Relativistic massive vector condensation

Physics Letters B 527 (2002) 142–148 www.elsevier.com/locate/npe

Relativistic massive vector condensation Francesco Sannino, Wolfgang Schäfer NORDITA, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Received 16 November 2001; received in revised form 18 December 2001; accepted 27 December 2001 Editor: H. Georgi

Abstract At sufficiently high chemical potential massive relativistic spin-one fields condense. This phenomenon leads to the spontaneous breaking of rotational invariance while linking it to the breaking of internal symmetries. We study the relevant features of the phase transition and the properties of the generated Goldstone excitations. The interplay between the internal symmetry of the vector fields and their Lorentz properties is studied. We predict that new phases set in when vectors condense in Quantum Chromodynamics like theories.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction The Quantum Chromodynamics (QCD) phase diagram as a function of temperature, chemical potential and the number of light flavors is very rich and highly structured. QCD should behave as a color superconductor for a sufficiently large quark chemical potential [1]. This leads to new phenomenological applications associated with the description of quark stars, neutron star interiors and the physics near the core of collapsing stars [1–3]. Much is known about the phase structure of QCD at nonzero temperature through a combination of perturbation theory and lattice simulations while the phase structure at nonzero chemical potential and for large numbers of flavors has been less extensively explored [4]. Standard importance sampling methods employed in lattice simulations fail at nonzero chemical potential for Nc = 3 since the fermi-

E-mail addresses: [email protected] (F. Sannino), [email protected] (W. Schäfer).

onic determinant is complex. For Nc = 2, the situation is very different since the quarks are in a pseudoreal representation of the gauge group and lattice simulations can be performed [5] At nonzero chemical potential Lorentz invariance is explicitly broken down to the rotational subgroup SO(3) and higher-spin fields can condense, thus potentially enriching the phase diagram structure of QCD and QCD-like theories. Indeed, in [6] it has been suggested that some of the lightest massive vectors at nonzero baryon chemical potential may condense. Recent lattice simulations seem to support these predictions [7]. In the context of the Electroweak theory vector condensation in the presence of a strong external magnetic field has also been studied in [8]. In this Letter we explore the condensation of relativistic massive vectors when introducing a nonzero chemical potential via an effective Lagrangian approach. In order for the vectors to couple to the chemical potential they must transform under a given global symmetry group. In view of the possible physical applications we consider the vectors to belong to the ad-

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 1 5 2 1 - 0

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joint representation of the SU(2) group. We choose the chemical potential to lie along one of the generator’s directions explicitly breaking SU(2) to U (1) × Z2 . When the vector field condenses the rotational SO(3) invariance breaks spontaneously together with part of the flavor (i.e., global) symmetry of the theory. The condensate locks together space and flavor symmetries. Following the standard mean-field approach we identify our order parameter with the vector field itself. In order to elucidate the mechanism we construct the simplest effective Lagrangian at zero chemical potential. This consists of a standard kinetic type term and a tree level potential constructed as a series expansion in the order parameter. We truncate our potential at the fourth order in the fields while retaining all the terms allowed by Lorentz and SU(2) symmetry. The introduction of the potential term allows us to describe the phase transition in some detail while our results will be general. We prove that the number of gapless states emerging when the flavor×rotational symmetry breaks spontaneously is insensitive to the choice of the coefficients in the effective potential. However, the dispersion relations of the gapless excitations are heavily affected by the choice of the potential. Here a crucial role is played by a mismatch between the symmetries of the non-derivative terms and the full Lagrangian. The full vector Lagrangian (i.e., kinetic potential) always possesses the Z2 × U (1) × SO(3) (flavor×rotational) symmetry which at high enough chemical potential breaks spontaneously to Z2 × SO(2). The gapless modes are linked to the 3 broken generators. One state is an SO(2) scalar while the other two states consti-

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tute a vector of SO(2). The scalar Goldstone possesses linear dispersion relations independently of the choice of the vector potential. However, the SO(2) vector state can have either quadratic or linear dispersion relations. In particular (as summarized in the Table 1), the quadratic ones occur when the potential parameters are such that it possesses an enhanced SO(6) symmetry. When condensing the vector breaks the SO(6) symmetry of the potential down to SO(5) while the kinetic-type term still has only a U (1) × SO(3) invariance. In this case the potential has 5 flat directions (i.e., 5 null curvatures) and we would count in the spectrum 2-gapless vectors and an SO(2) scalar. However, the reduced symmetry of the kinetic term prevents the emergence of two independent gapless vectors while turning the linear dispersion relations of one vector state into quadratic ones. This result is also in agreement with the Chadha–Nielsen counting scheme. However, since the Goldstone modes with quadratic dispersion relations must be counted twice relatively to the ones with linear dispersion relations we saturate the number of broken generators associated with the symmetries of just the potential term which is larger than the ones associated with the symmetries of the full Lagrangian. It is certainly important to investigate, in the future, if the choice of the parameter space leading to an SO(6) symmetry for the potential term is stable against quantum corrections. We note that our Lagrangian approach allows an analytical simple analysis of the relativistic vector condensation at nonzero chemical potential. The robustness of our approach, based entirely on symmetry considerations, and its predictions is tested against the general theorem on the number of Goldstone modes

Table 1 Summary of the relevant symmetries and spectrum of the Goldstone modes for the parameter choices λ > λ and λ = 0 of the fourth-order potential. We also indicate in the first row the symmetries (after the arrows) which are not broken by the vector condensate

Lagrangian Potential # of broken generators: Lagrangian Potential Spectrum of Goldstone modes: Type I (E ∝ |p |) Type II (E ∝ |p |2 )

λ > λ

λ = 0

Z2 × U (1) × SO(3) −→ Z2 × SO(2) Z2 × U (1) × SO(3) −→ Z2 × SO(2)

Z2 × U (1) × SO(3) −→ Z2 × SO(2) Z2 × SO(6) −→ Z2 × SO(5)

3 3

3 5

1 SO(2)-scalar + 1 SO(2)-vector –

1 SO(2)-scalar 1 SO(2)-vector

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by Nielsen and Chadha [9]. However, the Nielsen and Chadha theorem alone (which relies only on Lorentz breaking) cannot capture all of the relevant features related to the chemical potential phase transition. In particular, in this Letter, as explained above, we demonstrate the existence of an intimate relation between the velocities of the gapless modes and the symmetries of part of the theory associated with the non-derivative terms in the action. Our results are not limited to the vector condensation and shed new light on the superfluid phenomenon and more generally on the nature of the phase transition at nonzero chemical potential. There are also a number of relevant physical applications of topical interest. More specifically, our model Lagrangian describes the physics of 2-color QCD at nonzero baryon chemical potential as well as 3-color QCD at finite-isospin chemical potential. Indeed, in recent lattice simulations for 2-color QCD with even number of flavors at nonzero matter density [7] massive vector condensation was observed calling for such a detailed investigation. Here we show that the phase transition is very rich. Indeed, we suggest that by studying the dispersion relations of the gapless excitations lattice studies will be able to pin point the symmetries of the non-derivative terms of the vector theory which encode information of the underlying strongly interacting dynamics. Finally, in this Letter we also predict that vector condensation can be the leading mechanism for a novel superfluid phase transition for 2-color QCD with one flavor which was not considered at all in [6] or any other place in literature. Indeed, in this case the global symmetry is SU(2) and it remains intact at zero chemical potential and no Goldstone bosons emerge. However, the massive spectrum of the theory contains an SU(2) spin-one massive multiplet. At sufficiently high nonzero chemical baryon potential the vector condensation will lead to a new type of superfluid phase transition. Clearly, our model is also applicable to QCD with 3 color at nonzero-isospin chemical potential [10–12]. This Letter is structured as follows. In the next section we introduce a simple effective Lagrangian for vectors which allows us to uncover the main features related to the condensation of relativistic massive spinone fields. We conclude and point out some physical applications in Section 3. In particular, we suggest that new phases should set in for QCD like theories.

Finally, we extend our simple model to D − 1 number of space dimensions (with Euclidean SO(D) Lorentz symmetry).

2. Vector condensation at nonzero chemical potential There are different ways to introduce vector fields at the level of the effective Lagrangian (for example, the hidden local gauge symmetry of Ref. [13], or the antisymmetric tensor field of Ref. [14]) and they are all equivalent at tree-level. We choose to introduce the massive vector fields following the method outlined in [15–18]. This method also allows a straightforward generalization of our model to an arbitrary number of space dimensions. We adopt the following model Lagrangian for describing relativistic spin-one fields (in 3 + 1 dimensions) belonging to the adjoint representation of SU(2) 1 a aµν m2 a aµ A A F + L = − Fµν 4 2 µ 2 λ
a aν 2 λ
A A − Aaµ Aaµ + , 4 4 µ

(1)

a = ∂ Aa − ∂ Aa , a = 1, 2, 3, and metric conwith Fµν µ ν ν µ vention ηµν = diag(+, −, −, −). Here m is the tree level mass term and λ and λ are positive dimensionless coefficients with λ > λ . Other possible choices of the parameters do not guarantee stability of the potential and will not be considered in the following. The effect of a nonzero chemical potential associated to a given conserved charge—related to the generator (say B)—can be readily included [6] by modifying the derivatives acting on the vector fields:

∂ν Aρ → ∂ν Aρ − i[Bν , Aρ ],

(2)

 The with Bν = µδν0 B ≡ Vν B, where V = (µ, 0). vector kinetic term modifies according to:        Tr Fρν F ρν → Tr Fρν F ρν − 4i Tr Fρν B ρ , Aν    − 2 Tr Bρ , Aν B ρ , Aν    − Bρ , Aν B ν , Aρ . (3)

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The terms due to the kinetic term, after integration by parts, yield [6]    1 Lkinetic = Aaρ δab g ρν ✷ − ∂ ρ ∂ ν 2  V ρ∂ν + V ν∂ρ − 4iγab g ρν V ∂ − 2

  + 2χab V · V g ρν − V ρ V ν Abν (4)

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 a 1 a + ϕM δMN ✷E − ∂M ∂N ϕN 2  VM ∂N + VN ∂M b a ab3 ϕN δMN V ∂ − − ϕM ε 2 a a − ϕM [V V δMN − VM VN ]ϕN . (9) According to the value of the chemical potential we distinguish two phases. 2.1. The symmetric phase: µ
m

with

   γab = Tr T a B, T b ,    χab = Tr B, T a B, T b .

(5)

For B = T 3 we have i γab = − ab3 , 2 1 χ33 = 0. χ11 = χ22 = − , (6) 2 The chemical potential induces a “magnetic-type” mass term for the vectors at tree-level. The symmetries of the vector Lagrangian are more easily understood using the following Euclidean notation: 
a ψM = A3M , = A1M , A2M , ϕM (7) with AM = (iA0 , A ) and with metric signature (+, +, +, +). In these variables the potential reads:  m 2 − µ2 m2  2 VVector = + |ϕ0 |2 + ψM |ϕI |2 2 2  λ 2 2 + |ϕM |2 + ψM 4 2 λ  ϕM · ϕN + ψM ψN − (8) 4 with I = X, Y, Z while M, N = 0, X, Y, Z and repeated indices are summed over. At zero chemical potential VVector is invariant under the SO(4) Lorentz transformations while only the SO(3) symmetry is manifest at nonzero µ. The SU(2) symmetry is also explicitly broken, at nonzero chemical potential, to U (1). For the reader’s convenience we summarize the quadratic term in the fields Lagrangian in the new variables.   1 Lkinetic = ψM δMN ✷E − ∂M ∂N ψN 2

Here the SO(4) Lorentz and SU(2) symmetries are explicitly broken to SO(3) and U (1) respectively by the presence of the chemical potential but no condensation happens (i.e., ϕM  = ψM  = 0). The curvatures of the potential in the vicinity of the vacuum at µ
m are: Mϕ2a = Mψ2 M = m2 , 0

Mϕ2a = m2 − µ2 . I

(10)

The dispersion relations obtained by diagonalizing the 12 (4 space-time × 3 SU(2) states) by 12 quadratic matrix leads to 3 physical vectors (i.e., each of the following states has 3 components) with the following dispersion relations:  Eϕ ∓ = ±µ + p 2 + m2 ,  Eψ = p 2 + m2 . (11) This shows that when approaching µ = m the 3 physical components associated with Eϕ + become massless signaling an instability. Indeed, we now show that for values of the chemical potential larger than the critical value µc = m, a vector type condensation sets in. 2.2. The spin–flavor broken phase: µ > m In this phase the global minimum of the potential is for ϕ0a  = ψM  = 0, while we can choose the vev to lie in the (spin–flavor) direction:  1 µ2 − m 2 ϕX = (12) . λ − λ We have a manifold of equivalent vacua which are obtained rotating the chosen one under a Z2 × U (1) × SO(3) transformation. The choice of the vacuum partially locks together the Lorentz group and the internal symmetry while leaving unbroken only the

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subgroup Z2 × SO(2). Two generators associated to the Lorentz rotations are now spontaneously broken together with the U (1) generator. To compute the full dispersion relations of the theory we first need to provide the curvatures of the potential evaluated on the new vacuum: Mψ2 0 = Mψ2 Y = Mψ2 Z = m2 + λ

µ2 − m 2 , λ − λ

Mψ2 X = µ2 ,

(13) 
Mϕ21 = 2 µ2 − m2 ,

Mϕ21 = µ2 ,

X

0

Mϕ21 = Mϕ21 = 0, Y

(14)

Z

µ2 − m 2 , λ − λ µ2 − m 2 = λ . λ − λ

Mϕ22 = m2 + λ 0

Mϕ22 = Mϕ22 Y

Z

Mϕ22 = 0, X

(15)

In general three states have null curvature (specifically M 22 = M 21 = M 21 = 0), however, for the special ϕX

ϕY

ϕZ

case λ = 0 we have 5 zero curvature states. To explain this behavior we note that for λ = 0 the potential possesses SO(6) global symmetry which breaks to SO(5) when the vector field condenses. The associated 5 states would correspond to the ordinary Goldstone modes in the absence of an explicit Lorentz breaking. Using these curvatures we compute the dispersion relations. Four of the ϕ states have (2 per each sign) the following dispersion relations:  Eϕ2± = p2 + ∆2 ∓ 4µ2 p 2 + ∆4 V

with

∆2 = 2µ2 +

λ (µ2 − m2 ) , 2 λ − λ

(16)

and to each sign we associate a vector (with two components) with respect to SO(2). In the limit of small momenta for the gapless mode: Eϕ2+ = V


6 λ µ2 − m2 2 2µ4 4 p  + | p  | + O p 2∆2 (λ − λ ) ∆6

= vϕ2 − p 2 + · · · , V

where we denoted by vϕ − the superfluid velocity asV

sociated to the state ϕV− . For the last two ϕ physical

Fig. 1. Mass gaps E(0) as a function of the chemical potential µ. The phase transition occurs for µ/m = 1. The curves were obtained for the values λ = 1, λ = 0.33 of the potential coefficients. For µ/m < 1, all the gaps are threefold degenerate, and show the splitting of the different charge states for nonzero µ. For µ/m > 1 the modes with a nonvanishing gap split further. The solid lines are twofold degenerate (SO(2)-vectors), while the dashed lines correspond to SO(2)-scalar states.

modes (scalars of SO(2)) we show directly the dispersion relations as a momentum expansion.
 µ2 − m 2 2 p + O p4 = vϕ2 − p 2 + · · · , 2 2 3µ − m S
2
4  2 2 = 2 3µ − m + γ p + O p ,

Eϕ2+ = S

Eϕ2− S

(17)

where the subscripts V , S stand for vector and scalar respectively, and γ > 0. Finallythe three ψ type state

dispersion relations are EψI = p 2 + Mψ2 I , with MψI defined in Eq. (13). In Fig. (1) we show the mass gaps Ei (p = 0) for all the states. A relevant result is that the velocities of our gapless modes can be expressed directly in terms of the curvatures in the directions orthogonal to the gapless modes as follows: vϕ2 + V

=

M 22 ϕY

M 22 + 4µ2 ϕY

,

vϕ2 + S

=

M 21

ϕX

M 21 + 4µ2

.

(18)

ϕX

We find that the number of gapless states emerging when the flavor×rotational symmetry spontaneously breaks is insensitive to the choice of the coefficients in the effective potential. However, the dispersion relations depend crucially on the symmetries of the potential. The full vector Lagrangian (i.e., kinetic potential) possesses the Z2 × U (1) × SO(3) (flavor×rotational)

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symmetry which breaks, above the critical chemical potential, when the vector condenses to Z2 × SO(2). The gapless modes are associated to the 3 broken generators. One state is a scalar of SO(2) while the other two fields constitute a vector of SO(2). The scalar Goldstone possesses linear dispersion relations independently of the choice of the vector potential. This is the Goldstone associated to the U (1) breaking. However, for the dispersion relations of the SO(2) vector state we have either quadratic or linear dispersion relations. The quadratic ones occur when the potential possesses an enhanced SO(6) symmetry (i.e., for λ = 0). When condensing the vector breaks the potential symmetries from SO(6) to SO(5) while the derivative part of the Lagrangian is still invariant under the continuous U (1) × SO(3) only. In this case the potential has 5 flat directions (i.e., 5 null curvatures (occurring for λ = 0)) and we would count in the spectrum 2-massless vectors and a scalar of SO(2). However, the reduced symmetry of the derivative terms prevents the emergence of another gapless vector type mode and instead turns the linear dispersion relations of one of the vector states into quadratic ones. It is important to observe that at the second order phase transition point, all the velocities of the gapless modes vanish regardless of the value of the coupling constants. This is so since at the second order phase transition point also the potential curvatures (which at zero density would correspond to the physical masses of σ -like fields) in the directions orthogonal to the spontaneously broken generators ones vanish. This behavior is a natural consequence of the extended conformal symmetry of potential at this point. However, at nonzero chemical potential part of the theory is not conformal and the information encoded in the conformal part of the potential (as for the enhanced global symmetry case) is nicely and efficiently transferred to the momentum dependence of the dispersion relations via the velocities of the gapless modes (see Eq. (18)). Clearly, the continuity of the dispersion relations is still ensured. Our result is in agreement with the Nielsen–Chadha counting scheme, and allows for a number of relevant predictions which can be tested, as we shall discuss, by lattice simulations. It is amusing to note that, for λ = 0, the Nielsen– Chadha counting rule yields a number which is larger than the number of broken generators of the full theory. To the best of our knowledge this is the first time

147

that such a phenomenon has been discussed in the literature.

3. Conclusions and physical applications We summarized our findings in the Table 1 and conclude our work by considering some possible physical applications of massive relativistic vector condensation at high chemical potential.

Acknowledgements It is a pleasure for us to thank M.P. Lombardo, H.B. Nielsen, and P. Olesen for helpful discussions. We are also indebt to J. Schechter for discussions and careful reading of our manuscript. The work of F.S. is supported by the Marie-Curie Fellowship under contract MCFI-2001-00181, while W.S. acknowledges support by DAAD.

Appendix A. 2-color QCD Two-color QCD at nonzero chemical potential recently attracted a great flurry of interest since lattice simulations can be reliably performed at nonzero quark chemical potential. In [6] vector condensation at nonzero quark chemical potential has been predicted for two-color QCD with an even numbers of flavors. The predictions are supported [7] by lattice studies. Here we provided a detailed study of the vector condensation phenomenon at nonzero chemical potential while predicting new features related to the physics of the phase transition. A direct application of our results is for QCD with two colors and one light flavor. This theory has global quantum symmetry group SU(2). The extra classical axial UA (1) symmetry is broken by the Adler–Bell– Jackiw anomaly. At zero chemical potential Lorentz invariance cannot be broken and the simplest bilinear condensate (see Ref. [6] for conventions) is of the type: c1 c2 αβ

QIα,c1 QJβ,c2 EI J ,

(A.1)

with E = 2iT 2 the antisymmetric matrix in the flavor space and QIαc is a Weyl spinor with α = 1, 2 the

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spinorial index, c = 1, 2 the color and I = 1, 2 the flavor indices. The generators of SU(2) in the fundamental representation are T a = τ a /2 with τ a the standard Pauli matrices. It is easy to check that SU(2) T remains unbroken since we have T a E + ET a = 0 for all of the SU(2) generators. So we have no Goldstone bosons at all in the theory. However, we do have physical vectors (i.e., spin-1 fields) and massive scalar states at zero density. These states should be comparable in mass (recall the η versus the octet of ordinary vector bosons in 3-color QCD). The massive spin-one fields (i.e., the hadrons) must transform according to the regular representation of SU(2) and carry nontrivial baryon number which is proportional to T 3 . Our analysis can be immediately used to predict that when turning on a nonzero baryon chemical potential a superfluid phase transition will set in for µ larger then the mass of the vector fields. This is so since we always have at least one gapless mode with linear dispersion relations. We also predict the spontaneous breaking of rotational invariance and the emergence of a vector state (i.e., doublet of SO(2)) with either linear or quadratic dispersion relations. It is straightforward to extend and apply our analysis to the case of different number of flavors and to ordinary QCD at finite-isospin chemical potential. [6].

lations and one vector of SO(D − 2) with either linear or quadratic dispersion relations.

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Appendix B. Extra-dimensions Finally, we extend our model to an SU(2) charged vector in D − 1 number of space dimensions (with SO(D) Euclidean Lorentz symmetry) with an associated nonzero chemical potential. In this case the chemical potential first breaks explicitly SO(D) to the spatial SO(D − 1). At sufficiently high chemical potential SO(D − 1) breaks spontaneously to SO(D − 2). The gapless states consist of a scalar field under the SO(D − 2) transformations with linear dispersion re-

[13] [14] [15]

[16] [17] [18]

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