KCL-PH-TH/2012-41 Maxwell Construction for Scalar Field Theories with Spontaneous Symmetry Breaking 3 1 0 2 J. Alexandrea1 and A. Tsapalisb,c 2 n a J a King’s College London, Department of Physics, WC2R 2LS, UK 6 2 b Hellenic Naval Academy, Hatzikyriakou Avenue, Pireaus 185 39, Greece c Department of Physics, National Technical University of Athens ] h Zografou Campus, 157 80 Athens, Greece t - p e h Abstract [ 2 Usinganon-perturbativeapproximationforthepartitionfunctionofacomplexscalarmodel, v which features spontaneous symmetry breaking, we explicitly derive the flattening of the effec- 1 2 tive potential in the region limited by the minima of the bare potential. This flattening occurs 9 in the limit of infinite volume, and is a consequence of the summation over the continuous 0 . set of saddle points which dominate the partition function. We also prove the convexity of 1 1 the effective potential and generalize the Maxwell Construction for scalar theories with O(N) 2 symmetry. Finally, we discuss why the flattening of the effective potential cannot occur in the 1 Abelian Higgs theory. : v i X 1 Introduction r a The convexity of the effective potential for a scalar theory has been known for a long time [1], and is a consequence of its definition in terms of a Legendre transform [2]. In the situation where the bare potential features spontaneous symmetry breaking (SSB), convexity is achieved non-perturbatively, and cannot be obtained by a naive loop expansion. The effective potential becomes flat between the two minima of the bare potential, as a consequence of the competition of the two non-trivial saddle points [3]. By analogy with the Maxwell construction for a Van [email protected] [email protected] 1 de Waals fluid, the corresponding flattening can be understood from the so-called spinodal instability: no restoration force suppresses fluctuations of the vacuum for the quantized system, which is a superposition of the two bare vacua [4]. A PhD thesis has been written on the topic [5], where many aspects are studied, and a detailed literature review is given. We note that the equivalence between the effective potential defined via the Legendre trans- form and the Wilsonian effective potential is valid in the limit of infinite volume only [6], as the Wilsonian effective potential is not necessarily convex at finite volume. Lattice simulations of scalar SSB models in [6] show the gradual flattening for the Wilsonian effective potential (the “constrained effective potential”) as the number of lattice sites increases. On the other hand, as shown in the present work, the effective potential obtained by the Legendre transform is convex for finite volume too. A linear effective potential was explicitly derived in [7], for a real scalar field in a SSB bare potential. Using a simple approximation, it was shown that the Maxwell construction arises from the dominant contributions of both saddle points in the partition function. These dominant contribution are homogeneous and do not take into account the kink solution, which is stable in 1+1 dimensions only if no other field is present [8]. We consider here two space dimensions at least (a recent work on the quantization of the 1+1 dimensional kink can be found in [9]). Quantum corrections to the (Legendre) effective potential are calculable in series of ~ via functional techniques [10] as fixed-loop diagrammatic series. One- and two-loop corrections, computed in [10] for the O(N) theory, do not suffice to restore convexity for a tree-level sym- metry breaking potential. A large-N analysis, performed at the leading 1/N order in [11], leads to a real, convex potential in four dimensions albeit for a restricted range of field amplitudes, raising thus questions to the consistency of the scheme in that order. It is interesting though, that in lower dimensions the same approximation leads to consistent description of the phases of the model [11]. The absence of SSB in the large-N ground state was also argued in [12]: using RG invariant quantities, a stable symmetric saddle point was supported in 4-d, albeit with imaginary contributions to the effective potential. Further studies [13], [14], of the large-N saddle point in the ǫ-expansion concluded that a second order phase transition is present in d > 2 dimensions. In addition, it was shown that the O(N) invariant mass vanishes at the critical point with a critical exponent ν = 1/(d 2), for 2 < d < 4 dimensions. − Having in mind the convexity as the principal characteristic of the effective potential to hold in a SSB theory, we focus in the present article to a complex scalar field as well as the O(N)-symmetric model, and show that the Maxwell construction arises from the summation over all the saddle points which constitute a valley of minima, and dominate the partition function. This semi-classical approximation leads to a convex effective potential, which for large volumes is universal, in the sense that it does not depend on the coupling constant of the bare theory. The potential becomes flat in the limit of infinite volume. We believe that, for a 2 SSB potential, it is essential to first calculate the partition function, taking into account the whole set of minima, and then derive a convex effective action via the Legendre transform. We describe in section 2 the detailed steps of the quantization of a complex scalar model which features SSB. We first show convexity as a general property, and we define our semi- classical approximationforthecalculationofthepartitionfunction. Wethenderivetheeffective potential and show how the Maxwell construction occurs in the region where the scalar field modulus ρ is smaller than the vev v. The semi-classical approximation applied to the outside region ρ > v leads to an effective potential identical to the bare potential, which is convex in this region. Section 3 generalizes the resulting Maxwell construction for an O(N) model, for which the steps are similar to those followed for the complex scalar field. Note that the Maxwell construction obtainedhereisa result ofthefull quantization oftheN degrees offreedom, unlike the effective field theory approach for a linear sigma model, which consists in integrating the massive degree of freedom only, in order to describe the infrared dynamics of the remaining Goldstonemodes. The latterprocedure leadstoa non-trivialeffective theory, asnicely reviewed in [15]. In the Appendix, we treat the real field potential with two equivalent vacua (Z2) in the same approximation, and we demonstrate that the form of the effective action is reproduced from the O(N) action when N = 1. Finally, section 4 comments on the Abelian Higgs model, where no Goldstone mode is present because of gauge fixing, such that the partition function is dominated by one saddle point only. The semi-classical approximation is then equivalent to a tree-level approximation, and the usual Higgs mechanism occurs. We also explain why the general argument of convexity does not hold in the presence of a vector field. 2 Self-interacting complex scalar field 2.1 Construction of the effective action We review here the basic features of path integral quantization, in order to introduce our notations. We consider a model with a self-interacting complex scalar field φ = ρexp(iα), in which the bare potential U depends on ρ = √φφ⋆ only. The partition function is, using a Euclidean bare metric, Z[j,j⋆] = [φ,φ⋆]exp S[φ,φ⋆] jφ+j⋆φ⋆ , (1) D − − Z (cid:18) Zx (cid:19) where j = reiθ,j⋆ = re−iθ are source which parametrizes the system, and which will eventually be replaced by the classical fields φ ,φ⋆, defined as c c 1 δZ e−iθ δZ φ ρ eiαc = = (2) c c ≡ −Z δj − 2Z δr 3 1 δZ eiθ δZ φ⋆ ρ e−iαc = = . c ≡ c −Z δj⋆ −2Z δr In terms of the polar coordinates (ρ ,α ), the above definitions are equivalent to c c 1 δZ π ρ = , α = θ + (1+sign(δZ/δr)) . (3) c c 2Z δr − 2 (cid:12) (cid:12) (cid:12) (cid:12) We note that the partition fun(cid:12)ctio(cid:12)n Z depends only on the modulus of the source j. Indeed, (cid:12) (cid:12) the source term can be written jφ+j⋆φ⋆ = 2 rρcos(α+θ) , (4) Zx Zx and the summation over all the configurations φ implies that, for a fixed source j, one can change the variable α α θ, such that Z[j,j⋆] = Z[r]. Nevertheless, we keep explicit the θ → − dependence, in order to take into account the two degrees of freedom present in the model. The effective action Γ is defined as the Legendre transform of W[r] = ln(Z[r]) with respect − to the sources j,j⋆ Γ[φ ,φ⋆] = W[r] jφ +j⋆φ⋆ , (5) c c − c c Zx where the sources have to be understood as a functionals of the classical fields, after inverting therelations (2). Fromthedefinition (5), one finds thatthe equations ofmotionfor theclassical fields are δΓ δΓ = j , = j⋆ . (6) δφ − δφ⋆ − c c We are interested in the effective potential of the theory, which is obtained from the momentum independent part of the effective action 1 U (ρ ) = Γ[φ ,φ⋆] , with φ = constant , (7) eff c V c c c where V is the volume of space time. In this case, the effective action depends on the modulus ρ only, and, taking into account the equations of motion (6), we obtain c 1 δΓ = r . (8) 2 δρ (cid:12) c(cid:12) (cid:12) (cid:12) Finally, for constant fields, the functional d(cid:12)eriva(cid:12)tives become (cid:12) (cid:12) δ( ) 1 ∂( ) δ( ) 1 ∂( ) ··· ··· and ··· ··· . (9) δr → V ∂r δρ → V ∂ρ c c 4 2.2 Convexity of the effective potential The convexity of the effective action Γ is a consequence of its definition as the Legendre trans- form of the connected graph generating functional W[j,j⋆], as we explain here. Let us define the operator δ2W δ2W δ2W δjδj⋆ δjδj , (10) ≡ δ2W δ2W δj⋆δj⋆ δj⋆δj ! with the functional derivatives applied at a pair of spacetime points x,y as e.g. δ2W = φ (x)φ⋆(y) φ(x)φ⋆(y) , (11) δj(x)δj⋆(y) c c −h i where 1 ( ) [φ,φ⋆]( )exp S[φ,φ⋆] jφ+j⋆φ⋆ . (12) h ··· i ≡ Z D ··· − − Z (cid:18) Zx (cid:19) Using the invariance of the Euclidean action under translations and O(4) rotations, the distri- bution (11) is a real function of x y . In addition, it is also the opposite of a variance and | − | therefore the diagonal elements of the Hermitian operator (10) are equal and negative. The eigenvalues of this operator are δ2W δ2W , (13) δjδj⋆ ± δjδj (cid:12) (cid:12) and are negative, since they can be written in t(cid:12)erms(cid:12)of variances as (cid:12) (cid:12) (cid:12) (cid:12) var(Re φ ) var(Im φ ) var2(Re φ )+var2(Im φ ) . (14) − { } − { } ± { } { } As a consequence, W is a concave functional. p In order to study the properties of the effective action Γ, we introduce the operator δ2Γ δ2Γ δ2Γ ≡ δφδ⋆c2δΓφc δφδ⋆c2δΓφ⋆c . (15) δφcδφc δφcδφ⋆c ! Taking into account the definitions (2), and the equations of motion (6), the operators δ2W and δ2Γ are: δφ⋆c δφc δj δj⋆ δ2W = δδφj⋆c δδφjc , δ2Γ = − δδφj⋆c δδφj⋆⋆c , (16) δj⋆ δj⋆ ! δφc δφc ! and it can easily be seen that they satisfy the relation3 δ2W δ2Γ = 2 11 , (17) · − × 3We remind that j and j⋆ are independent variables. Hence diagonal elements of the product δ2W δ2Γ ⋆ ⋆ ⋆ ⋆ · involve (δj/δφ)(δφ/δj )=δj/δj =0. Similarly, we also have (δφ/δj)(δj/δφ )=δφ/δφ =0. R R 5 where 11 is the unit operator. δ2Γ is therefore proportional to the inverse of δ2W, and has positive eigenvalues: Γ is a convex functional of the classical field. To see the consequence for the effective potential U (ρ ), a constant configuration for the eff c scalar field is enough, and we have δ2 1 1 d4x U (ρ ) = U′′ + U′ δ4(x y) (18) δφ δφ⋆ eff c 4 eff ρ eff − c c Z (cid:18) c (cid:19) δ2 (φ⋆)2 1 d4x U (ρ ) = U′′ U′ δ4(x y) δφ δφ eff c 4ρ2 eff − ρ eff − c c Z c (cid:18) c (cid:19) δ2 φ2 1 d4x U (ρ ) = U′′ U′ δ4(x y) , δφ⋆δφ⋆ eff c 4ρ2 eff − ρ eff − c c Z c (cid:18) c (cid:19) where a prime denotes a derivative with respect to ρ . It is then straightforward to calculate c the eigenvalues of δ2Γ, which are U′′ /2 and U′ /(2ρ ). The convexity of Γ implies then that eff eff c ′ ′′ the effective potential is necessarily an increasing (U 0) and convex (U 0) function of eff ≥ eff ≥ ρ . c 2.3 Semi-classical approximation From now on we consider the following symmetry breaking potential λ U (ρ) = (ρ2 v2)2 , (19) bare 24 − with a minimum at ρ = v. The semi-classical approximation for the partition function (1) consists in taking its dominant contribution only, arising from the minima of the functional Σ[φ,φ∗] = S[φ,φ⋆]+ jφ+j⋆φ⋆ . (20) Zx Since we are interested in the effective potential, we consider only homogeneous sources, for which the minima of Σ are homogeneous fields. Non-homogeneous fields of solitonic type can also contribute to the partition function but since their action is finite, their contribution is negligible compared to the one of homogeneous configurations. We are therefore interested in the minima of Σ(ρ,α) V [U (ρ)+2rρcos(α+θ)] , (21) bare ≡ for a given angle α+θ and modulus r. One therefore looks for the real and positive solution ρ of the equation 0 λ (ρ2 v2)ρ +2rcos(α+θ) = 0 , (22) 6 0 − 0 6 which goestov when the source term rcos(α+θ) vanishes. Defining thecritical source modulus λv3 r = , (23) crit 18√3 we will study independently the two cases: r r : Equation (22) has three real solutions, among which the local minimum is given crit • ≤ by 2v π 1 r ρ = cos arccos cos(α+θ) . (24) 0 √3 3 − 3 r (cid:26) (cid:18) crit (cid:19)(cid:27) The function Σ(ρ,α) takes the form of a ’tilted Mexican hat potential’ (Fig. 1), with the set of corresponding saddle points forming a valley. The absolute minimum of the resulting valley corresponds to cos(α+θ) = 1, while the top of the valley is located at − cos(α+θ) = 1. It is also easy to verify that the local maximum ρ of Σ(ρ,α), which goes 1 to 0 when the source vanishes, reads 2v π 1 r ρ = cos + arccos cos(α+θ) . (25) 1 √3 3 3 r (cid:26) (cid:18) crit (cid:19)(cid:27) An equivalent criterion for the existence of the valley is that its top is further away than the local maximum in the radial direction, i.e. ρ < ρ for cos(α+θ) = 1. This condition 1 0 is indeed satisfied if r < r . Notice that when r = r , the top of the valley point crit crit merges with the local maximum ρ and becomes an inflection point (Fig. 2). 1 r > r : The equation (22) has the real and positive solution (24) for restricted values of crit • α, which satisfy cos(α+θ) r /r (see Fig. 2). Nevertheless, this set of local minima crit | | ≤ are much higher than the absolute minimum which is obtained for cos(α+θ) = 1 and − which reads 2v 1 r ρ = cosh cosh−1 . (26) 0 √3 3 r (cid:26) (cid:18) crit(cid:19)(cid:27) Since Σ sharply deepens in the neighborhood of the absolute minimum (26), the semi- classical approximation consists in taking into account the contribution of the latter point only. 7 Σ(ρ,α) Re(Φ) Im(Φ) Figure1: ThefunctionΣ(ρ,α), givenineq.(21), forexternal sourcemoduler < r . The’tilted crit Mexican hat’ shape possesses a valley of minima which define the semiclassical approximation. 2.4 Maxwell construction As discussed in the previous subsection, in the case where r < r the partition function is crit dominated by: 2π dα Z[r] exp( V [U (ρ )+2rρ cos(α+θ)]) (27) bare 0 0 ≃ 2π − Z0 2π dα = exp( V [U (ρ )+2rρ cosα]) , bare 0 0 2π − Z0 where ρ depends on the source r and is given by eq.(24). We note that the summation over 0 all the points in the valley ensures that the partition function Z depends on the modulus r of the source only. After introducing the dimensionless quantities Vv4 ρ r 0 A λ , ρ˜ , r˜ , (28) 0 ≡ 24 ≡ v ≡ r crit the minimum ρ˜ can be expanded in powers of r˜, and we find, up to fourth order, 0 √3 1 4√3 35 ρ˜ = 1 r˜cosα (r˜cosα)2 (r˜cosα)3 (r˜cosα)4 + (r˜5) . (29) 0 − 9 − 18 − 243 − 1944 O 8 Σ(ρ,α) Re(Φ) Im(Φ) Figure 2: The function Σ(ρ,α), given in eq.(21), for external source module r > r . The crit minima are located sharply around the cos(α + θ) = 1 point. The second local minimum − located at cos(α + θ) = 1 degenerates to an inflection point at r = r and disappears for crit r > r . crit We obtain then exp( V [U (ρ )+2rρ cosα]) (30) bare 0 0 − 8√3 4 = 1 Ar˜cosα+ A(1+8A)(r˜cosα)2 − 9 27 4√3 64 + A 1 8A A2 (r˜cosα)3 243 − − 3 (cid:18) (cid:19) 8 1 16 64 + A A+ A2 + A3 (r˜cosα)4 + (r˜6) , (31) 243 3 − 3 9 O (cid:18) (cid:19) and the integration over α leads finally to 2A A 1 16 64 Z(r˜) = 1+ (1+8A)r˜2 + A+ A2 + A3 r˜4 + (r˜6) . (32) 27 81 3 − 3 9 O (cid:18) (cid:19) 9 This last expression will now be used to calculate the effective potential, around the origin. From the definition (3) of the classical field, one finds 3√3 ∂Z ρ = v , α = π θ , (33) c c 8AZ ∂r˜ − such that the expansion (32) leads to ρ √3 √3 1 11 8 32 ρ˜ c = (1+8A)r˜+ A+ A2 A3 r˜3 + (r˜5) . (34) c ≡ v 18 81 2 − 6 3 − 3 O (cid:18) (cid:19) One can see that the regime r < r leads to a polynomial expansion of the classical field crit around ρ = 0. We then invert the last series, by expanding r˜ in powers of ρ˜ , and we find c c 18 144√3( 1/2+11A/6 8A2/3+32A3/3) r˜= ρ˜ + − − ρ˜3 + (ρ˜5) . (35) √3(1+8A) c (1+8A)4 c O Since Γ = VU is an increasing function of ρ , the equation of motion (8) can also be written eff c ∂Γ ∂Γ 8A = = r˜ , (36) ∂ρ˜ ∂ρ˜ 3√3 (cid:12) c(cid:12) c (cid:12) (cid:12) (cid:12) (cid:12) Together with the expansion (35), the integration over ρ˜ gives (cid:12) (cid:12) c 8A 48A( 1+11A/3 16A2/3+64A3/3) Γ[ρ˜ ] = ρ˜2 + − − ρ˜4 + (ρ˜6) , (37) c 1+8A c (1+8A)4 c O c where the constant of integration is disregarded. In the limit of large volume A >> 1, the effective action is then ρ 2 1 ρ 4 c c Γ[ρ ] = + + , (38) c v 4 v ··· (cid:16) (cid:17) (cid:16) (cid:17) and the effective potential is finally obtained after dividing by the volume 1 1 ρ 2 1 ρ 4 c c U (ρ ) = Γ[ρ ] = + + . (39) eff c c V V v 4V v ··· (cid:16) (cid:17) (cid:16) (cid:17) U therefore vanishes in the limit of infinite volume eff U (ρ ) 0 (infinite volume) , (40) eff c → and has the form of a flat disc for ρ << v. Few interesting remarks can be made here: c 10