Physics Letters B 300 (1993) 241-244 North-Holland

PHYSICS LETTERSB

Dimensional regularisation in Chern-Simons theory

R. D e l b o u r g o a n d A.B. W a i t e s Department of Physics, University of Tasmania, GPO Box 252C, Hobart 7001, Australia

Received 20 October 1992; revised manuscript received 8 December 1992

We consider a dimensional continuation of the 3D Chern-Simons term which properly defines the regularised integrals in the theory and allows the three-dimensional limit to be safely taken.

There has been much recent interest in Chern-S i - mons theories in three dimensions , due to their abil- ity to explain certain behav iour in the fract ional quan tum Hall effect and high Tc superconduct iv i ty and also due to their connect ion [ 1 ] with knots and conformal field theories in 2D.

One impor tan t considera t ion when studying Che rn -S imons theories is the method employed to regularise ul t raviolet divergent integrals. Several at- tempts have been made to successfully regularise the theory, including Paul i -Vi l la rs [ 2 ], analyt ic regular- isation [ 3 ] and dimensional regularisation [ 4,5 ], but the results have not been totally conclusive.

We wish to use d imens iona l regularisation, as it avoids the UV divergences in a very natural and un- obtrusive way, wi thout compl ica t ing the Feynman rules, and allows simple access to higher orders of per turba t ion theory. Dimens iona l regularisat ion in- volves setting up the field theory in arbi t rary n-di- mensional space- t ime. A problem arises whenever a theory involves objects whose proper t ies depend ex- plicitly on the d imension, such as 73 matr ices in even dimensions, and e tensors in odd d imens iona l theo- ries such as we are interested in. In 4D, this problem has been overcome by replacing the 73 by the fourfold an t i symmetr ic product of the other gamma matr ices [6] , and axial vectors by threefold an t i symmetr ic tensors

73~71UY~TaY~l ' 7aY3~YluTvT~I '

/~, V, e, r = 0 , 1,2 ..... 2/--1 .

In 3D, things are not quite as simple. Even if no term is included in the lagrangian, it has the potent ial to be generated dynamical ly, so in a d imensional context the E tensor must be generalised to 2 l+ 1 di- mensions. (I t is not good enough ~L to leave it as eu~a and cont inue to 3 + e d imensions [ 4,5 ]. )

Several methods to "soak up" the extra indices on the ( 2 / + 1 ) -d imensional e tensor suggest themselves. We could write the Chern-Simons contr ibution to the

lagrangian as

(.#lU2~u3 ...U21,u21+ l m Ul F u2lz3. . .F 'u2t'u2l+ l ,

retaining A u as the natural gauge potent ia l and F u"

as its associated field strength. This would indeed be a genuine ( 2 / + 1)-dimensional term, yielding the correct 3D limit. Unfor tunate ly this approach re- quires the considerat ion of new processes and dia- grams, since each F corresponds to an extra photon line. It would then be necessary to sum over l photon lines, complicat ing the formulat ion of the theory considerably, and would be akin to using a 75 which is the an t i symmetr ic product of a / / 7 matrices for treat ing the chiral anomaly.

Instead we will depar t from the normal approach and regard the "gauge" field as a n / - c ompone n t anti- symmetr ic tensor. Now the Chern -S imons term re- tains the bilinear form

5°cs = em...u2,+, A m U ' F ul+, ....u2/+ 1 , ( 1 )

a~ This technique works to one loop, but only because at that order the term proportional to ~ does not need to be regular- ised, as can be seen by power counting.

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where

Fut+ 1 ---~/21+ l = 0 [ ]'//+ Im ,Ut+ 2...,u2t+, I

is the antisymmetric curl of A u 'u t . Our Chern-Si- mons term has the advantage that even in 2l+ 1 di- mensions, the processes of the theory are unchanged. It is not obvious what is the physical significance of the Amm, particularly as it couples to a non-con- served tensor current qTytu,...yu, l ~u, but it is not cru- cial to visualise it for arbitrary l since it reverts to a bona fide gauge field in l= 1, i.e. in three dimensions.

We therefore generalise the Chern-Simons (CS) lagrangian in 3D

Lf3D = -- ~VU/1F~ + ~ I t ~ x Fu/1A~ (2)

to arbitrary 2l+ 1 dimensions by making the exten- sion described above:

5e= ( - -1) l - - F ~ l--a/+ ~ Fz~...,tt+~ 2( l+ 1)!

~'~ ~ lff' )- l . ..)L l + l z~ ~. l + 2.. .~. 2l + l + 2(l+l)!(l!)2~z~.. .~2t+~ - ~"

+ gauge fixing terms. (3)

Taking functional derivatives with respect to A Ul...u~ and A/1'/1~, we obtain the inverse tensor propagator

( D - 1 ),Ul...ul,/1l.../11(k)

= ( - 1 )l(k2tla,[vl ...llaw d -k[u l ?l ,uzc/12. . .?Lull /1tk/11z)

+ i#ka%u,...ul ...... i , (4)

where the [ ] in qu, t/1,'"t/u~l denote antisymmetry in the vi and the [ ] and V- B in ktu,~lu2c~v..~luo~tk/1,3 denote antisymmetry in the/~, and v~ respectively. The propagator, when inverted, yields

( - l ) t D,u l . . .m . . . . . . . ,(k) = 1! [kZ+ ( - 1 )1#21

X ( q u , t/1,...qu~/1,, - ~ kt,,qu2r-/1v"llu,,~,k/1,~ )

ilt -- l ! k Z [ k 2 + ( -- 1 )1#2] k°%u,...ut/1,. ...,

( - 1 )~ . "~" ~ /([il l ~.tL2 V-//2 ""r],ut]vlk/11 ~ " ( 5 )

This will reduce to the normal 3D gauge propagator, Du/1, by collapsing to l= 1, yielding the standard re-

sult [ 2 ]. It should be noted that if I is even, the pho- ton propagator becomes tachyonic. This would be a serious problem, but it can be seen that for even l, ( 1 ) becomes a total divergence, which will not con- tribute to the theory. Using the usual free fermion propagator, we may now calculate the vacuum polar- ization, the one-loop correction to the bare gauge propagator,

(~')L!..u~ ....... , = (~);,!..u,.~,...~,+Hu,...u;.~,.../1t, (6)

Hu,...u,.~l...,,( k ) = ( - 1 )u/Zl ie2

f d 21 + lp × J (2~) 2;+, t r[y tu l . . .u , lS(p+k)Tt , , - . /1 t]S(P)] •

( 7 )

Here [ l /2] is the integer part of 1/2, and Ytu~...utJ- 7tu, Yu2...Tud/l] is the antisymmetric product of I Yu,, which themselves are just the 2l non-parity-doubled (2 ;×2 t) gamma matrices, as well as the full "75", which acts as the last one, 72;+ 1. Using the methods associated with dimensional regularisation, includ- ing introduction of a Feynman parameter, and eval- uating the (2/+ 1 )-dimensional momentum integral, we find an expression for the "vacuum polarisation",

. . . . . . . ( k ) T ,,...u,.,,_./1, / / .~ . , . , / 1 , ( k ) = / 7 ~ ~

+ H 2 ( k 2 ) 2 3 2 3 T #l..4ll, Vl.../11 • Tu,...u, . . . . . . . i d l - H ( k ) ( 8 )

Here,

1 T,ul . . . lq , Vl...m - - k [ v l ?]/12E,uz . . . l~v lLulk l~ l~

- - k2?],Ul [/11'"l~,utvd ,

2 " a T ul..,,ul, vt . . .m - - l m k Ea,Ul...pl/1t...vt ,

Tam. uz,/1t./1t = ~lm t/1~-"~l,~ml ,

and

i P ( k 2 ) = eZ( - 2 )t+ l F ( ~ - l ) (4n)t+l/2

f d a o r ( l - o r ) X i r a 2 k 2 0 t ( l _ a ) ] 3 / 2 _ l ,

e2( -- 1 ) U/2121F(3 --l) H 2 ( k 2 ) = (4g)t+ t/2

f d a [ ( - - 1 ) l ( 1 --00 - - a ] X [ m 2 _ k Z a ( l _ a ) ]3/2_1,

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Volume 300, number 3 PHYSICS LETTERS B 11 February 1993

H 3 ( k 2 ) = e 2 ( - 2 ) t F ( ½ - / ) (4n)t+i/2

× [ m 2 _ k Z a ( l _ a ) ] w z - t

m2(½-/)[l+(-1)q~. X(l-t- 3 /

We would like Hu. to satisfy the Ward identity for the vacuum polarisation, kuHu.=O, when l= 1. We would also have liked our Hu~...~,.,.... ~ to obey this relation, but that cannot be for arbitrary l, since the current to which A couples is not conserved except for l= 1 In (8) above, kU~T ~ and

• / / I . . . ] Z l , p i . . . p I

kU'T2~...uj.,~...,t are always equal to zero, but kU'T3~...ut,,~...,~ persists a n d / / 3 = 0 only if l= 1. This can be summarized by saying that effectively ku'//,,...u~,,~...,~ contains an evanescent factor of l - 1. The important point is that the odd dimensions, since //1, //2 and //3 do not contain 1 / ( l - l ) diver- gences ~2, the Ward identity is always satisfied in the limit, and the theory is free of anomalies. It should also be noted that if l is even , / /2 contains a factor of 1 - 2a, which causes the ¢ contribution to Hu, to dis- appear. This is to be expected, since Lfcs becomes a total divergence for even l, as we discussed earlier.

Since the Ward identity becomes satisfied, we can indeed calculate the gauge invariant 3D vacuum po- larisation by letting I ~ 1 without fear. After perform- ing the ce integration, Hreduces to

Hu . (k ) = (r/u~ k~,kf~ k 2 j I P ( k 2 )

+ imeu~,,U'H 2 (k 2 ) , (9)

where

H l ( k 2)

- - e2 [ ( X ~ q- 4 rn 2 "~ ln(2m + " ~ "~ - 4m ] 16n ~=2] \ 2 m _ x / ~ ] '

e 2 [ 2 m + v / ~ ln[ /75 ] / / 2 ( k 2 ) = ~ \ 2 m - ~ / k ' ] "

Studying the asymptotic behaviour of 1-Iu., we find that as k--.0, and provided m # 0 ,

#2 Just such a divergence leads to the axial Adler-Bell-Jackiw anomaly in 4D chiral gauge theory.

e2k 2 e a H I ( k 2 ) - 1 2 n m ' H Z ( k 2 ) - 4 n m " (10)

We also note that if m = 0

H t ( k 2 ) = - e2x/C~2 H2(k2) = e 2 16 ' ~ '

(11)

but since the coefficient o f / /2 contains a factor o f m, t he / /2 contribution disappears, and we are left with the equivalent o f the usual parity-doubled vacuum polarisation [ 7 ].

To place these results in perspective we will now look briefly at the asymptotic behaviour of t h e / / u , obtained by others. Firstly, Stain's [8] expressions for//1 a n d / / 2 correspond exactly to ours. The/-/u, obtained by Pimentel, Suzuki and Tomazelli [ 3 ], us- ing analytic regularisation, also shows exactly the same behaviour as (10) and ( 11 ) above, provided that the limiting procedure is properly undertaken ,3 The results of Appelquist, Bowick, Karabali and Wijewardhana are also of precisely the same form as the present work [9]. Finally, comparing with the work of Deser, Jackiw and Templeton [2] , who used Pauli-Villars regularisation, we find some interest- ing differences. Again/ /1 shows the usual behaviour, but their ~ contribution does not disappear for m = 0. It has been suggested that this is a product of the Pauli-Villars regularisation scheme, but it is our be- lief that it arises due to their method of expanding H l and/72 around p2 = 0, and discarding the / /2 (0) term - believed to be zero - which in actuality cancels this persistent term ,4. We fully agree with the absorptive parts o f their integrals, but these discontinuities do not specify the subtraction constants - indeed none is needed, since all integrals are ultraviolet conver- gent. In that connection one can work out other am- plitudes perturbatively (including pure odd-photon- number processes which do not vanish in this model) and easily find that no infinite renormalisation con- stants are needed as 1~ 1.

~3 instead of setting k2_~ 0 in their vacuum polarisation, then let - ting 2~ 0, the correct behaviour is seen by performing the mo- mentum integration, letting 2~0, and then taking asymptotic limits.

~4 This aspect of the analysis in ref. [2], and its problems, have been alluded to in the non-abelian case by Pisarski and Rao [1o].

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The nonper tu rba t ive features o f the C h e r n - S i m o n s

theory are m o r e tanta l i s ing [8] in the l imi t o f small

rn and small k since the v a c u u m pola r i sa t ion wou ld

appea r to be d o m i n a t e d by t h e / / 2 term. Tha t l imi t is

a del ica te one and we plan to s tudy its nonper tu rba -

r ive impac t using the gauge t echn ique wi th a dressed

p h o t o n propagator , as has a l ready been done for the

pa r i ty -doub led theory when the C h e r n - S i m o n s t e rm

is absent.

O n e o f the au thors (A .B .W.) acknowledges the fi-

nanc ia l suppor t o f an A.P.R.A.

References

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[2] S. Deser, R. Jackiw and S. Templeton, Ann. Phys. (NY) 140 (1982) 372.

[3] B.M. Pimentel, A.T. Suzuki and J.L. Tomazelli, Vacuum polarisation tensor in three dimensional quantum electrodynamics, IFT preprint IFT-.028/91.

[4] C.P. Martin, Phys. Len. B 241 (1990) 513. [5] W. Chen, G.W. Semenoff and Y.-S. Wu, Two-loop analysis

of non-abelian Chern-Simons theory, University of Utah preprint UU-HEP/91 / 12.

[ 6 ] D.A. Akyeampong and R. Delbourgo, Nuovo Cimento A 17 (1973) 578;A 18 (1973) 19; P. Breitenlohner and D. Maison, Commun. Math. Phys. 52 (1977) 11; G. Bonneau, Phys. Lett. B 96 (1980) 147; Nucl. Phys. B 177 (1981) 523.

[ 7 ] R. Jackiw and S. Templeton, Phys. Rev. D 23 ( 1981 ) 2291. [8] K. Stam, Phys. Rev. D 34 (1986) 2517. [9]T. Appelquist, M.J. Bowick, D. Karabali and UC.R.

Wijewardhana, Phys. Rev. D 33 (1986) 3774. [ 10] R. Pisarski and S. Rao, Phys. Rev. D 32 (1985) 2081.

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