Anomalous chiral superfluidity

doi:10.1016/j.physletb.2010.01.015

Physics Letters B

Volume 684, Issues 2–3, 8 February 2010, Pages 119-122

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Abstract

We discuss both the anomalous Cartan currents and the energy–momentum tensor in a left chiral theory with flavor anomalies as an effective theory for flavored chiral phonons in a chiral superfluid with the gauged Wess–Zumino–Witten term. In the mean-field (leading tadpole) approximation the anomalous Cartan currents and the energy–momentum tensor take the form of constitutive currents in the chiral superfluid state. The pertinence of higher order corrections and the Adler–Bardeen theorem is briefly noted.

Introduction

Quantum anomalies play an important role at low energy where they condition the character of the anomalous decays such as $π^{0} \to γ γ$ , $γ \to π π π \dots$ decays. They translate the high energy content of a gauge theory in a way that is protected from radiative corrections and non-perturbative phenomena. A salient example of these anomalies is the Wess–Zumino–Witten term and its relevance to low energy chiral dynamics whether in meson or meson–hadron physics. Its form follows solely from geometry and gauge invariance. The importance of anomalies in dense QCD with an emphasis on the superfluid phase was noted in [1]. Recently, similar anomalies have surfaced in holographic QCD at finite density in the context of a hydrodynamical analysis [2].

Hydrodynamics is an effective description of long-wavelength physics that encapsulates the constraints of general conservation laws and symmetries. It describes the flow of the energy–momentum tensor and charged currents beyond the realm of perturbation theory. Although phenomenological in character, with flow and dissipation encoded in terms of transport parameters, hydrodynamics has been successful in describing many phenomena ranging from the fundamental such as ultrarelativistic heavy ion collisions to more mundane such as water flow.

An interesting question regarding the role of anomalies in the hydrodynamical set up was recently raised in [2], [3], [4], with a critical discussion in [5]. Using arguments based on triangle anomalies and thermodynamics they were led to an amendment of the constitutive currents [4]. Specifically, they have found that the constitutive but anomalous currents support additional terms as dictated by global anomalies that involve new transport parameters. We show in this Letter, that the amendments of the constitutive currents are in general expected from the Wess–Zumino–Witten action in the superfluid state [1]. Here, we derive these amendments in the mean-field (leading tadpole) approximation. We also note the relevance of the Adler–Bardeen non-renormalization theorem at the quantum level.

Section snippets

Anomalous current

Most of our discussion of anomalies will be centered on the QCD flavor anomalies and their transcription to the constitutive flavor currents in a superfluid. A specific example would be QCD at high fermion density and low temperature in the superfluid CFL phase with global $SU {(N = 3)}_{c + L + R}$ symmetry [1]. Another, would be just low temperature QCD with global $SU {(N)}_{L + R}$ symmetry, or the standard model in a superfluid state with only left handed fermions at low temperature.

For simplicity and notational

Energy–momentum tensor

The energy–momentum tensor associated to (3) follows canonically. Straightforward calculations yield ${\tilde{T}}_{ν λ} = T_{ν λ} - Tr (A_{L λ} \frac{δ Γ}{δ A_{L}^{ν}})$ with the symmetric energy–momentum $T_{ν λ} = - 2 F^{2} Tr {(L + A_{L})}_{ν} {(L + A_{L})}_{λ} + g_{ν λ} F^{2} Tr {(L + A_{L})}^{2}$ All the contributions from the Wess–Zumino term are in the anomalous current $δ Γ / δ A$ . (10) is gauge invariant under both a rotation of U and $A_{L}$ . The canonical stress tensor obeys the equation of motion $\partial^{ν} {\tilde{T}}_{ν λ} = - \partial_{λ}^{A} L$ with $L$ the Lagrangian density associated to (3) and $\partial^{A}$ acting only on the $A_{L}$ fields in $L$ .

Mean-field approximation

To analyze the anomalous current along with the symmetric energy–momentum, we define the chiral field as $U = e^{i π_{L}}$ with $π_{L} = π_{L}^{a} T^{a}$ a generic $SU (N)$ valued field. In the superfluid state, $π_{L}$ plays the role of the phonons which are excited either quantum mechanically or through temperature. For convenience we also define $2 Tr T^{a} (L + A) = i e^{a b} (π_{L}) (d π_{L}^{b} + {(e^{- 1} A_{L})}^{b}) \equiv i e^{a b} (π_{L}) Π^{b}$ after using $A_{L} = i T^{a} A_{L}^{a}$ . At tree level in the phonon fluctuations with $〈 e^{a b} (π_{L}) 〉 \approx δ^{a b}$ , (5), (10) are tied by the equation of motion (13), with

Hydrodynamics

We now identify the expectation value of the renormalized phonon field $Π_{r}^{a}$ with the local superfluid 4-velocity v through the Cartan chemical potentials $μ^{a}$ for the left charges in the absence of the external field (conserved left currents), $〈 Π_{r}^{a} (x) 〉 \approx μ^{a} v (x)$ We observe that this identification is in general irrotational, and similar to $J = | ψ^{†} ψ | \frac{(d ϕ - e A / c)}{m} = n v$ for the normal current contribution to a charged and non-relativistic $U (1)$ superfluid state with wavefunction $ψ = | ψ | e^{i ϕ}$ . A similar observation

Conclusions

We have shown how global non-Abelian flavor anomalies can be translated to the global constitutive currents from the microscopic currents using the anomaly equation in a superfluid state. In the mean-field or tadpole approximation, the gauge-invariant and symmetric part of the energy–momentum tensor renormalizes to an ideal fluid form. The gauge-invariant and anomalous flavor current renormalizes to a normal and ideal fluid contribution plus anomalous corrections. The latter are similar to a

Acknowledgements

We thank Edward Shuryak for discussions. This work was supported in part by US-DOE grants DE-FG02-88ER40388 and DE-FG03-97ER4014.

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Anomalous chiral superfluidity

Abstract

Introduction

Section snippets

Anomalous current

Energy–momentum tensor

Mean-field approximation

Hydrodynamics

Conclusions

Acknowledgements

Phys. Lett. B

Phys. Lett. B

Phys. Rev. D

JHEP

Phys. Rev.

Nucl. Phys. B

Nucl. Phys. A

JHEP