Quantum equivalence of four-dimensional nonlinear σ-model and antisymmetric tensor model

Nuclear Physics B332 (1990) 391-410 North-Holland

Q U A N T U M EQUIVALENCE OF FOUR-DIMENSIONAL NONLINEAR o-MODEL AND ANTISYMMETRIC T E N S O R M O D E L N.K. NIELSEN

Fysisk Institut, Odense University, DK-5230 Odense M, Denmark Received 20 August 1987 (Revised 17 August 1989)

An equivalence proof of the nonlinear o-model in four dimensions and the antisymmetric tensor model of Freedman and Townsend is given in the context of the loop expansion up to two-loop order. The one-loop effective actions of the two models are regularized by the proper time method and shown to deviate by a finite local counterterm. The two-loop proof is carried out on the formal (nonregularized) level.

1. Introduction

Some time ago, the nonlinear o-model in four dimensions on a group manifold was reformulated as a gauge theory of rank-2 antisymmetric tensor fields by Freedman and Townsend [1]. The model had also been formulated by Thierry-Mieg [2], who, however, did not look into its connection with the o-model. A number of authors have subsequently investigated the model, both from the viewpoint of quantization [3-10] and from the viewpoint of quantum equivalence [9-11]. Nevertheless, some loose ends remain: (i) No comparison of the loop expansions of the two models has been carried out; (ii) No regularized proof of equivalence of the two models has been given; (iii) Conflicting statements on the quantization of the antisymmetric tensor model have appeared in the literature, where refs. [3,10] are at variance with refs. [1,2,4-9]. These problems are treated in the present paper. We find loop expansions of the two models that formally agree up to two-loop order, and we show using proper-time regularization that the one-loop effective actions of the two models only differs by a finite and local counterterm (that, however [12], may contain a W e s s - Z u m i n o Witten-type action [13]). An explicit determination of this counterterm is not carried out. Finally, on the basis of the analysis of the two-loop effective action, the ghost structure of the antisymmetric tensor theory is analyzed and found to be in 0550-3213/90/$03.50 © Elsevier Science Pubfishers B.V. (North-Holland)

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N.K. Nielsen / Nonlinear o model

agreement with refs. [1, 2, 4-9], to the extent that this can be verified from two-loop graphs. Classical equivalence of the two models follows from the fact that in both models a Yang-Mills-type composite field A ti exists that is a pure gauge and has vanishing divergence by its structure and by the field equation [1]:

F;~i = O~A.i - O.Ai~ + gf

ijk

j k_ A.A~ - O,

O.A i" = 0.

(1.1), (1.2 /

Here f ijk are the structure constants of the group in question (only semi-simple groups are considered) and g is a coupling constant. For the o-model the action is

so = f d4x { - 1 A ( ° ) i A(°)i, } , i

A(°)i~, = 2e , ( ~ ) 0 7 " .

(1.3) (1.4)

Here ~r" is the dynamical variable which is an x-dependent coordinate on the group manifold, while e. i(~r) is the vielbein field on the group manifold that fulfills the C a r t a n - M a u r e r equation: O b e ai

- O.e bi + 2gxfijkebJea k = O.

(1.5)

The antisymmetric tensor field, denoted B~, has according to refs. [1, 2] the action (in second order form)

f d4x

-- ~

G r i G ~J( K -

1) ~ i j ,

(1.6)

where ni G t~i = ~ ~,.o~ o,,D~,,j,

Kp,~ij = ~ 8 'j'' - gfijke~uooBkoa

(1.7) (1.8)

SAT is invariant under gauge transformations: i

_

_

(1.9)

where

( D(AT))iJ= 0~,8ij + gfikJA(hT)k.

(1.10)

A(AT)' = ( K-1)~iJG ~j ,

(1.11)

In this case,

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and the two versions of A~ in eqs. (1.4) and (1.11), both fulfill eqs. (1.1) and (1.2) because of their structure and the classical field equations of the two models, and can thus be identified [1]. So the two models are classically equivalent. The layout of the paper is the following: sect. 2 contains preparatory material on propagators and ghosts, in sects. 3 and 4 one- and two-loop equivalence, respectively, are considered, while sect. 5 contains concluding remarks.

2. Propagators and ghosts This section contains preparatory material for sects. 3-5. We determine the propagators of the quantum fields (including ghosts) of the two models in the presence of on-shell background fields, and we recapitulate the ghost and ghost-forghost actions. We first consider the antisymmetric tensor field theory. A background-quantum field splitting is carried out according to i B ~ + B;,,i + ~,~,

(2.1)

where B~, now denotes the background field fulfilling eqs. (1.1) and (1.2). The corresponding splitting for A~ in the version (1.10) is A,( AT)i ~ A.i + ~ ;

i

,

(2.2)

..c]~ = ( K-1)~,/s~"XP°Dx~oo , (Dx) ° = clx8i~ + gfikJA~.

(2.3) (2.4)

Note that the covariant derivative now involves only the background field, in contrast to eq. (1.11). Consequently, by eq. (1.1) we have

=0,

(2.5)

which will be used repeatedly in the rest of the article. The part of SAT bilinear in ~i is by eq. (1.6) __

SAT , b i l -

1 f,cl4v,

8J

. . . .

ije.Jil~jv

j~ /~v

~

~

"

(2.6)

Since SAT is gauge invariant under the transformation (1.9), a gauge breaking action

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term must be added to (2.6). For convenience we use a background gauge:

Sbr= ½f d4x( Ot'~i~) 2.

(2.7)

The propagator determined by eqs. (2.6) and (2.7) fulfills

-(D~Do) ,~o+(D~Do)'%o ij

(2.8)

= 2i8ik8( x - x')( 0~,~.~, - ~l~,n'.,o).

The propagator is split into a transverse and a longitudinal part: ( ~ ° ( x ) ~ , ok , ( x ' ) ) = @ o °J ( ) ~X , o ,k( ) )X t

tr

+

(~Jo(X-

k,-,lon )~5,o,(x)) ,

D°(~/o(x ) ~ , ( x ' ) ) = O,

(2.9) (2.10)

where, from eq. (2.8),

= 2iSikB(x - x')(~,~,V,n - ~ , ~ ) _

ik X ' ) D~'w + 2D~,D2H~n(x,

2D, D : H,~(x,x')D~ ,~ + 2D.V:H;i(x.x.)V; i

~ 2D~D : HArt(x, x')D~, (2.11)

D,H~n(x,x)D,~- D,H;,(x,

~' x ')D,~

ik ik ~', - D,I-IL(x. x')b; + D#L(x. x')D~

(2.12)

where ( D2)2 Hik( x, x') = i S'kS(x - x ' ) ,

(2.13)

H:~(x, x') = n~oWk(x, x').

(2.14)

H ik and H ~ are ill defined because of infrared divergences, but this is not so for their derivatives, which is what we need. ik In the following sections we shall often set D 2Hen(x, x') and D2Hik(x, x'), as well as their covariant derivatives, equal to zero at coinciding points. This can be justified e.g. by the rules of dimensional regularization, if we observe that the

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background field A~ is a pure gauge field by eq. (1.1). Hence it is possible by eqs. (2.13) and (2.15) to write D 2Hun(x, ik x ' ) and D 2 H i k ( x , x ' ) as gauge transformed free massless propagators that vanish at coinciding points in dimensional regularization. In eq. (2.9) the transverse propagator is related to the o-model scalar propagator. We show this by analyzing the two-point function of ~¢~, defined in eq. (2.3). By eq. (2.11) it satisfies e~,,x,,DX(xffi~-(x),xffZO(x'))Kof

k = 4i6 ik e . . x p D X 3 ( x

(2.15)

- x'),

whence, by [ D , D.] = 0, (aff,'(x)~f(x'))=4i(K

z),jkS(x-

x') +4D, G'k(x,x')Do,

(2.16)

where G i k ( x , x ' ) according to eq. (23) is determined by DI'K ,

ij

(

) = o,

(2.17)

i.e. D Iz~' ", . . i j r ~ v G j k ( x , x ' ) = i a i k ( x - - x ' ) .

(2.18)

Eq. (2.19) is in fact the equation determining the o-model propagator in an on-shell background field. To see that, we consider the part of So (eq. (1.3)) bilinear in quantum fluctuations. The most convenient starting point of a loop expansion of the o-model is provided by the following expression for the variation of the action (1.3): 3S o _ -- ~! j f- cl4.rd(o)i,.91,d~i .~-~, v "v , -

-

~i = e ai3 rr a ,

(2.19),(2.20)

where A~ °)i is given by eq. (1.4). It is convenient to use the quantity ~i as a quantum field since it carries a group manifold tangent space index (it is a normal coordinate difference). The variation of ~,~(°)~ /J, , because of a variation of the group manifold coordinate ~r~, is by the Cartan-Maurer equation (1.5) 3A~ ° , ` = 2D~O)q, ',

D~ °) gp'= O~,q~'+ gf'JkA~°)Jq~k.

(2.21), (2.22)

Consequently, the o-model action is, in the bilinear approximation that governs one-loop quantum fluctuations, given by Sa ,bil -

~

f dax 0 ~~D~q~ ' ',

(2.23)

N.K. Nielsen / Nonlinear

396

o

model

with the covariant derivative defined in eq. (2.4) by the on-shell condition and the classical equivalence of the two models. The propagator of the field ~ is determined by

½( a~D~ + D~a~)~dj(x)dpJ(x ') )= iSiJS(x - x').

(2.24)

From the definition (1.8) and [D,,D~] = 0 follows

D~'K~,/JD ~ = ( D 2) ;J + gfOkAl'~'D~,,

(2.25)

that is the same differential operator as in eq. (2.24). Hence, by comparison of eqs. (2.18) and (2.24):

(

x

x') )

=

c'J( x, x') .

(2.26)

Eq. (2.16) (with eq. (2.26) taken into account) is represented graphically in fig. ld. We then turn to the ghost action, as determined in refs. [1,2,4-9], where we record the Feynman rules to the extent necessary for the following sections. In the background gauge, where the gauge fixing action term is (2.7), the vector-ghost action is

Sv_gh=fd4x~;;'D2c;~+f d~(e"D"-e'~D')ji'k~/c~.

(2.27)

The vector ghost is complex and has Fermi statistics. The vector-ghost action needs gauge fixing and requires a complex scalar ghost c; and a real scalar field d;, both with Bose statistics, and with the action

Ss-gh ~

f d4x V2c' + f d4xe i'k V /c k + I f d4xdV2d '-

(2.28)

The ghost action also contains a triple ghost coupling term, dictated by BRS invariance but not contributing to vacuum diagrams. From eqs. (2.27) and (2.28)

"r.i.x t

~

p.k.x'

"=.i.x

i,x

k,x' (c)

, • , p.k.x'

(b)

(a)

,x,,vvvw

= 4i

L • (d)

,+

>

<

Fig. 1. Representation of the two-point functions (a) (sJ~'(x)~ok(x')), (b) (K-1),/ka(x--x') and (c) (q~i(x)~k(x')). (d) A graphical version of eq. (2.16).The arrows denote covariantderivatives.

N.K. Nielsen / Nonlineara model

397

follow by comparison with eqs. (2.13) and (2.14) the ghost propagators

(

} = z) w, (x, x'),

(c'(x)U(x') )= D2HiJ(x,x').

(2.29) (2.30)

3. One-loop equivalence In this section we prove one-loop equivalence. We first give a brief formal proof that serves as a prelude both to the regularized proof in the rest of this section and to the formal proof on the two-loop level in sect. 4. The one-loop induced effective action corresponding to eq. (2.23) is denoted/'otll; its variation due to a variation of the background fields is by the Schwinger action principle according to eqs. (2.18) and (2.26)

(~ff'o[1]= __ 1 f d4x (~Kp.gik)(( Dl~,~i)(x)(DvdDk)(x))

-- f d4xKtLvikgfijl( ~AJ)(dJ(x)(Dvd~k)(x) }.

(3.1)

Similarly, the one-loop effective action /~IA~4of the antisymmetric tensor theory has by eqs. (2.3) and (2.6) and the Schwinger action principle the following variation due to a variation of the background fields:

•~rm=-~fd4x(SK.. ij )(d il~(x),~ jv (x)} v*

AT

j k ~f d4xeIxup~Jgf ijk (SAp)(~,,(x)d:(x)}.

(3.2)

Sbr [eq. (2.7)] does not contribute here because of the background on-shell condition and the remark in the paragraph following eq. (2.14). In the first term of eq. (3.1) we combine eqs. (2.16) and (2.26) to get

f d4x(8KttviJ)(J~fip'(y)J~Cjv(y))

~- -½ f d4y(~Kp.viJ)~( Dttd~i)(x)(DV~J)(y) },

(3.3) where the singular term from the 8 function in eq. (2.16) has been discarded. In the second term of eq. (3.2) we use that also under variation the background fields

N.K. Nielsen / Nonlinear o model

398

should be solutions of the field equation. This means that we can set

6AJo = 2DoSq#

(3.4)

and get by partial integration and use of eqs. (2.3) and (2.16)

: f d'xe~'~°°gf'Jk(SA[)@~,(x)sd,~(x) __ _

f d4xg f i j k ( 8 , j )

)

K.f;(s~C'~(x) ~¢i. ( x ) )

--- - 2 f d 4 x g f i J k ( 8 , J ) K . f t ( ( D~,t)(x)(D"eoi)(x)) -- f d4xgf'Jk(rAS")K.f'((O~q/)(x)q,'(x)),

(3.5)

where again terms proportional to 8(0) were discarded. By comparison of eqs. (3.3) and (3.5) with eq. (3.1) we find

8F[IIAT----8F~ u ,

(3.6)

establishing one-loop equivalence between the two models in a formal sense. We then turn to the regularized version of the one-loop equivalence proof. We use proper time regularization, where the one-loop effective actions have the representations

1. l"°° d'c Tr(^i~i ~ ~ 1 • goo

(3.7)

T

dT

ro['] = - ~,j0 -T-Tr e"i.

(3.8)

is the proper time variable. The traces run over space-time variables as well as internal space indices. The differential o p e r a t o r s / ~ , xo and /~ are, by virtue of eqs. (2.8) and (2.25) 1 I&=.oo=sl-%.x.D

X

(K 1 ) ~'~ e.~poD~+ D, Dp%.

-V.Vfll~.o - D~.Do~I.o + D~Vo~?,p] .

= D K~,~D .

(3.9) (3.10)

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Nonlinear o model

399

We here find it convenient to use a matrix representation where tB ~, ixv l~k= d[ijkBJ Itv, etc. Since the background field A. is a pure gauge, the covariant derivative is V, = e-2g"O, e 2g~ ,

(3.11)

where (~r) ~k =f~/k~r/, and 7r/ is a coordinate in the group manifold. Hence, the cyclicity of the trace allows the covariant derivatives in the operators of eqs. (3.7) and (3.8) to be replaced by ordinary derivatives: F AT .I = -

f -

~0

~dr - - Tr(ei'a~ t ] ~ ,ttv

/~[1] =

__

½ifom -7-Trei'a, dr

(3.12),(3.13)

'r

with /~v

~ ,oo

1[ ~-

--

%~x.OX( lg-~)'~e.roo8r + O, Oon~. - O.Oon,. - O, Oon.o + O~Oon,o]. (3.14)

(3.15)

~ = O"/( O" where I(~,u = e 2 g ~ r g ~ v e - 2g~r •

(3.16)

We shall also need, in our proof, the auxiliary operator /~v ~

( K~)- 1,

X

([3~x.- OxO.) + 8,8XKx.

(3.17)

The structure of these three differential operators imply a number of Ward-like identities that are useful below: ( ~ , a ,/~..x.~:x~p~u. = -%~x.@ X(ei,~) °p .

(3.18)

following from

3~° - O..O° ) .

A.~.xoe

(3.19)

as well as 0"/(,x(ei'a) x~= ei'a0x/( x~, (e ir/~ ) Izv0 , =

0 IZei'r A .

(3.20) (3.21)

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N.K. Nielsen / Nonlinear a model

The equivalence proof amounts to relating the variation of /'04 and of F~11 under changes of / ~ . From eq. (3.12) one gets by the elementary Baker-Hausdorff formula 8 e A = fo1dt e tA 8A e (1

t)A

(3.22)

and cyclicity of the trace: AT = - ~

d r Tr e,,x, 0X(6K -1)

(e'~a) °",~"

= - ¼ ~ d r Tr(6/~)°"(/~-l),,e~°x°O~o(e ''a)

Xo,ojrll~"rc°'Or(I~ - 1 ) vO"

(3.23)

Here we use eqs. (3.17), (3.18) and (3.20) to get ,(/~ 1]] p.we,~0xo9 (ei,a]l vop \

~ ~.

= (K

d,,on0 r (/~-1), °

Xo, co'q

1)/,~r(["]8~rr/--

O~'On)(ei~'a)nv(R-1),,o

O -i

or,ei'a,,',l(-1,,

o-

owe

i,a a- o

,

(3.24)

by means of which eq. (3.23) becomes 8/"[11AT----8A + 8/"[11 ,

(3.25)

i 8A = - -2 r-+01imTr(SK) PU(ei'a),"(/~ t),o

(3.26)

with

which is a finite local expression. From eqs. (3.25) and (3.26) one concludes that counterterm A,

ct14 and

F~11 differ by a finite local

/-[11 AT = A + F.[11

(3.27)

82 (81A) = 81(82A).

(3.28)

provided eq. (3.26) is integrable:

In order to prove eq. (3.28), one applies the Baker-Hausdorff formula (3.22) on

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N.K. Nielsen / Nonlinear o model

eq. (3.26). We obtain i lim Tr(8,/~-1) ~p(Sz/()m'(ei'/~)~,~ 82(81A) = -2~4o ~ ,+o l i m ' f lJod a T r

((81/(-1)/~)~°(ei"~al°x((82/~-l)/()x"(/~ei'(1-~)A)~'~

l lim~.fldaTr((81I~-1~I~] -

"r~O

.t o

I

(ei-~a) °x / v p ~

×[(e 1(82e))~ o,OOeo.+ 0~0 (82~)o.1(e""-°'~) ~ °

~

.

(3.29)

In the second term of eq. (3.29) we use /~e~,,1 ~)5_ 1 ( 0 0 ) - i-~ z ~ - a 0-~ ei'(1-")a'

(3.30)

and integrating by parts in r and a, we get: ~lim~.fldaTr((81/~ r40 ¢0

1)/~)

(e~,,A),x((82/~ 1)/~)x,(/~ei,(~ ~)a) "~ up.

i O 1 ~ 1 =-2 ,4olim~ ' f o daTr((8aK- ) e

/ ~°(e'°~/°~((8~e ~)e) " " e ' " - ° ' ~ ) " ~

i + 2 , ~lim o - Tr( 82I~-1)~ o (81/£ )o, (ei,~) ~"

(3.31)

The third term in eq. (3.29) is reformulated by means of eqs. (3.20) and (3.21). Using also cyclicity of the trace, one finally obtains ~ ( 8 , A) = ~ , 4 o

,(

)~o( ~ )

+

+-/ lira O f ) d a T r ( ( 8 ~ X 2 ,~o O~" o

1)X)~p(e~"5)°x((~2/( a)X)x,(e~"a ~"5) "~

-~ l i m q ld~v~(0~(~lx/

" "°~

r~O 'tO

q._0 X( ~2/~)

up

~-1) I ~ ) . o ) o ° e i " l x,( e,ra/~ ) "~'( ( S 1 K

a)A,

(3.32)

which is manifestly symmetric in 81 and 82. This proves the correctness of eq. (3.28)

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N.K. Nielsen / Nonlinear o model

The explicit value of the finite counterterm A is not determined here, and also complications connected to the choice of the correct integration measure have been neglected. For the simpler case of the corresponding two-dimensional model a treatment of these points can be found in ref. [12].

4. Two-loop equivalence In this section we prove equivalence at the two-loop level; the section is partitioned into four subsections.

4.1. T H R E E - A N D F O U R - P O I N T V E R T I C E S

To construct two-loop graphs one needs three- and four-point vertices. In the o-model they are obtained from eq. (3.5) by a power series expansion. By substituting A~ with 2 Du0~ in eq. (2.23) the triple coupling is

So, tri =

(4.1)

~ [ d 4 x g f ijk[

whence the quartic coupling is found in a similar way: Solquar t --

~fd4xg2f'/kfJ/"(O~#)*k(DJ)#".

(4.2)

For the antisymmetric tensor model one proceeds analogously from eq. (3.11) and gets, using also the definition (2.3)

,tri

= I f A4~ ~lLv~'r~f ijk ~ i e¢'/J~k 8j

~ .~c

~

o~ ~O~,r ~.p,p

(4.3)

which, with the d propagators as indicated in fig. 1 and the double line representing ~, is represented graphically in fig. 2. In eq. (2.6), both the covariant derivatives and K t contribute to SAx, tri. From eq. (4.3) one gets, using eqs. (2.3) and (1.8) implying (4.4)

1

gfijk£p'oAp

~'J

Fig. 2. Vertex c o r r e s p o n d i n g to the triple coupling (4.3) in the a n t i s y m m e t r i c tensor model.

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N.K. Nielsen / Nonlinear o model

gfijk

~- j

Fig. 3. Vertex corresponding to the effective triple coupling (4.6) in the antisymmetric tensor model.

the following quartic coupling:

SAT.quart =

! i ' d 4 X [/¢,--1~ ijol.te?vr,~cikl,~ck ~` opow~p~£jmnr/Irn~n gj \xx }~p c /N/ ~ ' ~ ~ r c ~i/ "~o ~o~oqo"

[A.K] k--r ..,/

The analysis of propagators carried out in sect. 2 and resulting in eq. (2.16) (with (2.26)) suggests that an effective triple coupling in the antisymmetric tensor model can be obtained by the substitution z~¢~---, 2 D,# which, combined with integration by parts, produces the expression

s e,,T,tri

f d4x go(iJ%iD,q:K,Xf t D x d J

'

(4.6)

which is represented graphically in fig. 3, using again the propagators and conventions of fig. 1. This is indeed the case, as demonstrated below by analysis of the two-loop vacuum graphs of the antisymmetric tensor model. (Eqs. (4.6) and (4.1) are shown to be identical in subsect. 4.3.) On the other hand, no such simple relationship exists between eqs. (4.2) and (4.5). To obtain the effective quartic coupling of the antisymmetric tensor theory corresponding to eq. (4.6), one has to carry out a detailed analysis of all two loop graphs in the antisymmetric tensor model. This analysis is carried out in subsect. 4.2, while the emerging effective couplings are shown to be the same as those in eqs. (4.1) and (4.2) in subsect. 4.3.

4.2. T W O - L O O P G R A P H S

The two-loop vacuum graphs of a generic field theory are shown in fig. 4; they arise from the path integral Z (2) = f

o

d ~ eiSbit(iSqu 1

b

2 5Stri),

(4.7)

c

Fig. 4. Two-loop vacuum graphs of a generic field theory.

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N.K. Nielsen / Nonlinear o model

-@ lal

(b) Fig. 5. Two-loop vacuum graphs in the antisymmetric tensor theory corresponding to fig. 4a and b.

where Sbil, Str i and Squ are bilinear, trilinear and quadrilinear, respectively, in the quantum fields, whose measure is denoted d/~. Inserting eq. (4.3) into (4.7) one gets, corresponding to fig. 4a and fig. 4b in the antisymmetric tensor model, the two-loop vacuum graphs of fig. 5. Note that some of the internal lines are "mixed", corresponding to a correlation function {~(x)d~(x')). Finally, from eqs. (4.5) and (4.7) one gets the part of Z (2) in the antisymmetric tensor model corresponding to the graph in fig. 4c: 7(2) = ~AT, c

.

(4.8) In the diagrams of fig. 5 the identity of fig. ld corresponding to eq. (2.17) is used wherever possible, and terms proportional to 8(0) are discarded. The &function term on the right-hand side of eq. (2.17) will then give rise to three terms that cancel ZA~2) T,c, and the remainder is represented graphically in fig. 6, where the two new vertices are taken from that of fig. 2 by substituting d / ~ 2 D,¢ ~ once or twice.

-@ -2(b (a)

(b) Fig. 6. The graphs of fig. 5 after use of fig. l d (with the graphs from the first term on the left-hand side left out).

N.K. Nielsen /

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Nonlinear o model

(a)

Icl

--2

(b)

(d)

- i

--'~ - i (e)

Fig. 7. The graphs of fig. 6 after carrying out a partial integration and using eqs. (2.11) and (2.16).

The next step in the two-loop equivalence proof is to show that the graphs in fig. 6 can be re-expressed in terms of the scalar propagator in fig. lc and the vertex in fig. 3. This takes place by partial integration, by making use of the identity of fig. ld, and of eq. (2.11) combined with the definition (2.3). In this way the graphs of fig. 6 are transformed into those of fig. 7. Besides the terms corresponding to the right-hand sides in fig. 7a-e, there are additional terms arising either from the 3 functions on the right-hand sides of eqs. (2.11) and (2.16), or from the last four terms on the right-hand side of eq. (2.11) containing ghost propagators. The latter possibility is only realized for fig. 7b, which is analyzed in more detail in subsect. 4.4. We discard everywhere terms proportional to 3(0), or ghost propagators and their derivatives at coinciding points. Below we give the algebraic expressions for the terms arising from 3 functions for the graphs depicted in fig. 7a-e respectively.

d4xg2fiikf"~"(eoi(x)eo'(x))K,o

kn

1" j (Deo (x)D°eom(x)),

Ca)

-if

(b)

- 4 , f d'x g 2/'iV'mJ(,'(x),'(x)}K.~k" ( V ~" (x / D'Om(~))

+i f d°~uV'V'mJ(~'(x)D,~'(x))~'~o°(G(x)d:(x) ), (c)

0.

(4.9)

(4.10) (4.11)

(d)

5' f

(e)

i f d4xgZ 2

(4.12) ijk

i f IronI,:,okn @p(x)D'q)J(x))(eO'(x)DOeOm(x)).

(4.13)

N.K. Nielsen / Nonlinearo model

406

Applying the Jacobi identity for the structure coefficients together with eqs. (2.11) and (2.16) to the second term of eq. (4.10), we get

/S d4x' =

i

7 f d4xgVkjmfilj(+i(x)Do+l(x))t~t~vO°(~v(X)j~am(x)) + 2ifd4xg2f'Skf'mj(oi(x)*'(x))KookP(DP*P(x)D°*m(x)),

(4.14)

where the first term obviously cancels eq. (4.12). Thus, adding up the contributions (4.9)-(4.13) we obtain

-i s d4xg'S'SV'm"K.ok"[( ,'( x ) ,'( x ) ) ( D',J( x ) D°,m( x ) ) + l(,i(x)Dr, J(x))(,l(x)Do,

m(x ) )]

-- 2i f d4xg2fi'Vb"J(*i(x)eJ(x))K,xkP(DX*P(x)D'*m(x)). The right-hand sides of fig. 7a-e are essentially the insertion of the effective triple coupling (4.6) into the However, in order to prove equality, one has to carry out use eq. (2.18), producing in this way a term to be added to

(4.151

diagrams obtained by general formula (4.7). partial integrations and (4.15).

½if d4x g2SijVlmnKrok"[ (*i( x ),l( x ) ) ( Dr, J( x )D°Om( x ) ) + (,i(x)D°,'(x))(D'*J(x)*'( -2iSd4xg2f'ikfS'=(D','(x),i(x))K.xkP(DX,

x ) )]

p(x),m(x)).

(4.16)

Adding the last term of (4.15) to the last term of (4.16) one gets by integrating by parts, using eq. (2.19), deleting terms proportional to 8(0), and using the Jacobi identity of structure constants:

--2i f d4xg2fijVl'mK,xkP[(*i(x)*'(x))(DX*p(X)D'*"(x)) +

= }i f d4xg2i'JkixmXK~xkP(,i(x)DV(x)VX, p(x),m(x)).

(4.171

N.K. Nielsen / Nonlinearo model

407

Comparison of the sum of (4.15) and (4.16) (using eq. (4.17)) with the general formula (4.7) makes it now possible to identify the effective quartic coupling of the antisymmetric tensor theory: SAelf

T quay,= f d4x

+ }g2fiJV'maK,xk"*'*m( D'qJ)(Da* ") }. 4.3. R E F O R M U L A T I O N

(4.18)

OF E F F E C T I V E C O U P L I N G S

It remains to be proven that So,in is the same as veer OAT,t~ and that S Oquart is the K,eff same as OAT, quart. First consider ~'AT.tri cefr (eq. (4.6)). By the definition (1.8) it can be written as SAelf T,tri

= - f d4x g2fi-i/'fk/",i( D.fpa)e"'"°B~( Do, t)

(4.19)

This can be reformulated in two ways. By the Jacobi identity for the structure constants: SAatv,tn =

-- ~' f

d4xg2fik

"fkjle,~,°B~*i( D,,')(

D~O')

(4.20)

and by partial integration and use of eq. (1.11):

s~f~,tri = f d4xg2fi'kfk/"g'""B~( DoOi)( D./o')eJ

f

+ d4xg2fiJkf/'b~*iA'~(

D x&i)* ' .

Equating the expressions (4.20) and (4.21) one learns that of eq. (4.1), s/~fTf,tri -~- So,tr i .

S~,tr i is

(4.21) the same as So,tri (4.22)

The reformulation o f S~ff quart follows the same pattern. Inserting K~ak" from the definition (1.8) into eq. (4.18) one gets SAcff T,quart

= I f d4xg2fijkflrnk,i( D)~,j),I( DX,m)

-

f

-} f d4xg3f'k'fk-/"f"~?g'":XB~:e(D,.,J)(Dx,~), (4.23)

408

N.K. Nielsen / Nonlinear a model

where also the Jacobi identity for the structure constants has been used. Using here partial integration, eq. (1.11), and again the Jacobi identity one gets from the last term of eq. (4.23)

- ~f dnxg3i'k'iki"i""ee~<"XBJ'*'(V'q>s)(Oxq>m)+s' =

_

• f dax g3fiklfkJnfnmPAidpiJ(Dxq$m)

+~

f d4xg3(3f'kTkJ"f

"me + f'k"fkJif"'~P)e""X'B~,eoi( D,eoJ)( DxeOm)eo p , (4.24)

by means of which eq. (4.23) becomes SAoff "r' q<'a't =

1 f d4x g 2S,Sk/,,,%, (Dx+J) +,(DX+~)

-~ f d4xg3f'kTks"f""PAi+'+S(DX+")+'.

(4.25)

Comparing with eq. (4.2) we finally get S~lT,quart = So,quart'

(4.26)

With the proof of eqs. (4.22) and (4.26) the proof of two-loop equivalence is complete apart from some complications due to the gauge nature of the antisymmetric tensor model that are dealt with in subsect. 4.4.

4.4. G H O S T T W O - L O O P D I A G R A M S

In this subsection we show how the two-loop diagrams due to the ghost actions of eqs. (2.27) and (2.28) enter the equivalence proof. As mentioned above, gauge complications in the two-loop equivalence proof of the previous section only arise in connection with fig. 7b. This diagram will now be treated in more detail. One integration by parts is visualized in an obvious way in fig. 8. The resulting two terms are reformulated by eqs. (2.11) and (2.16), where the &function terms give rise to the graph representing expression (4.10) (the second

2@ Fig. 8. A partial integration carried out on the left-hand side in fig. 7b.

409

N.K. Nielsen / Nonlinear o model

Fig. 9. A partial integrationand use of eq. (2.16) in the secondterm on the left-hand side of fig. 8.

term and half of the first term). The second term of fig. 8 also contains a part that is reformulated according to fig. 9 and in the process gives rise to the remainder of the right-hand side of (4.10). Thus fig. 7b is accounted for, but in addition there is from the first term on the right-hand side of fig. 8 a contribution from the non-8-function terms in eq. (2.11), the equation defining the antisymmetric tensor field propagator:

f d'.f d 4 x /.2ftlkflmn~pawep( ~l(X)004(xt)}D'D2nJn,l'~( x,x')~)co( ~.(x)~om(x')}

f

= -2f d4. d'.'.':'W"' (D.~'(.)D.~'(.'))D'.,','.(.,x') ~,

1.,

. .. ~ x

,

,-, ,-, -

. . t x

. x ) b.

--D~D''"'~,O(x,x)6.+WD''m~,.(x,x)b.) t

p

p

=_4fd4xfd4x.g.f,.kf,.,,,(DUq,,(x)DOq,t(x,))D.H,.( x , x ) DDo , 2,mk ( X ' , x ) D , (4.27) where in the last step eq. (2.14) was used. The expression (4.27) is supposed to be cancelled by the two-loop graphs arising from the ghost action in eqs. (2.27) and (2.28), and using the general formula (4.7) as well as eqs. (2.17), (2.29) and (2.30) one indeed obtains, when the statistics of the ghosts (Fermi for vectors, Bose for scalars) is taken properly into account

2f d'xf d 4 x 'g2fijAflmn(D~)t(x)Dp~jl(x,))(D.D2HJn,.c/(x'x;)- D~:D2HJn,'~p(X, x')) X(O

p t' ~) -vH ''mk

, ~ , Ai X ; , x )x - DCPD2H~k,P,,(x , x ) )

2f d"x f d'~' gV'"W"'"(D.,'(~) D#(x'))D.D't;""(~, x') D'D't;'~(x', ~), (4.28) which, by partial integration and use of eq. (2.14), is seen to cancel (4.27). Complete

410

N.K. Nielsen / Nonfinear o model

cancellation requires second generation scalar ghosts that give rise to the second term of (4.28).

5. Concluding remarks In this paper it has been indicated how equivalence between models related by a duality transformation is obtained in the loop expansion of the models. The formal equivalence proof has been worked out up to two-loop order, but there seems to be no reason why it should not run the same way in higher-loop orders. The main ingredients are eqs. (2.11) (that is essentially a Ward identity) and (2.16). In the regularized one-loop calculations an explicit evaluation and analysis of the finite local counterterm A, defined in eq. (3.26), remains to be done. It is a pleasure to acknowledge helpful discussions with M.T. Grisaru and A.A. Slavnov and a useful correspondence with P.K. Townsend at an early stage of this work.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

D.Z. Freedman and P.K. Townsend, Nucl. Phys. B177 (1981) 282 J. Thierry-Mieg, Harvard University preprint HUTMP-79/B86 (1980) L. Baulieu and J. Thierry-Mieg, Nucl. Phys. B228 (1983) 259 S.P. DeAlwis, M.T. Grisaru and L. Mezincescu, Phys. Lett. B190 (1987) 122; Nucl. Phys. B303 (1988) 57 A. Diaz, Phys. Lett. B203 (1988) 408 C. Batlle and J. Gomis, Phys. Rev. D38 (1988) 1169 R. Potting, H.-S. Tsao and C. Taylor, Phys. Lett. B215 (1988) 537 S.A. Frolov, Theor. Math. Phys. 76 (1988) 886 A.A. Slavnov and S.A. Frolov, Theor. Math. Phys. 75 (1988) 470 T.E. Clark, C.-H. Lee and S.T. Love, Nucl. Phys. B308 (1988) 379 E.S. Fradkin and A.A. Tseytlin, Ann. Phys. (N.Y.) 162 (1985) 31 N.K. Nielsen, Induced WZW-type term in dual field theory, Odense preprint (1989) J. Wess and B. Zumino, Phys. Lett. B37 (1971) 95; E. Witten, Nucl. Phys. B223 (1983) 422