Absence of the London limit for the first-order phase transition to a color superconductor

Defu Hou (Hua-Zhong Normal U.); Dirk H. Rischke (Frankfurt U.; FIAS); FIAS & Frankfurt U.); Hai-cang Ren (Rockefeller U. & Hua-Zhong Normal U.); Ioannis Giannakis (Rockefeller U.); Jorge L. Noronha (Frankfurt U.

REVIEW

Absence of the London limit for the first-order phase transition to a color superconductor

Jorge L. Noronha (Frankfurt U., FIAS), Hai-cang Ren (Rockefeller U. & Hua-Zhong Normal U.), Ioannis Giannakis (Rockefeller U.), Defu Hou (Hua-Zhong Normal U.), Dirk H. Rischke (Frankfurt U., FIAS & Frankfurt U.)

High Energy Physics - Phenomenology · Submitted 23 Feb 2006 · 27 references

Pith's verdict: A real factor-of-3 typo in Eq. (23) that does not actually inflate the final numbers; the paper's main results hold, and it deserves a proper refereeing.

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arXiv IDhep-ph/0602218 (version 1)

Pith IDE3YQQQOJ, Pith's permanent record of this paper and its review

CitationDefu Hou (Hua-Zhong Normal U.), Dirk H. Rischke (Frankfurt U., FIAS), FIAS & Frankfurt U.), Hai-cang Ren (Rockefeller U. & Hua-Zhong Normal U.), Ioannis Giannakis (Rockefeller U.), Jorge L. Noronha (Frankfurt U.. "Absence of the London limit for the first-order phase transition to a color superconductor." arXiv:hep-ph/0602218 (2006). https://pith.science/paper/E3YQQQOJ

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Quick take: the factor-of-3 stress-test is half right. Eq. (23) as printed is off by 3—the Pippard limit of F(z) is −(A/3) z ln z, not −A z ln z—but that is a typo, not a physics error. The paper's reported Δ^2_c∗, latent heat, and tHLM are all consistent with the corrected Eq. (23), not with the printed one. So the headline claims stand as they appear in Eqs. (30), (41), (43). The stress-test's conclusion that the latent heat is three times too large does not hold up; those final numbers are internally consistent with the correct asymptotic form.

What is actually new: the metastability temperature Tsh, the latent heat, the transition-strength estimate, and the numerical extrapolation of ξ0/λ to moderate μ. The physical picture is clear and worth taking seriously: Meissner masses suppress long-wavelength gluonic fluctuations relative to the normal phase, so the superconducting phase is stabilized; the transition is first order and T*_c > T_c. The authors explicitly explain why their sign differs from Ref. [20] (the missing exchange subtraction), and that is a plausible many-body statement, though the two-loop derivation is only sketched in the appendix.

Soft spots: the printed Eq. (23) will trip up any reader; it needs a factor 1/3. A referee should ask for that to be fixed, and for a sentence in the text that flags the correction. The numerical extrapolation to moderate μ is honestly labeled as uncontrolled, which is fine, but it means the quantitative claims there should be taken as indicative. The sign issue relative to Ref. [20] is the real substantive question; I would want a referee familiar with the CJT formalism to check the exchange subtraction carefully. The paper's argument is plausible, but the appendix is a sketch, not a step-by-step derivation.

Worth refereeing: yes. The core effect was already in Ref. [16], but the thermodynamic quantities and the London-limit discussion are genuine additions. The factor-of-3 typo is easily fixed and does not invalidate the results. I would not cite the numbers until the corrected version appears, but I would bring the paper to our reading group for the sign discussion.

Referee report

Major objections (2)

[Section III, Eqs. (23) and (29)-(30)] The asymptotic coefficient in Eq. (23) is a factor of 3 too large. In the Pippard limit f(y) = A/y with A = 3 pi^3/(28 zeta(3)), the leading large-z behavior of F(z) in Eq. (19) is F(z) approximately -(A/3) z ln(Az), not -A z ln(Az). Consequently, for the combination -F'(z)/z + F(z)/z^2 used in Eq. (29), one obtains approximately A/(3z), not A/z. Inserting A/(3z) into Eq. (29) reproduces Eq. (30), whereas using the printed Eq. (23) gives Delta^2_c* = pi^2/(21 zeta(3)) g^2 Tc^2, a factor of 3 larger than Eq. (30). Because Eq. (41) (latent heat) and Eq. (43) (the strength measure) inherit Delta^2_c*, those formulas would also be a factor of 3 too large if Eq. (23) were used; the printed Eqs. (41) and (43) are consistent with Eq. (30), so the inconsistency is localized in Eq. (23). The O(g) upward shift in Eq. (31) is unaffected because it comes from the logarithm, not from the prefactor, but the weak-coupling asymptotic formula must be corrected.
[Section II, Eqs. (12) and (14)] The central qualitative result, that gauge fluctuations stabilize the superconducting phase and raise T_c, depends entirely on the sign of Gamma_fluct, which in turn depends on the exchange-energy subtraction in Eq. (12c). The paper states that Ref. [20] omitted this subtraction and therefore obtained the opposite sign, but it does not provide a closed-form derivation of the subtraction or a detailed identification of the error in Ref. [20]. The appendix sketches the diagrammatic origin of Eq. (14) but leaves several intermediate steps implicit, including the explicit ring-sum evaluation and the power-counting estimates in Eqs. (A5)-(A7). Given that the sign determines whether T*_c is larger or smaller than Tc, the authors should supply a complete derivation of the subtracted ring resummation, or at minimum a step-by-step demonstration that the un-subtracted expression is not the stationary-point free energy.

Minor objections (3)

[Section III, Eq. (29)] The combination -F'(z)/z + F(z)/z^2 in Eq. (29) is typeset identically to the function F(z) defined in Eq. (19); please use a distinct symbol, such as a calligraphic or barred F, to avoid confusion.
[Section III, Eq. (36)] In Eq. (36), the factor g^2 T_c*^2 should read g^2 T_c^2 at leading order; as written, the equality with Delta_c*^2/2 is only correct up to O(g^3).
[Section IV, Figs. 2-5] The captions of Figs. 2-5 should state explicitly that the low-chemical-potential portions rest on the extrapolation assumption introduced in Section IV, namely that the generalized GL free energy remains numerically reliable at intermediate densities.

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The paper

The paper claims

Gluonic fluctuations make the color-flavor-locked transition in dense quark matter strongly first order, with critical temperature $T^*_c = (1 + \pi^2/(12\sqrt{2})\,g) T_c$, and eliminate the London limit of magnetic interactions.

The reading

This paper studies how gauge-field (gluonic) fluctuations alter the transition into the color-flavor-locked (CFL) phase of dense quark matter. It claims that these fluctuations stabilize the superconducting phase, turning what would be a second-order pairing transition into a strongly first-order phase transition with critical temperature $T^*_c = (1 + \pi^2/(12\sqrt{2})\,g) T_c$, larger than the pairing-instability temperature $T_c$. In weak coupling it derives the latent heat $L = 2 g^2/(21\zeta(3))\,\mu^2 T_c^2$ and shows that the coherence length to penetration-depth ratio at the transition is enormous, so magnetic interactions never become local: the London limit is absent. If the paper is right, cooling of compact-star matter through this transition would release latent heat discontinuously and the superconducting phase would be stabilized over a wider temperature window than pairing alone allows.

What carries the argument

The load-bearing object is the generalized Ginzburg-Landau free energy of Eq. (18), built from the CJT (Cornwall-Jackiw-Tomboulis) effective potential with two-loop dressed quark and gluon propagators. Its nontrivial ingredient is the fluctuation functional $F(z)=\int_0^\infty dx\,x^2\{\ln[1+z f(x)/x^2] - z f(x)/x^2\}$, where $f(y)$ describes the momentum dependence of the chromomagnetic Meissner mass, $m^2(T,k) = (1/\lambda^2) f(k\xi_0)$. The subtraction of the exchange energy inside the brackets is what makes the fluctuation term negative, and the large-argument (Pippard) asymptotics of $F$ control the size of the gap jump and the absence of the London limit.

Extended reading notes

Core claim

The central claim is that gluonic fluctuations lower, rather than raise, the free energy of the CFL phase, because the gluonic Meissner mass suppresses long-wavelength magnetic fluctuations compared with the normal phase, where transverse gluons are massless. In the generalized Ginzburg-Landau free energy obtained from the CJT (Cornwall-Jackiw-Tomboulis) effective potential with two-loop dressed propagators, the fluctuation piece is the negative ring-resummation integral $8T\int d^3k/(2\pi)^3 [\ln(1+m^2(T,k)/k^2) - m^2(T,k)/k^2]$, whose sign is fixed by subtracting the exchange energy from the ring sum. Consequently the transition temperature is shifted upward by a factor $(1 + \pi^2/(12\sqrt{2})\,g)$, the latent heat is $L = 2 g^2/(21\zeta(3))\,\mu^2 T_c^2$, and the diquark condensate jumps discontinuously at $T^*_c$. Since $\xi_0/\lambda_{c*}\gg1$ persists from weak coupling down to intermediate densities, the paper concludes that color superconductors never enter the London limit: diquark-gluon interactions remain nonlocal across the phase diagram.

Load-bearing premise

The result stands or falls on the bookkeeping rule that the exchange-energy term must be added to the condensate part of the free energy and subtracted from the fluctuation part; if that subtraction is invalid, the fluctuation energy changes sign and the predicted upward shift of $T^*_c$ and the strong first-order transition disappear.

Editorial extensions

If this is right

The CFL transition is strongly first order in weak coupling: $T^*_c = (1 + \pi^2/(12\sqrt{2})\,g) T_c$, so the superconductor can be superheated up to $T_{\mathrm{sh}} = [1 + g^2/72\,(1-\ln 2)] T^*_c$.
The latent heat released at the transition is $L = 2 g^2/(21\zeta(3))\,\mu^2 T_c^2$, far larger than the tiny values typical of electronic superconductors.
The diquark condensate jumps discontinuously at $T^*_c$, with $\Delta_{c*}^2 = \pi^2/(63\zeta(3))\,g^2 T_c^2$ in weak coupling.
Magnetic interactions remain in the Pippard limit throughout the CFL phase and its metastable superheated extension, so the London limit and local diquark-gluon couplings are never realized.
Extrapolated to intermediate densities with a running coupling, $T^*_c/T_c$ stays above $1$ and $\Delta_{c*}/\Delta(0)$ remains sizable, so the qualitative effects survive at densities relevant for compact stars.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: If the exchange-energy subtraction is correct, competing calculations that omit it (and conclude fluctuations lower $T^*_c$) are not merely quantitatively off but have the wrong sign; the discrepancy is a direct testable check of the two-loop effective potential.
Editorial inference: A strongly first-order CFL transition with latent heat $L\propto \mu^2 T_c^2$ would make neutron-star cooling non-monotonic, with heat released discontinuously as a CFL core forms; this may be observable in the thermal evolution of older compact stars.
Editorial inference: The same generalized-Ginzburg-Landau treatment applied to two-flavor color superconductors or to charge-neutral, massive strange quark matter would likely find a similar suppression of long-wavelength magnetic fluctuations, so the absence of the London limit may extend beyond CFL.
Editorial inference: Because local diquark-gluon interactions are excluded, vortex-lattice and supercurrent calculations near $T_c$ would need a nonlocal reformulation, which would alter predicted responses of color-superconducting matter to magnetic fields and rotation.

Original abstract

We study the effects of gauge-field fluctuations on the free energy of a homogeneous color superconductor in the color-flavor-locked (CFL) phase. Gluonic fluctuations induce a strong first-order phase transition, in contrast to electronic superconductors where this transition is weakly first order. The critical temperature for this transition is larger than the one corresponding to the diquark pairing instability. The physical reason is that the gluonic Meissner masses suppress long-wavelength fluctuations as compared to the normal conducting phase where gluons are massless, which stabilizes the superconducting phase. In weak coupling, we analytically compute the temperatures associated with the limits of metastability of the normal and superconducting phases, as well as the latent heat associated with the first-order phase transition. We then extrapolate our results to intermediate densities and numerically evaluate the temperature of the fluctuation-induced first-order phase transition, as well as the discontinuity of the diquark condensate at the critical point. We find that the London limit of magnetic interactions is absent in color superconductivity.

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.27 · human participants 0.01 · controlled comparison 0.35 · randomized assignment 0.01 · states quantitative result 0.13 · compares with prior results 0.86 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the effects of gauge-field fluctuations on the free energy of a homogeneous color superconductor in the color-flavor-locked (CFL) phase. Gluonic fluctuations induce a strong first-order phase transition, in contrast to electronic superconductors where this transition is weakly first order. The critical temperature for this transition is larger than the one corresponding to the diquark pairing instability. The physical reason is that the gluonic Meissner masses suppress long-wavelength fluctuations as compared to the normal conducting phase where gluons are massless, which stabilizes the superconducting phase. In weak coupling, we analytically compute the temperatures associated with the limits of metastability of the normal and superconducting phases, as well as the latent heat associated with the first-order phase transition. We then extrapolate our results to intermediate densities and numerically evaluate the temperature of the fluctuation-induced first-order phase transition, as well as the discontinuity of the diquark condensate at the critical point. We find that the London limit of magnetic interactions is absent in color superconductivity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the effects of gauge-field fluctuations on the free energy of a homogeneous color superconductor in the color-flavor-locked (CFL) phase.	contribution 0.99	0.92	0.03	asserted 0.57
s1	Gluonic fluctuations induce a strong first-order phase transition, in contrast to electronic superconductors where this transition is weakly first order.	result 1.00	0.75	0.18	asserted 1.00
s2	The critical temperature for this transition is larger than the one corresponding to the diquark pairing instability.	result 1.00	0.74	0.06	asserted 1.00
s3	The physical reason is that the gluonic Meissner masses suppress long-wavelength fluctuations as compared to the normal conducting phase where gluons are massless, which stabilizes the superconducting phase.	interpretation 1.00	0.56	0.10	asserted 1.00
s4	In weak coupling, we analytically compute the temperatures associated with the limits of metastability of the normal and superconducting phases, as well as the latent heat associated with the first-order phase transition.	method 0.94	0.96	0.05	asserted 0.99
s5	We then extrapolate our results to intermediate densities and numerically evaluate the temperature of the fluctuation-induced first-order phase transition, as well as the discontinuity of the diquark condensate at the critical point.	method 0.90	0.96	0.05	asserted 0.86
s6	We find that the London limit of magnetic interactions is absent in color superconductivity.	result 0.97	0.95	0.87	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from proof or derivation 0.56.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
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levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: The two-loop approximation to Γ

**Figure 2.** FIG. 2: Comparison between the different temperatures invol

**Figure 3.** FIG. 3: (a) Comparison between the critical temperatures at

**Figure 5.** It is instructive to express the right-hand side of Eq. (29) in terms of the GL parameter (3) and compare it with the corresponding equation for a metallic superconductor, whose generalized GL free energy was given in Ref. [16]

Showing 4 of the paper's 5 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/0602218 by the authors.

References

Works cited (27)

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T. Matsuura, K. Iida, T. Hatsuda, and G. Baym, Phys. Rev. D 69, 074012 (2004)

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[16]

I. Giannakis, D. Hou, H.-c. Ren, and D.H. Rischke, Phys. Rev. Lett. 93, 232301 (2004)

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H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973); D.J. Gro ss and F. Wilczek, Phys. Rev. D 8, 3633 (1973); Phys. Rev. D 9, 980 (1974)

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J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34, 1353 (19 75)

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M. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys. B 537 , 443 (1999); R. Rapp, T. Sch¨ afer, E.V. Shuryak, and M. Velkovsky, Phys. Rev. Lett. 81, 53 (1998); K. Rajagopal and F . Wilczek, hep-ph/0011333, to appear as Chapter 35 in the Festschrift in honor of B.L. Ioﬀe, At the Frontier of Particle Physics/Handbook of QCD , M. Shifman ed., (World Scientiﬁ...

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I.A. Shovkovy, Found. Phys. 35, 1309 (2005); M. Huang, In t. J. Mod. Phys. E 14, 675 (2005); T. Sch¨ afer, hep-ph/050906 8

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T. Sch¨ afer and K. Schwenzer, Phys. Rev. D 70, 114037 (200 4); H. Grigorian, D. Blaschke, and D. Voskresensky, Phys. Rev. C 71, 045801 (2005); D.N. Aguilera, D. Blaschke, M. Buba lla, and V.L. Yudichev, Phys. Rev. D 72, 034008 (2005); P. Jaikumar, C.D. Roberts, and A. Sedrakian, nucl-th/05090 93; A. Schmitt, I.A. Shovkovy, and Q. Wang, Phys. Rev. D 73, 0...

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J. Madsen, Phys. Rev. Lett. 85, 10 (2000); C. Manuel, A. Do bado, and F.J. Llanes-Estrada, JHEP 0509, 076 (2005)

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D.T. Son, Phys. Rev. D 59, 094019 (1999)

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[11]

T. Sch¨ afer and F. Wilczek, Phys. Rev. D 60, 114033 (1999 ); R.D. Pisarski and D.H. Rischke, Phys. Rev. D 61 051501 (2000); D.K. Hong, V.A. Miransky, I.A. Shovkovy, and L.C.R. Wijewardhana, Phys. Rev. D 61 056001 (2000), [Erratum- ibid. D 62, 059903 (2000)]; W.E. Brown, J.T. Liu, and H.-c. Re n, Phys. Rev. D 61, 114012 (2000), Phys. Rev. D 62, 054016, 05...

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[12]

I. Giannakis and H.-c. Ren, Nucl. Phys. B 669, 462 (2003)

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[13]

B.I. Halperin, T.C. Lubensky, and S. Ma, Phys. Rev. Lett . 32, 292 (1974)

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[14]

See, for instance, E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics (Pergamon Press, New York, 1980); see also V.P. Gusynin and I.A. Shovkovy, Nucl. Phys. A 700, 577 (2002) for t he case of a color superconductor

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I. Bonalde, B.D. Yanoﬀ, M.B. Salamon, and E.E.M. Chia, P hys. Rev. B 67, 012506 (2003)

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[18]

K. Iida and G. Baym, Phys. Rev. D 63, 074018 (2001)

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[19]

I. Giannakis and H.-c. Ren, Phys. Rev. D 65, 054017 (2002 )

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[21]

D.H. Rischke, Phys. Rev. D 62, 054017 (2000)

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[22]

Review of Particle Physics by Particle Data Group, Phys . Rev. D 66, 010001 (2002)

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[23]

J.L. Noronha, I. Giannakis, D. Hou, H.-c. Ren, and D.H. R ischke, nucl-th/0511047, to appear in the proceedings of 18th International Conference on Ultrarelativistic Nucle us-Nucleus Collisions: Quark Matter 2005 (QM 2005), Budape st, Hungary, 4-9 Aug 2005

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[24]

K. Iida and G. Baym, Phys. Rev. D 66, 014015 (2002), [Erra tum-ibid., Phys. Rev. D 66, 059903(E) (2002)]

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[25]

P.T. Reuter, Q. Wang, and D.H. Rischke, Phys. Rev. D 70, 1 14029 (2004); Erratum-ibid. D 71, 099901 (2005)

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[26]

K. Iida, T. Matsuura, M. Tachibana, and T. Hatsuda, Phys . Rev. D 71, 054003 (2005)

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[27]

J.L. Noronha, I. Giannakis, D. Hou, H.-c. Ren, and D.H. R ischke, in preparation

View on Pith

How the literature uses it

3 arXiv papers cite this in 4 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 2 sentences without a confident label

Uses its method or tool 2 papers

One can also make an expansion in terms of the condensate $Δ$ near the critical temperature where $Δ$ is small [ref] .

BCS-BEC crossover in a relativistic boson-fermion model beyond mean field approximation · CJT formalism · 0.91
In order to proceed, we rewrite the full propagators as [ref] [equation] The full quark propagator $S$ generally depends on the gap energy $Δ$.

Speed of sound and trace anomaly in a unified treatment of the two-color diquark superfluid, the pion-condensed high-isospin matter, and the 2SC quark matter · Formalism · 0.90

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [20]

T. Matsuura, K. Iida, T. Hatsuda, and G. Baym, Phys. Rev. D 69, 074012 (2004)

View on Pith 2004

[2] [16]

I. Giannakis, D. Hou, H.-c. Ren, and D.H. Rischke, Phys. Rev. Lett. 93, 232301 (2004)

View on Pith 2004

[3] [1]

D. Ivanenko and D.F. Kurdgelaidze, Astroﬁz. 1, 479 (1965 ); Lett. Nuovo Cim. 2, 13 (1969); N. Itoh, Prog. Theor. Phys. 44, 291 (1970); F. Iachello, W.D. Langer, and A. Lande, Nucl. Phys. A 219, 612 (1974)

View on Pith 1969

[4] [2]

H.D. Politzer, Phys. Rev. Lett. 30, 1346 (1973); D.J. Gro ss and F. Wilczek, Phys. Rev. D 8, 3633 (1973); Phys. Rev. D 9, 980 (1974)

View on Pith 1973

[5] [3]

J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34, 1353 (19 75)

View on Pith

[6] [4]

For early works, see: D. Bailin and A. Love, Phys. Rept. 10 7, 325 (1984), and references therein

View on Pith 1984

[7] [5]

M. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys. B 537 , 443 (1999); R. Rapp, T. Sch¨ afer, E.V. Shuryak, and M. Velkovsky, Phys. Rev. Lett. 81, 53 (1998); K. Rajagopal and F . Wilczek, hep-ph/0011333, to appear as Chapter 35 in the Festschrift in honor of B.L. Ioﬀe, At the Frontier of Particle Physics/Handbook of QCD , M. Shifman ed., (World Scientiﬁ...

View on Pith Pith review arXiv 1999

[8] [6]

D.H. Rischke, Prog. Part. Nucl. Phys. 52, 197 (2004)

View on Pith 2004

[9] [7]

I.A. Shovkovy, Found. Phys. 35, 1309 (2005); M. Huang, In t. J. Mod. Phys. E 14, 675 (2005); T. Sch¨ afer, hep-ph/050906 8

View on Pith 2005

[10] [8]

T. Sch¨ afer and K. Schwenzer, Phys. Rev. D 70, 114037 (200 4); H. Grigorian, D. Blaschke, and D. Voskresensky, Phys. Rev. C 71, 045801 (2005); D.N. Aguilera, D. Blaschke, M. Buba lla, and V.L. Yudichev, Phys. Rev. D 72, 034008 (2005); P. Jaikumar, C.D. Roberts, and A. Sedrakian, nucl-th/05090 93; A. Schmitt, I.A. Shovkovy, and Q. Wang, Phys. Rev. D 73, 0...

View on Pith 2005

[11] [9]

J. Madsen, Phys. Rev. Lett. 85, 10 (2000); C. Manuel, A. Do bado, and F.J. Llanes-Estrada, JHEP 0509, 076 (2005)

View on Pith 2000

[12] [10]

D.T. Son, Phys. Rev. D 59, 094019 (1999)

View on Pith 1999

[13] [11]

T. Sch¨ afer and F. Wilczek, Phys. Rev. D 60, 114033 (1999 ); R.D. Pisarski and D.H. Rischke, Phys. Rev. D 61 051501 (2000); D.K. Hong, V.A. Miransky, I.A. Shovkovy, and L.C.R. Wijewardhana, Phys. Rev. D 61 056001 (2000), [Erratum- ibid. D 62, 059903 (2000)]; W.E. Brown, J.T. Liu, and H.-c. Re n, Phys. Rev. D 61, 114012 (2000), Phys. Rev. D 62, 054016, 05...

View on Pith 2000

[14] [12]

I. Giannakis and H.-c. Ren, Nucl. Phys. B 669, 462 (2003)

View on Pith 2003

[15] [13]

B.I. Halperin, T.C. Lubensky, and S. Ma, Phys. Rev. Lett . 32, 292 (1974)

View on Pith 1974

[16] [14]

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