Journal ArticleWe argue that the p function for the statistics parameter in (2+1)-dimensional electrodynamics is zero to all orders in perturbation theory beyond one loop. We show that there can be finite radiative corrections from massless charged scalar fields

Semenoff, Gordon W.

Wu, Yong-Shi

English

The University of Utah: J. Willard Marriott Digital Library

VOLUME 62, NUMBER 7 PHYSICAL REVIEW LETTERS 13 FEBRUARY 1989 Renormalization of the Statistics Parameter in Three-Dimensional Electrodynamics Gordon W. Semenoff, (I) Pasquale Sodano, (2).(a) and Yong-Shi Wu (3) (J)Department of Physics. UniL'ersity of British Columbia, Vancoul'er, Canada V6T 2A6 (2) Department of Physics, Boston Unil'ersity, 590 Commonwealth AL'enue, Boston, Massachusetts 02215 (3) Department of Physics, Unit'ersity of Utah, Salt Lake City, Utah 84112 (Received 17 October 1988) We argue that the f3 function for the statistics parameter in (2+ I)-dimensional electrodynamics is zero to all orders in perturbation theory beyond one loop. We show that there can be finite radiative corrections from massless charged scalar fields. PACSnumbers: 12.20.Ds,05.30.-d, II.IO.Gh, 11.15.Bt Gauge theories in (2+ 1) dimensions have several features not seen in their 0+ I)-dimensional counter-parts. These center on the possible appearance of the Chern-Simons three-form in the action I and are connect-ed with parity anomalies 2 and exotic charge,2 angular momentum, 3 and statistics. 4 Original motivation for their study was their resemblance to the high-tem-perature limit of gauge theories in 0+ 1) dimensions. More recently, there has been interest in (2+ 1)-dimensional electrodynamics in models of certain condensed-matter phenomena, particularly high-Tc su-perconductors 5- 7 and the quantized Hall effect. 8 The Euclidean action of scalar electrodynamics with both Chern-Simons and Maxwell kinetic terms for the gauge fields is s= J d3X[(DIl¢0)*<D1l¢0)+m6¢6¢0+i4ao fIlVAA~avAAO+~F~vF~v], 7r 4eo (I) where DIl = all + igoA~. This is a prototype for the effective field theory of the CP I model discussed in Ref. 9 (see also Ref. 10). The Chern-Simons term gives the ¢ quanta fractional statistics. 4 (With the Maxwell term this is only true at large distances') The gauge field and e6 have dimension 1 and go and ao are dimensionless. This field theory is superrenormalizable with comput-able, finite physical parameters. There is considerable interest in a model with only Chern-Simons kinetic terms. 4- 10 It is defined by the limit e6 --> 00 where it is renormalizable with dimensionless bare coupling go I ao I -1/2 and F~vF~v is an irrelevant operator. Since all physical parameters are not finite in this limit, renor-malization is required. In Ref. 9 it was argued that the dimensionless statistics parameter g6/ao can have an in-teresting renormalization flow. We show in this paper that this is not the case. When the renormalized mass m 2~0, the corresponding f3 function is zero to all orders in loop expansion. For the critical case m 2 =0 (although there is no symmetry which prevents radiative mass gen-eration, it is natural at a critical point for the scalar to be massless) we demonstrate that f3 is zero to two-loop order and we conjecture that it is zero to all orders. We also show that, when m 2 =0, g6/ao has finite renormal-ization in higher loops. With ¢o = Z 1/2¢, A~ = Z jI2 All' go=Z -IZ3-1/2Z2g, ao =Z3-IZaa, mJ =m 2+ 8m 2, and the Ward-Takahashi identity Z =Z2, the action is Za is fixed by the requirement that the renormalized Chern-Simons term has coefficient a/47r. The physical statistics parameter is g2/a=ZagJ/ao. Propagators, vertices, and counterterms are summarized in Fig. 1. We use eJ as a gauge-invariant cutoff to be taken to 00 at the end of computations. To see that when m 2> 0, Za receives no contribu-tions beyond one loop, II consider the irreducible Euclidean N-photon correlation function for N > 4 in the one-loop ap-proximation rlll .. IlN(PI; ... ;-Lt'-Ipk ) shown in Fig. 2. r is symmetric. Radiative corrections to Za with two or more loops are one-photon-irreducible graphs obtained either by sewing together all but two legs of r [see Fig. 3(a)1, then finding the term linear in the external momentum and antisymmetric in vector indices of the resulting two-point function, 715 VOLUME 62, NUMBER 7 PHYSICAL REVIEW LETTERS 13 FEBRUARY 1989 c?' k' o~v - k~ kv - JJ.E~VA kA o k'(k'+JJ.') )l. p~q )l.VV ~ (k'6~v - k~v) + (z.-1) "'- E~VAkA ~ 2w x (z - 1) p' + (z -1) m' + z 5 m' FIG. 1. Progagators and vertices for scalar electrodynamics. The topological mass is If =ae6/2Jr, ed -~ co, and y is the gauge-fixing parameter. FIG. 3. How nllv is constructed from r.it depends on p whereas 'j{ is independent of p. or by sewing two l's together in all possible (one-photon irreducible) ways [Fig. 3(b)), lim E/lvP-!- J dl l ... dql ... lIlA (p;/I; ... )lvp (-p;q,; ... )'itA .. p . (p;/ j ; ••• ;q2; ... ). P'O upp When m 2> 0,1 is analytic at pii =0. 'it depends on p where as 'H does not. The Ward-Takahashi identity is O=p;"TjJl v, jJN[p';''';Pi;''';-~'Pk]' Taking a derivative by Pi, setting Pi =0, and using analyticity yields 0=11'1 'Vi jJN[p,; ... ;o; ... ;_N~lpk]' This immediately implies that (4) vanishes. It further implies lim-!-ljJv (p;-p; ... )=lim [-!-ljJv(p;o; ... )--!-ljJv (O;-P; ... ») =0, p.oupp p~ouPp upp and (3) also vanishes. (This generalizes to other massive matter such as spinor and vector fields. It is also valid at finite temperature. There the Euclidean momentum vectors have one component the discrete Matsubara frequency pp =2;rk B Tni with T the temperature, nj an integer, and the other two components continuous variables. For N > 4, Euclidean N-photon functions have unique extensions to functions of complex momenta pP. With massive matter, these functions are analytic at zero frequency and momentum and the Ward-Takahashi identity also holds. The above 2 3 2 3 + + ... N N FIG. 2. The N-photon correlation function at one-loop order is the symmetrized sum with all combinations of both three-and four-point vertices. 716 reasoning shows that no two-photon vertex with one power of derivatives is generated by radiative corrections beyond one loop at finite temperature.) Thus, given the known finite one-loop corrections from both scalars and spinors,2 their contributions to the f3 function for the + FIG. 4. One-loop photon self-energy. VOLUME 62, NUMBER 7 PHYSICAL REVIEW LETTERS 13 FEBRUARY 1989 statistics parameter is zero. la) I b) Id) Ie) This leaves the possibility of a nonzero f3 function from massless charged fields where the N-point functions r cannot be assumed analytic at zero external momenta. The zero mass limit which defines a massless field theory may not commute with the zero momentum limit which determines the renormalization of a coupling constant. However, since the f3 function is independent of momen-tum, it will be zero in the massless limit as it is in mas-sive theory. This will be verified by explicit calculation to two-loop order in the following. We conjecture that it holds to all orders. This argument does not exclude finite renormalizations and we shall find explicit nonzero finite corrections at two-loop order from massless scalars. FIG. 5. Two-loop photon self-energy. The one-loop correction to the photon two-point function from charged scalars (Fig. 4) is { [] 1/2} £ 2.J;;;2 p2+4m 2 . p2 rr~~)(p)=(p28Ilv-PIlPv) ---2-+ arcsIn 2 2 . SJr P p 2..Ji2 P +4m Ie) It) (5) Although the loop integral is potentially divergent and requires a cutoff at intermediate stages of calculation, the result is finite. Its antisymmetric part vanishes and when m 2 > ° its symmetric part has a local limit (p 2811 v - PIlPV) xg 2/24Jr.J;;;2 which contributes to Z 3. Figure 5 depicts two-loop self-energy diagrams. The corresponding integrals are finite for e& < 00 and by power counting are logarithmically divergent when e& -- 00. The contributions of Figs. 5(b)-5(e) to (1/2p2)cllvl.rr~~)h cancel immediately upon combining the integrals. Figure 5(a) can be reduced to the Feynman parameter integral ~rldxl"'dx4e(l-xl-"'-X4) 1 SJr 2a J o [(I - X 1- X2)(X 1+ ... + X4) - (X3 + X4) 2]5/2 (6) where p =ae&/2Jr and Figure 5 (n reduces to ,,4 11 XIX2 - ---'2-- dx 1 dx 2 e(l - x I - X 2) -------'--=-------;-;:-SJr 2a 0 [(1-xl-x2)(xI+x2)-xlx2]5/2 P 2 (I - X 1 - X 2)( I - X I ) + m 2 (I - X 1 )(x I + X 2) + P 2 ii xln , m 2 (I - X 1 )(x I + X 2 ) + P 2 ii (S) (9) If we keep m 2 > ° and put P 2 =0, these two contributions cancel for all values of p2 > 0, consistent with the no-renormalization argument above. On the other hand, if we first put m 2 =0 and then p2 / P 2 large (since e& -- 00), the divergent leading terms proportional to In (p2 / P 2) cancel. Therefore, as expected, there is no infinite contribution to Za. After some standard manipulations, 12 the finite parts are ~ r l 'A1/2p 2 Jr dpd'AdxdyeO -y-p) [ ()]5/2 SJr a 0 I - 'A + 'Ap I - P xln pO -p)(I-'A)O -'Ap) = -2.74~ (10) [0 -'Ap)(px+y) -'Ay 2][1 -'A+'ApO -p)] - [pO -'Aph+O -'A)yF SJr 2a where the approximate result is obtained numerically. To couple spinors we add f IJIJ[ Yll (ill il - igoAJ) + M 0] lJIo to (I). Generalization of the arguments above shows that 717 VOLUME 62, NUMBER 7 PHYSICAL REVIEW LETTERS 13 FEBRUARY 1989 there are no contributions to Za beyond one loop at ei-ther zero or finite temperature when M~O. At one loop, the correction is known 2 at T =0 and can be obtained at T~O as . 1 (I) -L M hm --2 fllvATII'v PI. - tanh-k-- . p ·0 2p 8n BT When M =0 the one-loop correction vanishes but there could be finite corrections from higher loops. Equation (5) indicates a radiatively generated Maxwell term at one-loop (and higher) order. We deter-mine Z 3 so that the renormalized Maxwell term has coefficient 1/4e 2• Even when the bare action has only a Chern-Simons term, a Maxwell term is generated in the effective action. In that case it is possible to resume per-turbation theory using the renormalized Maxwell term in the Feynman rules which are then superrenormalizable. The present analysis also applies to the singular case where l/e 2 =0 which remains marginally renormaliz-able. In summary, for scalars when m =0, Za=l +2.74g 4/2na 2+ ... , Z3=e6le 2 + ... ; when m2~0, Za=l, Z3=e?;/e 2-e?;g2/24n.,J;;;2+ .... For spinors when M =0, 8Za = (finite corrections from >- two loops), Z 3 is affected at two and higher loops; when M2~0, 8Za =-(gl/2a)tanh(M/kB T), 8Z 3 = -e?;g2/12n.JM2+... from one- and higher-loop vacuum polarization. In the massive theories we have Za exactly. In both cases there are no infinite corrections to ao and the corresponding f3 function vanishes. However, there is finite renormalization from massless particles in higher loops and from massive spinors at one loop. There can be an infinite renormalization from charged massive vectors at one 100p.13 The latter [as well as the Cp I model to which (I) is approximate] is a nonrenor-malizable theory and the perturbative expansion is un-reliable. Also, our results for renormalizable theories are perturbative and would be improved by demonstrating Borel summability of the perturbation series. Finally, we speculate on hysteresis where we vary parameters, i.e., T, m 2, e 2 of scalar electrodynamics. The statistics parame-ter is unaffected if we avoid critical points. If we go to a critical point the statistics parameter gets finite comput-able corrections. We then leave the critical point with renormalized fractional statistics. G.W.S. is supported by the Natural Sciences and En-gineering Research Council of Canada. 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Renormalization of the statistics parameter in three-dimensional electrodynamics

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Renormalization of the statistics parameter in three-dimensional electrodynamics

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