This is a comment on Phys. Rev. Lett. 98, 180403 (2007) [arXiv:0704.2162].Comment: 1 page, comment on arXiv:0704.2162, published versio

[[corresponding]]Wang S. Y.

Wang S. Y.

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Shang-Yung Wang

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Comment on “Modified Coulomb Law in a Strongly Magnetized Vacuum”

A Comment on the Letter by Anatoly E. Shabad and Vladimir V. Usov [Phys. Rev. Lett.PRLTAO0031-9007 98, 180403 (2007)10.1103/PhysRevLett.98.180403]. The authors of the Letter offer a Reply.補正完畢SCI紙本電子

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Comment on ‘‘Modified Coulomb Law in a StronglyMagnetized Vacuum’’In a recent Letter [1], Shabad and Usov studied theelectric potential of a charge placed in a strong magneticfield B  B0  m2=e (m is the electron mass and e is itscharge), as modified by the vacuum polarization.According to these authors, in the limit B ! 1 the modi-fied potential becomes a Dirac delta function plus a regularbackground. With this potential they showed that the un-boundedness from below of the nonrelativistic hydrogenspectrum is removed. In this Comment, we point out thatShabad and Usov used an incorrect calculation to arrive attheir otherwise correct result and conclusion.It is firmly established that the magnetic catalysis ofchiral symmetry breaking [2] is a universal phenomenon.The general result states that a strong magnetic field leadsto the generation of a dynamical fermion mass even at theweakest attractive interaction between fermions. The real-ization of this phenomenon in the chiral limit in QED hasbeen studied in detail in the literature [3–5]. In particular,in Ref. [5], we obtained an asymptotic expression for thedynamical fermion mass, reliable in the weak couplingregime and the strong field approximation.The dynamical fermion mass in massless QED in astrong magnetic field, mdyn, is given by [5] mdyn ’  exp  log1 eBp; (1)where   e2=4 ’ 1=137 is the fine structure constant.In a moderately strong magnetic field B * B0, the ratiomdyn=m is vanishingly small; hence the calculation ofRef. [1] with the nonperturbative effect of magnetic ca-talysis completely ignored is justified. In an infinitelystrong magnetic field, however, the dynamical fermionmass mdyn approaches infinity linearly witheBp. Theuniversal nature of the magnetic catalysis of chiral sym-metry breaking implies that in realistic QED with massiveelectrons, the electron in an infinitely strong magnetic fieldwill acquire an effective dynamical mass that is muchlarger than m. As a result, the calculation of Ref. [1] isnot correct in the limit B ! 1.Taking into consideration the nonperturbative effect ofdynamical mass generation in realistic QED in a strongmagnetic field, we find the vacuum polarization that isquoted in Ref. [1] to be modified by 2k23; k2?  2eB exp k2?2eBTk234m2; (2)where the explicit dependence on eB has been restored. InEq. (2), m is the effective electron mass in a strongmagnetic field, yet to be determined by a gap equationthat couples to 2 through the full photon propagator. Atthis point it might appear that the only way to proceed is tofirst solve for m, which is a numerically intensive task.Fortunately, this is not the case. Since, as can be seen fromEq. (1), the effect of magnetic catalysis increases withincrease of B, it is conceivable that m has the asymptoticbehavior m  mdyn ! 1 as B ! 1. The key point is tonote that while both m and B approach infinity in the limitB ! 1, there is still a wide separation of scales in thisproblem, namely, m 	eBp. Furthermore, in the limitB ! 1 the ratio of these two scales, m=eBp 	 1, is aconstant independent of the magnetic field. Indeed, asshown in Ref. [5], the wide separation of scales mdyn 	eBpis the underlying physics behind the fact that Eq. (1) isreliable for weak couplings and strong fields.Based on the above analysis, the arguments of Ref. [1]that lead to an isotropic Yukawa law and, consequently, to aDirac delta function plus a regular background in the limitB ! 1, can be rendered valid, provided that we make thetwo corrections: (i) the electron Compton length, m1, ischanged to the effective electron Compton length in astrong magnetic field, m1 ; (ii) the separation of scalesLB 	 m1 is changed to LB 	 m1 , where LB  1=eBpis the electron Larmor length. The reason that Shabad andUsov obtained the correct result and conclusion is becausethe separation of scales LB 	 m1 	 m1 holds in thelimit B ! 1.Note added in proof.—The asymptotic behavior of mhas been numerically verified in a recent study [6].Shang-Yung WangDepartment of PhysicsTamkang UniversityTamsui, Taipei 25137, TaiwanReceived 24 July 2007; published 30 November 2007DOI: 10.1103/PhysRevLett.99.228901PACS numbers: 12.20.m[1] A. E. Shabad and V. V. Usov, Phys. Rev. Lett. 98, 180403(2007).[2] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys.Rev. Lett. 73, 3499 (1994); Phys. Lett. B 349, 477 (1995).[3] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys.Rev. D 52, 4747 (1995); Nucl. Phys. B 462, 249 (1996);D.-S. Lee, C. N. Leung, and Y. J. Ng, Phys. Rev. D 55,6504 (1997); E. J. Ferrer and V. de la Incera, Phys. Rev. D58, 065008 (1998).[4] V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys.Rev. Lett. 83, 1291 (1999); Nucl. Phys. B 563, 361 (1999);Phys. Rev. D 67, 107703 (2003); A. V. Kuznetsov andN. V. Mikheev, Phys. Rev. Lett. 89, 011601 (2002).[5] C. N. Leung and S.-Y. Wang, Nucl. Phys. B 747, 266(2006); Ann. Phys. (N.Y.) 322, 701 (2007).[6] S.-Y. Wang, arXiv:0709.4427.PRL 99, 228901 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending30 NOVEMBER 20070031-9007=07=99(22)=228901(1) 228901-1 © 2007 The American Physical Society

Comment on "Modified Coulomb Law in a Strongly Magnetized Vacuum"

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