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PHYSICAL REVIEW D

VOLUME 48, NUMBER 8

Nonrenormalization

15 OCTOBER 1993

of the statistical parameter in three-dimensional Department

QED

Chem-Simons

Deog Ki Hong of Physics, Pusan ¹tional University, Pusan 609-735, Korea Taejin Lee

Department

of Physics,

Kangwon

Center for Theoretical Physics, Department

¹tional

University,

Chunchon 200-701, Korea

Seon H. Park of Physics, Seoul National

University,

Seou1151 742, K-orea

{Received 29 October 1992) We compute explicitly the two-loop-order radiative corrections to the statistical parameter in the marrenormalizable fermion three-dimensional Chem-Simons QED which has only the kinetic Chem-Simons term for the gauge field. Whether or not the fermion is massive, there are no infinite corrections. However, we find some finite corrections in the theory with massless fermions. We also argue that the statistical parameter is not subject to the infinite radiative corrections from the ultraviolet divergences of the theory at all orders; thus, the P function for the statistical parameter vanishes exactly. ginally

PACS number{s): 12.20.Ds, 05.30. — d, 11.10.Gh, 11.15.Bt

(2+1)-dimensional physics is much enriched by the Abelain Chem-Simons term [1], which gives a gaugeinvariant mass to the gauge field in (2+1)-dimensional electrodynamics and introduces a long-range interaction between the charged particles. Changing their spin and statistics through this long-range interaction, the charged particles turn into anyons [2], particles with fractional spin and statistics, which have a wide range of applications in (2+1)-dimensional physics such as the fractional quantum Hall effect [3] and high-T, superconductivity

[4]. The Chem-Simons term has received considerable attention associated with many important issues among which is the radiative correction to it. It is well known that the Chem-Simons term is generated by radiative corrections at the one-loop order [5], and the issue is whether it receives radiative corrections beyond one loop. Explicit computations [6,7] exhibit that the radiative corrections at the two-loop order vanish in threedimensional electrodynamics even with the bare ChernSimons term. This suggests that the Chem-Simons term is not subject to radiative corrections beyond one loop; thus, the one loop result is exact, which is the nonrenormalization theorem for the Chem-Simons term [8,9]. On the other hand, Polyakov [10] argued that the Chem-Simons term may receive infinite radiative corrections and, hence, an interesting renorrnalization-group flow for the statistical in the (2+ 1)parameter dimensional CP' model with the Chem-Simons term. The infinite corrections have also been discussed in Ref. [11] in the charged vector-meson model. However, the CP' and the massive vector-meson models are nonrenormalizable theories, so it still remains doubtful whether the statistical parameter has a meaningful nontrivial renormalization-group flow. This point was further discussed in Ref. [12] by considering scalar threedimensional QED (QED&) in the limit where the Maxwell 0556-2821/93/48{8)/3918{4)/$06. 00

48

term is suppressed and the theory becomes marginally renormalizable: A nontrivial renormalization-group flow for the statistical parameter was not found, although there can be finite corrections from the massless scalar fields. Recently, one of the authors [13] has also dis' model with the Cherncussed this point in the CP Simons term and has argued in the framework of the large X expansion that the Chem-Simons term may receive some infinite radiative corrections. The purpose of this paper is to examine whether there are infinite (or finite) radiative corrections to the statistical parameter in Chem-Simons QED& [14] which has only the Chem-Simons term as the kinetic term for the gauge field: L

= —4o

—.B~

—e&A~ y" +me $0+

E

4

e"' F Ai,

where the subscript (or superscript) 0 denotes the bare quantities. It is this Chem-Simons QEDs that is often believed to describe the low-energy sectors of the condensed matter systems where the nobel phenomena of the fractional quantum Hall effect and the high-T, superconductivity may occur. We consider both cases of the massive fermion and the massless fermion (mo =0). Although the Chem-Simons QED~ has been discussed by numerous authors [2, 15] because of its implicative importance and simplicity, it still needs a thorough analysis. If the ferrnion is massive, the model shares the same infrared behaviors with the similar Maxwell-Chem-Simons QED3 which has both a Maxwell term and ChernSimons term, but has different ultraviolet behaviors. Chem-Simons QED3 is marginally so renormalizable that the statistical parameter may have a nontrivial flow which cannot be anticipated renormalization-group

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1993

The American Physical Society

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48

Maxwell-Chem-Simons the super-renormalizable QED2. We can easily see that the Chem-Siinons QED2 is not super-renormalizable but marginally renormalizable, counting the superficial degree of divergence co(G) of the given Feynman diagram G. If the diagram 6 of L loops and V vertices consists of Iz gauge boson lines and I~ fermion lines, co(G) is given by

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in

co(G)=3L

If

use

we

we IF = ,'g, f„—,

co(G) —

. Iq — IF — L =I&+I +1 —V,

(2)

Iz = ' g „a„and —,

find

V

3=+ (co, —3)

co„=a,+ f, Ez —EF, —

(3)

,

a„(f„)

where denotes the number of gauge boson (fermion) lines incident to the vertex v and E~ (EF) denotes the number of external gauge boson (fermion) lines in G. For the vertex of the minimal coupling, we have a, =1, and co„=3. Thus all diagrams contribute to a given function with the same degree of divergence and of Green's functions with only a finite number — ~ 0 a— re (superficially) divergent: The model is 3 renormalizable. Concurrently we observe that the model has only one dimensionless coupling constant e~/Q~p, if we scale the gauge boson field as A „~(1/Q~p)A terms of renormalized quantities, fp=+Zig, ep =Z2 /Z, e, and ap = Z„a,Eq. (1) can be written as

f„=2,

E„EF

„.

A„=A„,

"Q+Z—2eA„gy"g Zim L = Z, Q . d„y—

Pg—

FIG. 1. Two-loop-order contributions to the

vacuum polar-

ization.

.

—.e"',

II2(0) = lim

k-p

6i

a

II,(k),

II„,(k)= J (2~)'g 1„,(k, —k, q, where I" is the one-loop four-point

q)GF— (q),

p

photon function. Here G (q) is the photon propagator which can be decomposed into a parity even part and parity odd part:

(q) =G

G

(q)+G

(q), /pe~

l

q

P

2

K

2

+l6

K

O

K

+lE'

1P~

"-~ qg 2.

The explicit calculation shows [7] that the contributions to II2(0) with G and G for the photon propagator are separately vanishing before the gauge-field loop integration is carried out. Since Chem-Sirnons QED& and Maxwell-Chem-Simons QED& have the same one-loop npoint photon function and the photon propagator in the Chem-Simons QED& is given as

(4)

We define the renormalized statistical parameter in the theory by H=e /a=(Z, Z2 Z„)ep/Kp=Z 8p. The general form for the gauge-invariant vacuum polarization in (2+ 1) dimensions can be written as

II„(k)=(kg„„k„k, II2(k ), )II,(k )— +i@„„2k

(5)

where the first and the second terms are parity even and odd terms, respectively. The power counting shows that the parity even term induced by the radiative corrections, II, (k ), is finite, which iinplies no need for the counterterm of the Maxell term. The coemcient of the induced Chem-Siinons term is given by II2(0). The one-loop contribution to II2(0) is the same as in Maxwell-ChernSimons QEDi. The radiative correction to II2(0) at twoloop order is obtained by evaluating the integrals depicted by two diagrams (Fig. 1). Our explicit evaluation shows that the divergent contributions from both diaThis cancellation occurs grarns cancel each other. whether or not the fermion is massive. So there are no infinite radiative corrections, and, thus, we have a vanishing P function for the statistical paraineter. We may still speculate on finite radiative corrections. It is easy to find that the cancellation is precise if the fermion is massive. We recall that the corrections at the two-loop order to the Chem-Simons term in the Maxwell-Chem-Simons QED3 is written as

K

&pwX

q

it is clear that the cancellation is precise. The cancellation in the theory with a massive fermion is what the Coleman-Hill theorem [8] predicts. This nonrenormalization theorem does not require the finiteness of the theory but only the analyticity of the one-loop nChem-Simons photon function and gauge invariance. QED2 with a massive fermion certainly satisfies the requirernents. In the case of the massless fermion we cannot apply the Coleman-Hill theorem to the theory, since the theory fails to satisfy one of the requirements, i.e., the analyticity of the one-loop n-photon function in the infrared region. our explicit calculation shows that the Nevertheless, infinite corrections are still absent. In passing, we must note that the Feynman integrals representing the radiative corrections to the statistical parameter are free of infrared divergences, although one-loop n-photon functions are not analytic in the infrared region if the fermion is massless. However, we can argue that the absence of the infinite corrections at least from the ultraviolet divergences persists at all orders beyond one loop, extending the Coleman-Hill theorem. The theory with a massless fermion and the theory with a massive fermion share the same ultraviolet behaviors, so the leading corrections

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from the ultraviolet divergences are not altered if a finite mass for the fermion is introduced. Then the theorem ensures the cancellation and this explains the absence of the infinite corrections from the ultraviolet divergences in the theory with the massless fermion. What remain are the contributions from the integrals over the infrared region. At two-loop order the integrals representing the corrections to the statistical parameter are infrared finite, hence, if the corrections exist, they are finite, which can be confirmed in an explicit evaluation. The same arguments above also apply to the massless scalar QED3 considered by Semeno8; Sodano, and Wu [12].

IP"(k)=ie

I

d q d p (2n. ) 3 (2m ) 3

tr[2y"S(p)y S(p

S (p ) = i /jd

G~

—q+k)yI'S(p+k)]G

and

(q)= —— e~~i,

phd.

g

2 +EE'

denote the fermion propagator and the photon propagator, respectively. In the theory with the massless fermion the parity odd part of IP'(k) is proportional only to e" k&. Then it follows the corrections to the statistical parameter can be given in terms of

II„as

ll~. (k) e„„,

11,(0) = —.

k'

(9)

Our explicit evaluation yields

II~(0) =

Noting that the parity is explicitly violated when the Chem-Simons term is present, one may argue that the fermion mass may be generated radiatively at one-loop order and the Coleman-Hill theorem applies also to the massless Chem-Simons QED, hence, the absence of radiative corrections. However, this is not the case in the dimensional regularization scheme, as the following explicit calculation shows. One can easily calculate the one-loop fermion self-energy X(p) and find that X(0) =0; i.e. , the fermion mass is not generated radiatively. The contributions from the two-loop diagrams (Fig. 1) to the vacuum polarization tensor are

q)— y~S(p)y S(p+k)

+yi'S(p)y S(p — q)y"S(p

where

e4

48

(10)

The dimensional

regularization is employed to obtain the analytic result Eq. (10). We shall close this paper with a brief summary. We have seen that there are no infinite radiative corrections from the two-loop diagrams to the statistical parameters whether the fermion is massive or massless; therefore, we have no need for the renormalization of the statistical parameter. This also implies no need for the renormalization of the dimensionless expansion parameter of the perturbative theory e /4m'. If the fermion is massive, the two-loop-order radiative corrections vanish exactly and all higher-order corrections are expected to vanish. Certainly, it is a consequence of the nonrenormalization theorem [8,9]. Thus we write the physical statistical parameter in the massive theory as

[1] S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. (N. Y.) 140, 372 (1981);W. Siegel, Nucl. Phys. B156, 135 (1979).

[2] F. Wilczek, Phys. Rev. Lett. 49, 957 (1982); F. Wilczek and A. Zee, ibid. 51, 2250 (1983). [3] S. Crirvin, in The Quantum Hail Effect, edited by R. Prange and S. Girvin (Springer, Berlin, 1986).

=

(q),

2

&+

4m'

including the finite one-loop-order correction which is the only radiative correction. However, if the fermion is massless (m =0), the coeScient of the Chem-Simons term receives finite corrections from the two-loop diagrams Eq. (10) which may be written in terms of the radiative corrections to the statistical parameter as 2

0 h, = 1+ 4m+

+

2

4m'

1+

7T2

+

.

go

.

(12)

The one-loop-order correction is of course included in Eq. (12). Since there are only finite corrections in both cases, we may take them as quantum corrections to the classical statistical parameter 00. After submitting our work, we received from Dr. V. P. Spiridonov a paper [16] which discusses the similar subject in Maxwell-Chem-Simons QED3.

D.K.H. and T.L. would like to thank H. S. Song for the hospitality during their visit at the Center for Theoretical Physics, Seoul National University. T.L. was supported in part by the Korea Science and Engineering Foundation and in part by nondirected research fund, Korea Research Foundation (1992). D. K.H. was supported in part by KOSEF and also in part by a YonseiUniversity Faculty grant (1991). We also would like to thank C. Lee for useful discussions and comments.

[4] P. Anderson, Science 235, 1196 (1986); X. Wen and A. Zee, Phys. Rev. Lett. 60, 1025 (1988); D. Haldane, ibid. 61, 1029 (1988). [5] S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. (N. Y.) 140, 372 (1981); I. AfBeck, J. Harvey, and E. Witten, Nucl. Phys. B206, 413 {1982);R. Jackiw, Phys. Rev. D 29, 2375

BRIEF REPORTS (1984). [6] Y. Kao and M. Suzuki, Phys. Rev. D 31, 2137 (1985). [7] M. Bernstein and T. Lee, Phys. Rev. D 32, 1020 (1985). [8] S. Coleman and B. Hill, Phys. Lett. 159B, 184 (1985). [9] T. Lee, Phys. Lett. B 171, 247 (1986). [10] A. Polyakov, in I'ields, Strings and Critical Phenomena, Proceedings of the Les Houches Summer School, Les Houches, France, 1988, edited by E. Brezin and J. ZinnJustin, Les Houches Summer School Proceedings Vol. XLIX (North-Holland, Amsterdam, 1990), p. 305; S. Deser and N. Redlich, Phys. Rev. Lett. 61, 1541 (1988). [ll] C. R. Hagen, P. Panigrahi, and S. Ramaswamy, Phys.

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Rev. Lett. 61, 389 (1988). [12] G. W. Semeno(F, P. Sodano, and Y.-S. Wu, Phys. Rev. Lett. 62,' 715 (1989); W. Chen, Phys. Lett. 8 251, 415 (1990}. [13] S. H. Park, Phys. Rev. D 45, R3332 (1992). [14] C. R. Hagen, Ann. Phys. (N. Y.) 157, 342 (1984). [15] Y.-S. Wu and A. Zee, Phys. Lett. 147B, 325 (1984); A. PoGauge Fields and Strings (Hardwood, London, 1987); G. SemenofF, Phys. Rev. Lett. 61, 517 (1988). [16] V. P. Spiridonov and F. V. Tkachov, Phys. Lett. B 260, 109 (1991).

lyakov,