are determined in the case of negative m2 and all non-zero coupling constants It is shown that the chemical potential affects significantly on the phase transition and the condensated matter I INTRODUCTION Weinberg-Salam-Glashow theory is well known as a unification of weak and electromagnetic interactions In this model the SU (2) × SU (1) symmetry group is the minimal one [1, 2] However, the theory is only renormalizeable by Higgs mechanism, where the non - abelian gauge invariance is broken spontaneously (t’ Hooft 1971) [3, 4] This is the firsts realistic gauge theory that describes the experimental data with high accuracy Furthermore, the mechanism of spontaneous symmetric breaking provides a good investigation of Bose - Einstein condensation [5, 6] Our main aim is to present in detail the electroweak theory without fermions at finite temperature and non-zero chemical potential basing on the thermal field theory [7, 8] In this connection, it is possible to consider our work as being complementary to result previously at zero temperature and the U(1) coupling constant g = [7, 9] This paper is organized as follow In Section II the formalism is introduced in the presence of non-zero chemical potential µ and a source term Iν B ν Section III is devoted to the scenarios < m2 < µ2 In Section IV the scenarios m2 < ≤ µ2 is investigated In section V the critical temperature in the electroweak theory is derived Our conclusions are summarized in Section VI II FORMALISM We start from the Higgs sector of Lagrangian density in the electroweak theory basing on SU (2) × SU (1) symmetry L = [(Dµ − iµδ0µ ) Φ]+ (Dµ − iµδ0µ ) Φ − m2 Φ+ Φ − λ Φ+ Φ a aµν F − Bµν B µν (1) − Fµν 4 PHASE TRANSITION IN THE ELECTROWEAK THEORY 81 where µ is chemical potential, the coupling constant λ > τa Dµ = ∂µ − igAaµ − ig Y Bµ a Fµν = ∂µ Aaν − ∂ν Aaµ + g abc Abµ Acν Bµν = ∂µ Bν − ∂ν Bµ Here a, b, c = 1, 2, 3; µ, ν = 0, 1, 2, It is well known, the potential V (Φ+ Φ) = − µ2 − m2 Φ+ Φ + λ(Φ+ Φ)2 (2) is minimum at these values of scalar field Φ0 = µ2 − m2 v2 or Φ+ Φ = = φ = 0 2λ 2 2 2 If µ < m the solution stable is Φ0 = 0, if µ > m it is stable at Φ+ Φ0 = φ0 ν Introducing a source term Jν B into the Lagrangian (1), i.e L → L + Jν B ν (3) (4) where Jν = J0 δ0ν , < Bν >= The Lagrangian density (4) becomes L = (∂µ Φ)+ (∂µ Φ) + iµ Φ+ (∂0 Φ) − (∂0 Φ+ )Φ + + gτa Aaµ + g Bµ Φ+ gτa Aaµ + g B µ Φ + + i ∂µ Φ+ gτa Aaµ + g Bµ Φ − Φ+ gτa Aaµ + g B µ ∂µ Φ + − µΦ+ gτa Aa0 + g B0 Φ + (µ2 − m2 )Φ+ Φ + a aµν − λ(Φ+ Φ)2 − Fµν F − Bµν B µν + Jν B ν 4 (5) The equations of motion for scalar field Φ, gauge fields Bµ and Aµ , respectively, take the forms −Dµ Dµ Φ + µ gτa Aa0 + g B0 Φ + (µ2 − m2 )Φ − 2λ(Φ+ Φ)Φ = (6) ∂ µ Bµν + ig (Dµ Φ)+ Φ − Φ+ (Dµ Φ) − Jν = (7) τa g2 τa τa a c ∂ µ Fµν + g abc Aµb Fµν + ig Φ+ ∂ν Φ − ∂ν Φ+ Φ + Aaν Φ+ Φ + 2gµδν0 Φ+ Φ = (8) 2 2 The vacuum expectation value of gauge field Aµ is determined from Eqs (3) and (6) < Aa0 >= (9) The current J0 is derived from Eq (7) and the condition < B0 >= J0 = 2g µφ20 (10) 82 PHAN HONG LIEN, DO THI HONG HAI By shifting the scalar field by Φ→Φ= φ0 +√ χ =√ χ + v (11) where φ0 is the new ground state expectation value µ2 − m2 v =√ 2λ φ0 = (12) and χ is a real field (Higgs), which leads to the spontaneous breaking of symmetry As it well known, the gauge fields are usually defined by Wµ(±) = √ A1µ ∓ A2µ Zµ = Wµ3 = W 3µ = Aµ = gBµ + g A3µ (g + g )1/2 (13) gA3µ − g Bµ (g +g )1/2 = cosθA3µ − sinθBµ = sinθA3µ + cosθBµ (14) (15) where θ is the Weinberg angle, tgθ = g /g Define two new coupling constants gW = g e=g g (g + g )1/2 g (g + g )1/2 = gcosθ (16) = gsinθ (17) gW is just gauge coupling constant, a e is the electric charge The masses of gauge bosons W , Z are given by m2W = m2Z = m2W m2Z = v2 2 2 g φ0 = g v = gW + e2 4 2 2 2 g +e v g +g φ0 = W 2 gW gW + e2 gW (18) (19) (20) i.e the SU (2) symmetry breaking effect leads to mass different between the charge and neutral gauge bosons, Wµ± and Zµ PHASE TRANSITION IN THE ELECTROWEAK THEORY 83 The complete Lagrangian (5) that includes the Higgs sector and gauge part reads L= i ∂µ χ∂ µ χ − µ (∂0 χ+ )χ − χ+ (∂0 χ) + (µ2 − m2 )(v + χ)2 + 2 g + e2 λ − (v + χ)4 + µv W Z0 + 2eA0 χ + 4 gW + Wµ(+) ✷g µν − ∂ µ ∂ν + m2W Wν(−) + 1 + Zµ ✷g µν − ∂ µ ∂ν + m2Z Zν + Aµ (✷g µν − ∂ µ ∂ν) Aν + 2 g + W Zµ ∧ Wν(−) Z µ ∧ W ν(+) + g + e2 Wµ(+) ∧ Wν(−) W µ(+) ∧ W ν(−) + − W i − gW ∂µ Wν(+) − ∂ν Wµ(+) Z µ ∧ W ν(+) + i + gW ∂µ Wν(−) − ∂ν Wµ(−) Z µ ∧ W ν(−) + i + gW (∂µ Zν − ∂ν Zµ ) Wµ(+) ∧ Wν(−) + (+) + ie Fµν Aµ ∧ W ν(+) + (−) + Fµν Aµ ∧ W ν(−) ie e2 Gµν W µ(+) ∧ W ν(−) + Aµ ∧ Wν(+) 2 + Aµ ∧ W ν(−) (21) where (±) Fµν = ∂µ Wν(±) − ∂ν Wµ(±) ∓ igW Z µ ∧ Wν(∓) + e2 1 gW µ = µ − (g + g )1/2 Zµ = µ − Zµ 2 gW g + e2 (+) (−) g2 m2 = m2 + Wµ(+) Wµ(−) = m2 + W Wµ Wµ 2 (22) (23) (24) Let us consider in detail two scenarios m2 > and m2 < 0, the chemical potential µ play role as a parameter acting on the breaking of symmetry III FIRST SCENARIO: m2 > 0, e = g = In this case g = gW , mW± = mW0 = 21 gv If µ2 < m2 the SU (2) × U (1)Y × SU (3) symmetry is exact, the theory is relativistic If µ2 > m2 > 0, the U (1)Y symmetry is broken Take the ansatz as follow (+) W3 (±) W1,2 (−) = W3 = (±) W0 = I = 0, = (3) W1,2 = (3) Z0 = W0 (3) W3 =0 =K=0 (25) 84 PHAN HONG LIEN, DO THI HONG HAI In the Mean Field Approximation (M F A), where all field derivates had been setting to zero, the effective potential becomes g2 (−) Z0 ∧ W3 Z0 ∧ W 3(+) + g2 (+) (−) + Wi ∧ Wj W i(+) ∧ W j(−) + 1 (+) (−) µ − gZ0 − m2 + g W3 W3 − 2 V =− + (v + χ)2 + λ (v + χ)2 = −g I K − µ − gK 2 φ2 + m2 + g2I 2 φ2 + λφ4 (26) The physical processes satisfies the stationary condition at φ = φ0 ∂V ∂I |φ=φ0 ∂V = 0, ∂K |φ=φ0 = 0, ∂V =0 ∂φ |φ=φ which leads to the system of equations µ − gK K − φ20 I = (27) µ 2I + φ20 K = φ20 g (28) − m2 − g I − 2λφ20 φ0 = (29) For µ2 > m2 ( i.e v > 0), the ground state expectation value φ0 > 0, I = 0, Eq (27) yields √ K= φ0 > (30) Substituting (30) into Eq (29), we have (µ2 − m2 ) + 3gµ g2 − 2λ φ20 − √ φ0 = 2 (31) It’s solution reads φ0 = √ 2(8λ − g ) (g + 64λ)µ2 − 8(8λ − g )m2 − 3gµ (32) For g ≤ 8λ the potential has minimum at the solution stable φ0 At the critical value µ = m there is a second order phase transition PHASE TRANSITION IN THE ELECTROWEAK THEORY 85 IV SECOND SCENARIO: m2 < ≤ µ2 We focus on the dynamics in the second case m2 < in the presence of chemical potential µ ≥ Here, it is well known, the symmetry is broken spontaneously SU (2) × U (1)Y → U (1)EM Let us consider the effective potential in M F A It is defined from the Lagrangian (1) V = −LM F T IV.1 Case g = and g = 0, φ0 = const Firstly we consider a homogenous ground state solution with φ0 being constant, which does not break the rotational in variance, i.e Wa(±) = 0, Za = 0, In this case, due to definition in (17) χ0 = 0, (33) g = implies e = Therefore 1/2 gW + e2 = gW = m2Z = g φ20 g = m2W A0 = (34) (35) The effective potential in M F A takes the form (−) V = − gW Z0 ∧ W0 Z ∧ W 0(+) + 2 (+) (−) (+) + gW W0 ∧ W0 W0 ∧ W 0(−) + − µ2 − m2 φ20 + λφ40 (36) where µ = µ − gZ0 (+) (−) 2 m = m + g W0 W0 (37) (38) The equations of motion become (+) i∂0 W0 ∂V =g ∂Z0 ∂V (±) ∂W0 (+) + 2µW0 φ0 = (39) gZ0 − µ φ0 = (40) (±) = g φ20 W0 = (41) and ∂V = ∂φ0 g µ − Z0 2 i g (+) (−) − m2 − 2λφ20 − g∂0 Z0 + W0 W0 φ0 = 2 (42) 86 PHAN HONG LIEN, DO THI HONG HAI It is easily to find the symmetry solution φ0 = and other solutions of system of Eqs (40) - (42) 2µ Z0 = (43) g (±) W0 = (44) and m2 2µ (±) ; Z0 = ; W0 = (45) 2λ g i.e the ground state expectation values of vector fields are well determined physical quantities.The solutions (45) is shown that spontaneous U (1)Y symmetry breaking exits only for negative m2 Substituting (45) into (38), we obtain the effective chemical potential and the squared mass, respectively µ = 2µ, m2 = m2 φ0 = − IV.2 Case g = and g = 0, m2 < ≤ µ2 In this case, the ground state (45) describes spontaneous breaking SU (2) × U (1)Y to U (1)EM and preserve of rotational invariance The propagators for massive vector gauge boson is given by Sµν (k) = i kµ kν g − (1 − ξ) µν k − mW k − ξm2W (46) Next we consider finite temperature by ”imagine time” formalism The matrices corresponding to (46) in t’Hooft - Feymann ξ = takes the form →2 2 2− − ω k + g φ g 2iµω µν −1 Sµν (k) = (47) → − −2iµω ω − k + 12 g φ20 gµν It’s dispersion relations is determined from detS −1 = ω± = − → µ2 + k + g φ20 ± µ (48) In infrared it becomes µ2 + g φ20 ± µ (49) That means the chemical potential leads to spliting the quantum masses of two charged vector boson Similary, the inverse propagator of neutral gauge boson Zµ and photon Aµ is →2 2−− ω k g 2iµω µν G−1 (50) → − µν (k) = −2iµω ω − k + m2Z gµν ω± where 1 m2Z = (g + g )φ20 = (g + g )v 2 PHASE TRANSITION IN THE ELECTROWEAK THEORY 87 When k → 0, the equation detG−1 µν = reads detG−1 (k) = ω − ω (m2Z − 4µ2 ) = (51) Their dispersion relations are ω1,2 = m2Z − 4µ2 = (mZ + 2µ)(mZ − 2µ) ω1 = ω2 = (g + g )1/2 v + 2µ (g + g )1/2 v − 2µ (52) (53) (54) V PHASE TRANSITION IN THE ELECTROWEAK THEORY We consider the scalar field in the presence of non - zero chemical potential and temperature The equation of motion for φ reads g2 + g 2 Zµ + 2g µBµ )φ = (55) (µ2 − m2 − 4λφ20 − 3λχ2 − g Wµ(+) Wµ(−) + When φ → φ0 it is shown [7] that all effective fields and chemical potential are very smaller than the critical temperature TC When the quantum state is equilibrium, the fluctuations haven’t influences significant to physical properties of system If the conditions charge, these fluctuations could be spread and the quantum system becomes instable, then the phase transition leads it to new stable properties At the transition temperature φ0 = 0, due to the finite temperature part of propagators [10] one can determine the thermal average of squared scalar and gauge vector fields T2 T2 < χ2 >= ; < Wµi >= (56) 12 Substituting (56) into (55), we obtain the critical temperature TC2 = 16(µ2 − m2 ) 16λ + 3g + g or equivalently 4(µ2 − m2 ) (57) 2λ + e2 (1 + 2cos2 θ)/sin2 2θ It is shown that the phase transition depends significantly on the chemical potential and the electric charge TC2 = VI DISCUSSION AND CONCLUSION In the above sections the Weinberg Salam Glashow without fermions is considered at finite temperature and density The dispersion relations are obtained, where the chemical potential acts to mechanism of spontaneous breaking of symmetry and it leads to splitting the quantum masses of vector bosons The critical temperature has been directly derived in the mean-field approximation The phase transition is second order one 88 PHAN HONG LIEN, DO THI HONG HAI Linde [11] and Kapusta [7] have pointed out that at fixed temperature the condensation of the W ± mesons should occur at higher densities than the symmetry restoring density However, the finite density of charged fermions didnt affect on the results because its interactions and electromagnetic one are different Eventhough strong electromagnetic could lead to the deconfinement Miransky [9] and other authors [12] have investigated a similar model, which includes three massless vector boson Aaµ and two doublets K + , K and K − , K Our next paper is intended to be devoted to the Weinberg Salam model including both bosonic and fermionic parts and to numerical calculations at finite temperature and density In conclusion, we would like to emphasize that the spontaneous symmetry violation and symmetry restoration at high temperature depend on the dynamics of the theory that is concerned with the physical processes ACKNOWLEDGMENT One of the authors (PHL) would like to thank Prof Tran Huu Phat for helpful suggestion for this problem REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] S Weinberg, The Quantum Theory of Fields II, 1996 Cambridge University Press S Pokorski, Gauge Field Theory, 1987 Cambridge University Press G ’t Hooft, Nucl Phys B 33 (1971) 173 G ’t Hooft, Nucl Phys B 35 (1971) 167 T Haugset, H Haugerud, F Ravndal, Ann Phys (NY) 266 (1998) 27 J O Andersen, Rev Mod Phys 76 (2004) 599 I Kapusta, Finite Temperature Field Theory, 1989 Cambridge University Press; J I Kapusta, Nucl Phys B 148 (1979) 461 M Le Bellac, Thermal Field Theory, 1996 Cambridge University Press V A Miransky, I A Shovkovy, Phys Rev Lett 88 (2002) 111601 P Arnold, D Espinosa, Phys Rev D 47 (1993) 3456 A D Linde, Phys Lett B 86 (1979) 39 Tran Huu Phat, Nguyen Van Long, Nguyen Tuan Anh, Le Viet Hoa, Phys Rev D 78 (2008) 105016 Received 30-09-2011 ... (32) For g ≤ 8λ the potential has minimum at the solution stable φ0 At the critical value µ = m there is a second order phase transition PHASE TRANSITION IN THE ELECTROWEAK THEORY 85 IV SECOND... breaking effect leads to mass different between the charge and neutral gauge bosons, Wµ± and Zµ PHASE TRANSITION IN THE ELECTROWEAK THEORY 83 The complete Lagrangian (5) that includes the Higgs... )1/2 v − 2µ (52) (53) (54) V PHASE TRANSITION IN THE ELECTROWEAK THEORY We consider the scalar field in the presence of non - zero chemical potential and temperature The equation of motion for