(2) 214. M. Loewe et al. / Physics Letters B 635 (2006) 213–217. 3. Chemical potentials as background fields. classical action S[A] is given by 1 a d4 x Fμν Faμν , S=− 4 where. (2). a = ∂μ Aaν − ∂ν Aaμ + g abc Abμ Acν . Fμν. (3). According to the BFM, we must shift our field + Bμa , where Bμa is the background field. Then, the new generating functional will be a δ G̃ Z̃[J, B] = DA det δw b 1 × exp i S[A + B] − G̃ · G̃ + J · A , (4) 2ξ Aaμ. → Aaμ. where G̃a is the modified gauge fixing condition. In order to maintain gauge symmetry, we need a G̃a of the following shape: μ G̃a = ∂ μ Aaμ + g abc Bμb Aμ c ≡ D̄μ Aa .. (5). The new infinitesimal transformations, will be defined according to. (6). so that the gauge symmetry remains unbroken, including the G̃ · G̃ term. Notice that the gauge field transforms as a matter field and the gauge parameters appear associated to the background field . The sum of both fields transforms in the usual way: 1 δ Aaμ + Bμa = − abc w b Acμ + Bμc + ∂μ w a , (7) g i.e., as gauge fields, so that the classical action in (4) remains invariant. In the loop calculation of the effective action, the background field (Bμa ) appears as external amputated legs, whereas the quantum gauge fields (Aaμ ) and the ghost fields (ηa ) live only in internal lines. The modified Lagrangian, including the ghost fields will read (8). (13). where vμ is a 4-velocity with respect to the thermal bath, that allows us to keep a formal covariant language, although, finally we have to choose the frame of reference where the heat bath is at rest, i.e., vμ = (1, 0, 0, 0). We have chosen the third component of the internal SU(2) group. This restriction corresponds to a simple orientation in the group manifold. The appearance of the quotient μ/g is a consequence of keeping the usual BFM relation between ZB and Zg , the background field and coupling constant renormalization factors, respectively, given by −1/2. Zg = Z B. (14). .. (15). We would like to emphasize that the same gauge fixing condition was found in our previous Letter , using a complete different approach. In , we found an exact expression for the effective potential of the Weinberg–Salam model in the presence of chemical potentials and thermal effects. The idea was to diagonalize the effective potential to get separate contributions from each field. The splitting of the effective potential is not possible without our gauge fixing condition. For example, in  the author gives an expression for the effective potential, but only in the high temperature expansion. To show the efficiency of this method, we will proceed with the calculation in the one loop approximation of the effective potential for a pure gauge theory. As it is well known, the oneloop thermal effective action is given by. exp. β Γ1 [φc ]. β. =. D[Fields] exp. dτ. d3 x Lq (x̄) ,. (16). 0. such that 2 1 a Bμν + D̄μ Aaν − D̄ν Aaμ + g abc Abμ Acν , 4 Lghost = − D̄μ ηa∗ D̄ μ ηa − g abc ηb Aμ c , 2 1 LGF = − D̄μ Aμ a , 2ξ where. μ vμ δ a3 , g. Ga = ∂μ Aaμ + μ abc Acμ v μ δ 3b .. 1 δBμa ≡ − abc w b Bμc + ∂μ w a , g. Lmod = LAμ + Lghost + LGF ,. Bμa =. The gauge fixing conditions acquires the form. δAaμ ≡ − abc w b Acμ ,. δJμa ≡ − abc w b Jμc ,. The basic idea of this section is to introduce the chemical potential (μ) as the background field. This is different from the usual approach, where chemical potentials appear in covariant derivatives as constant external time component gauge fields. We will use the following prescription:. where we shifted to Euclidean metric (τ = it), with. LAμ = −. x̄ = (−iτ, x), (9). D̄μ Aaν ≡ ∂μ Aaν + g abc Bμb Acν ,. (10). D̄μ ηa ≡ ∂μ ηa + g abc Bμb ηc ,. (11). p̄ = (iωn , p),. (17). with ωn = 2πn/β for bosons and ωn = 2π(n + 1)/β for fermions. In Eq. (16) the Lq denotes the quadratic Lagrangian for the Aaμ and ηa fields. Now, as it is well known, the effective potential corresponds to the effective action, by taking the classical field as a constant. β β Γ1 [φc = constant] = −β d3 x Ωeff . (18). and a Bμν ≡ ∂μ Bνa − ∂ν Bμa + g abc Bμb Bνc .. (12). Since the internal lines in the Feynman diagrams are associated to the gauge and the ghosts fields, we will need to find the.
(3) M. Loewe et al. / Physics Letters B 635 (2006) 213–217. quadratic Lagrangian in both fields, in order to form β. . 1 μν dx̄ d x̄ Aaμ (x̄ )Bab (x̄ , x̄)Abν (x̄) 2 − dx̄ d x̄ ω∗ (x̄ )C(x̄ , x̄)ω(x̄). (19). d 3 x Lq = −. dτ 0. Now, the effective potential will be given by 1 −β d3 x Ωeff = − Tr ln B + Tr ln C. 2. 215. So, the final result for the effective potential is 1 β d3 k 2 ln 1 − e−β(|k|+μ) 1 − e−β(|k|−μ) Ωeff = β + 2 ln 1 − e−β|k| . (25) This is exactly the result we wanted to obtain. We can see that we a have massless and chargeless gauge boson with two degrees of freedom, and two massless and charged gauge bosons with two degrees of freedom each.. (20) 4. Scalar and fermion SU(2) gauge symmetry. The quadratic Lagrangian for a pure gauge theory is 2 1 1 D̄μ Aaν − D̄ν Aaμ − Baμν abc Abμ Acν 4 2 1 μ 2 ∗ − D̄μ Aa − D̄μ ωa D̄ μ ωa . 2ξ. First, we will concentrate our attention on a scalar SU(2) gauge-invariant theory, given by the following Lagrangian. Lq = −. (21). a = 0. Taking into account the choice (13), we notice that Bμν 1/2 a Since ZB multiplies Bμν , we need three different chemical a potentials μa associated to a particular flavor. Otherwise Bμν vanishes and we are not able to carry on the renormalization procedure . The idea is that renormalizability has already been proved, and then we are free to select one direction in the isospin space in order to compute the effective potential. This remind us the problem that appears when quantizing gauge field theories, between the Rξ and the unitary gauge. A very simple way to calculate the effective potential is to write each field explicitly. Let us choose, to simplify the calculations, ξ = 1. The sum of all contributions of the gauge fields will give. 1 ∂μ A1ν ∂ μ Aν1 + μ2 A1ν Aν1 2 1 − ∂μ A2ν ∂ μ Aν2 + μ2 A2ν Aν2 2 1 − ∂μ A3ν ∂ μ Aν3 2 − μvμ Aν1 ∂ μ A2ν − Aν2 ∂ μ A1ν .. T 1 a μν 1 Fa − V (φ), Dμ φ a D μ φ a − Fμν (26) 2 4 where φ belongs to the adjoint representation, i.e., it is given by a real scalar triplet φ1 φ = φ2 , (27) φ3 L=. and Dμ φ a = (∂μ − igAμ )φ a , a Fμν. = ∂μ Aaν. − ∂ν Aaμ. + g. (28) abc. Abμ Acν ,. (29). with Aμ = Abμ T b .. (30). The classical potential is given by. Lq = −. 2 m2 T λ (31) φ φ + φTφ , 2 4 In the SU(2) case, the group generators will be given by T a = ωa /2, where 0 0 0 ω1 = 0 0 −i , 0 i 0 0 0 i ω2 = 0 0 0 , −i 0 0 0 −i 0 ω3 = i 0 0 . (32) 0 0 0 V (φ) =. (22). As usual, we will write this Lagrangian in Euclidean metric, and the calculation of the effective potential will be given by taking the traces of (22). After evaluating the sums over the Matsubara frequencies, we find the gauge field contribution to the thermal effective potential 1 β d3 k 8 ln 1 − e−β(|k|+μ) 1 − e−β(|k|−μ) ΩAμ = 2β + 8 ln 1 − e−β|k| . (23) Although it seems that we have extra degrees of freedom, we must not forget that we still need to calculate the contribution of the ghost fields, which reads 1 β d3 k 2 ln 1 − e−β(|k|+μ) 1 − e−β(|k|−μ) Ωghosts = − β + 2 ln 1 − e−β|k| . (24). Now, following the BFM prescription, we will expand our fields in the following way: φ a → φ a + φ̄ a , Aaμ → Aaμ + Bμa , where 0 a φ̄ = 0 , ν. (33). (34).
(4) 216. M. Loewe et al. / Physics Letters B 635 (2006) 213–217. and. the masses of the fields involved are given by. μ Bμa = vμ δ a3 . g. (35). Here φ̄ a is the vacuum expectation value of the scalar fields and Bμa is the background field associated to the chemical potential. These two background fields are constant. The Lagrangian will now read T μ a 1 L = D̃μ φ a + φ̄ a D̃ φ + φ̄ a − V (φ) 2 2 1 − D̄μ Aaν − D̄ν Aaμ + g abc Abμ Acν , (36) 4 with D̃μ φ a = ∂μ − igT b Abμ + Bμb φ a , D̄μ Aμ a. = ∂μ Aμ a. + g. abc. Bμb Aμ c.. m2φ1,2 = m2 + λν 2 + ξ MA2 ≡ m21 , m2φ3 = m2 + 3λν 2 ≡ m23 , m2A1,2 = MA2 , m2A3 = 0, m2η1,2 = ξ MA2 , m2η3 = 0.. (47). Choosing ξ = 1, and writing the Lagrangian in the form β. dτ. d3 x Lq = −. 0. 1 T a 2 D̄μ Aμ a − igξ φ T φ̄ . 2ξ. Since T a , T b = i abc T c ,. for every unbroken symmetry,. iT φ̄ = 0,. for every broken symmetry,. a. (40). q. 1 ∂μ φ T ∂ μ φ + Bμ B μ φ T φ + i φ T Bμ ∂ μ φ − ∂μ φ T B μ φ , 2 (41) 1 2 q LAμ = − D̄μ Aaν − D̄ν Aaμ + φ̄ † Aμ Aμ φ̄, (42) 4 2 1 q b ν + 2C abc ∂μ Aμ ∂μ Aμ LGF = − a a Āν Ac 2ξ 2 2 (43) , + C abc Abμ Aμ + ξ 2 φ † T a φ̄ c q. Lφ =. (44). Because of the Goldstone theorem, the number of scalar bosons that acquire a gauge dependent mass and the number of massive gauge fields should be the same as the number of spontaneously broken symmetries. If we define ab 2 MA = φ̄ T T a T b φ̄, (45) so that MA2 =. g2ν 2 , 4. dx̄ φ a (x̄ )Aab (x̄ , x̄)φ b (x̄). A straightforward calculation leads us to the several thermal contributions from the different fields. For the φ3 boson we have 1 −β k2 +m23 β 3 d k ln 1 − e Ωφ 3 = (50) , β. Lghost , with. q. . (39). we can see that this gauge fixing condition will remove every quadratic mixing between the scalar and the gauge fields. For the calculation of the effective potential we only need q q q the quadratic terms in the Lagrangian, Lq = Lφ + LAμ + LGF +. Lghost = −D̄μ ηa∗ D̄ μ ηa + ξg 2 ηa∗ φ̄ T T a T b φ̄ηb .. dx̄ . we have that the thermodynamical effective potential is given by 1 1 −β d3 x Ωeff = − Tr ln A − Tr ln B + Tr ln C. (49) 2 2. (38). and the Goldstone theorem says iT a φ̄ = 0,. . 1 μν dx̄ Aaμ (x̄ )Bab (x̄ , x̄)Abν (x̄) − dx̄ 2 − dx̄ dx̄ ω∗ (x̄ )C(x̄ , x̄)ω(x̄), (48). (37). The gauge fixing condition should be treated in the same way as before, but this time incorporating the scalar fields LGF = −. 1 2. (46). and for φ1 and φ2 1 −β k2 +m21 + μ2 β 3 d k ln 1 − e Ωφ1,2 = β −β k2 +m21 − μ2 + ln 1 − e .. (51). q. Notice that LAμ is very similar to that calculated in (22), but now two of the gauge fields are massive, because of the φ̄Aμ Aμ φ term in (42). The contribution of these fields is 1 −β k2 +MA2 +μ β 3 d k 4 ln 1 − e Ω Aμ = β −β k2 +MA2 −μ × 1−e (52) + 4 ln 1 − e−β|k| . When we add the contribution of the Faddeev–Popov Lagrangian we obtain 1 −β k2 +MA2 +μ β ΩAμ ,η = d3 k 2 ln 1 − e β −β k2 +MA2 −μ × 1−e + 2 ln 1 − e−β|k| . (53) Notice that the number of degrees of freedom is the expected one. If we choose ν 2 = −m2 /λ we recover the usual Higgs– Kibble mechanism. In the presence of chemical potential, the number of Goldstone bosons is not necessarily the same as in the μ = 0 scenario, as was shown in [13–15]..
(5) M. Loewe et al. / Physics Letters B 635 (2006) 213–217. 217. The treatment of the fermion fields is equivalent to the usual procedure, where the chemical potentials appear as external zero component gauge fields forming a new covariant derivative. This a consequence of the fact that the conserved fermionic charge does not depend on the derivatives of the fields, i.e., there is no need to integrate over the conjugate momenta to pass from the Hamiltonian picture to the Lagrangian formalism. For the fermion fields we have. This gauge fixing condition has been only explored for small gauge field configurations. The analysis of the existence of Gribov copies will be carried on in a future work.. ∂ − igA / − igB/ )ψ + m2ψ ψ̄ψ, Lψ = i ψ̄(/. References. (54). where m2ψ is the mass of the fermions due to the Higgs–Kibble mechanism. The calculation of their contribution to the thermodynamical potential is straightforward. We found the wellknown result β. Ωψ = −. 1 β. . −β d3 k 2 ln 1 + e. −β + 2 ln 1 + e. . k2 +m2ψ + μ2 . k2 +m2ψ − μ2. . . .. (55). The final one loop effective potential in a SU (2) gauge theory with scalars and fermions will be given by the sum of Eqs. (50), (51), (54) and (55). In this Letter we have shown that the gauge fixing condition that enables to diagonalize the effective potential for a system including gauge, scalar and/or fermion fields emerges naturally from the description based on the BFM method.. Acknowledgements The authors would like to thank financial support from FONDECYT under grant 1051067..  L.F. Abbott, Acta Phys. Pol. B 13 (1982) 33.  P. Pascual, R. Tarrach, QCD Renormalization for the Practitioner, Springer, Berlin, 1984, Apendix D.  S. Weinberg, The Quantum Theory of Fields, vol. 2, Cambridge Univ. Press, Cambridge, 1996, pp. 95–109.  L.F. Abbott, Nucl. Phys. B 185 (1981) 189.  R. Ferrari, M. Picariello, A. Quadri, Ann. Phys. 294 (2001) 165.  A. Denner, G. Weiglein, S. Dittmaier, Nucl. Phys. B 440 (1995) 95.  M. Chaichian, M. Hayashi, hep-th/9509011.  J. Antikainen, M. Chaichian, N.R. Pantoja, J.J. Salazar, Phys. Lett. B 242 (1990) 412.  H. Matsumoto, Y. Nakano, H. Umezawa, Phys. Rev. D 29 (1984) 1116.  M. Loewe, S. Mendizabal, J.C. Rojas, Phys. Lett. B 617 (2005) 87.  J. Kapusta, Phys. Rev. D 24 (1981) 426.  J. Kapusta, Phys. Rev. D 42 (1990) 919.  V.A. Miransky, I.A. Shovkovy, Phys. Rev. Lett. 88 (2002) 111601.  S. Chadha, H.B. Nielsen, Nucl. Phys. B 217 (1983) 125.  T. Schafer, D.T. Son, M.A. Stephanov, D. Toublan, J.J.M. Verbaarschot, Phys. Lett. B 522 (2001) 67..