Beta functions and central charge of supersymmetric sigma models with torsion

Volume 186, number 2

PHYSICS LETTERS B

5 March 1987

BETA F U N C T I O N S A N D C E N T R A L C H A R G E OF SUPERSYMMETRIC SIGMA MODELS WITH TORSION E G U A D A G N I N I and M M I N T C H E V

Dtpartlmento dt Ftswa dell'Umverstta dl Ptsa, 1-56100,Ptsa, Italy and lstttuto Naztonale dt FtswaNucleate, Seztone dt Plsa, 1-56010,Ptsa, Italy Receaved 24 November 1986

We present a method for the computation of the renormahzatmn group fl-functlons and the central charge in two-dlmensmnal supersymmetnc sigma models in a gravitational background The two-loops results are exhibited We use the Pauh-Vdlars regulanzatlon which preserves supersymmetry and permits an unambiguous treatment of the model w~th torsion The central charge we derive for a general manifold Is in agreement w~th the expression found on group manifolds

Generahzed a-models [ 1 ] recexve recently much attention because of their relation with string theories [ 2-5 ] F r o m the point o f view o f the string, one is mainly interested m the conformal properties o f the a-models, more precisely m the renormahzat~on group fl-functlons and the central charge 2 o f the Vlrasoro algebra The purpose o f the present paper is to present a systematic framework for the derivation o f l / a n d 2, which is the supersymmetrlc generahzatlon o f a recently proposed method [ 6 ] for bosomc a-models We perform the computatxon by supergraphs, constructed in terms of superpropagators m a gravatat~onal background The theory is regularized m superspace a la P a u h - V d l a r s This regulanzat~on xs particularly convenient for several reasons Firstly, it provides a clear distraction between ultraviolet and infrared effects - a subtle point m two dlmensxons (see e g ref [ 3 ]) Secondly, it avoids the techmcal difficulty related to the use of the superconformal gauge In this gauge the classical action decouples from the gravitational background and the problem is how to determine the dependence o f the effective action F on ~t The Pauh-Vdlars regularlzat~on preserves the supersymmetry and although the theory is defined m two dimensions, the couphng with the superconforrnal background does not vanish Moreover one is able to determine the structure o f the grawtatlonal background dependence o f f even for strong background fields [ 7 ] Another motlvatxon o f our work ~s the following There exists in the hterature only one computation o f the two-loop fl-funct/ons o f the N = 1 SUSY a-model with torsion [ 8 ] It is based on dimensional reduction In ref [ 6 ] we have found however, that the prescnptlon for deahng with e~ m thxs scheme leads to an incomplete result forfl g m the bosomc model Th~s fact suggests an examxnatxon of the supersymmetnc case within a scheme which is free from the above ambiguity Clearly, the Pauh-Vfllars regularlzatlon has the last property We exhlbat below the exphclt two-loop computation o f the central charge and the fl-functlons for the N = 1 supersymmetnc a-model with torsion Concerning 2 we find agreement with the expression derived for the particular case o f group manifolds [ 9 ] As for the fl-functlons, we confirm the result o f ref [ 8 ] We start with the action m fiat space (dlag q~,= ( + , - )) 1

S( ~ ) = ~ ; d4z[g,s( ~ ) f ) ~ ' D ¢ ~ + h u ( ~ ) I ) ~ ' 7 3 D ~ ] , z=(xu, Oa), d4z=d2xdOdO,

y3_=),Oy~,

D,~=O/OO,~_l(yuO)aOu,

0370-2693/87/$ 03 50 © Elsevier Scxence Pubhshers B V (North-Holland Physics P u b h s h m g Division)

(1)

173

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where g,j and h,j are the metric and an antlsymmetrlc tensor field on the target space ~/d The coordinates q~'(z) on ~//a are N = 1 real scalar superfields q~' (z) = ~0'(x) + O~'(x) + ½00F'(x), where ~0' and F ' are real scalar fields and ~,' and 0 are Majorana spmors We use the background field method, combined wxth the normal coordinate expansion [ 10] o f S(q~)

S( qg, q) = 1 f d4z{g,j~)qb,Dqb ~+ h,j~)~,73Dq~j + 2g,j[)~' ~qJ + g , ~ q ' ~q~ + ~,k0I)t~'(1 + 73)Dt/~qkq ' + ] Tqkq' ~qJ73 ~qk + ~ ( ~k(o)t[)qg' ~q~qkqt

+

~ktqlli~t~3 ~q~qkqt)

+ ~ (V m~,k0 -- 2~,k/nTjmn)[)~'(l +73)D~I~jqkqtq m "~-~ ( ~,kIj "JrT m tk TOm) ~q'

~qj qk ql + ½~,kO ~q' 73~qj qk ql

+ ½(~ r. ~kt,j~ + ~pk~, T~.. o + ~jktp 7",..")D~' ~q~qkqtq.. dr ½( ~ m ~k[ql/-- ~pkh Tjm p -- ~)k/p Tim P)[)~)'73 ~qj qk ql qm + -~ (~ m~ n ~tkI) + ~tkl 0 ~jmnp "~ 3~,kl ° ~p,nnj q- 4~ n ~,ktp Tjm p + 4~,klp TOms TSjn) X I ) ~ ' ( 1 + T3)Dqgjqkqlqmq n +

}

(2)

Here

~q' = D q ' + I " j k D ~ k q I q- T ' j k 7 3 D ~ q k, Z,jk = ½(V ,hjk q-V :hk, d-V kh,:), ~,klj =Rtko + V j T~kt--V lZ~kJ + T m k t T , mj - Tmkjzm,l,

(3,4) (5)

and V (~') Is the c o v a n a n t denvatlve without (with) torsion T,jk on ~//d In order to apply the standard perturb a u o n theory, one has to rewrite (2) m terms of qa= e a,(~) q, where the e a, are the vaelbems assocmted to the metric g,j The massless superpropagator reads

Go(z', z) = 1 J d2k exp[ -xk(x' - x ) + O'l~O]Ao(k),

(6)

where dzk==-dZM(2rc) z,/~= k~'Tu and/Io(k) is the Fourier transform o f /Io(X ) = (a/4n)ln [( - x z + a~)(U2/4)]

(7)

The factor 4 m the logarithm m taken for convenience, whale/t ~s the scale parameter needed for the definltmn of any dlmensmnless q u a n t u m field As explained m details m ref [ 6 ], one has to be careful m performang the Fourier transform o f (7) In what follows we adopt the representatmn (see appendax A of ref [ 6 ] )

/to(k) =hm d {( _/z2 ) - " [ F ( I - q ) / F ( r / ) l ( k .40 dr/

z+

x~)"-' },

(8)

where r/--.0 as a weak hmat, a e at has to be taken only after all antegrations m the Feynman daagrams have been performed We shall see below that for the computation of the/~-functmns and the central charge ;t one needs an the Pauh-Vdlars regulanzatmn only one auxdmry mass, whmh we denote by m Th~s means that an the relevant supergraphs one has to perform the replacement 174

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O a

b

c

d

G

e

f

Fig 1 Graphs relevantfor the fl-functmnsat one and two loops

(9)

Go(z', z) ~ G(z', z ) - G o ( z ' , z ) - G ( z ' , z, m), where G(z', z, m) is the free massive scalar superpropagator Exphcltly,

G(z', z) =1 ~ dEk exp[ -lk(x'-x) + O'l~O][Ao(k)-Am(k) +lmAm¢~(O'--0)],

(10)

where Am(k) - (k 2 - m 2 + 1~) - 1 and ~(0) - (1/21) 0O Now we turn to the computation of the one- and two-loops supergraphs determlnmgfl 8 andfl h After performing the 0-algebra one is left with bosonlc integrals which have been computed already m appendix B of ref [ 6 ] for the bosonlc ~-model In particular, the divergent logarithms appear always m the combination

I-ln(m2/p 2) + 2~,

(11)

where y is Euler's constant The relevant supergraphs at one and two loops are given m fig 1 One has

i

graph 1a = - 4-~ 4 f d4z ~ v I ) ~ ' ( l + 7 3 ) D ~

(12)

The graph lb is manifestly proportional to 12 and therefore gives no contribution to fls and flh [ 1 ] Graphs I c-1 f cannot be ruled out a p n o n since they may have/-parts (simple logarithms) The explicit computation gives

1

graph l c = (4~t)2

I(I--2) 1-'---~fd4zRk/.~l,ktf)(lb'(1 "F~'3)Dt2~,

1~ 128 graph l d = (4~)-------1

graph l e = (4zt)2

12

f

d4z

~,,,kl ~(kt),,,I)~'(1

+y3)D~,

f d4z(cg,Tkt,~)(OjTk°")f)~'Dtl~,

1 5 1216f graph 1f = (4rt)-------

d4z~t,koTkmnTlm.[)~(l+y3)Dt~

(13)

(14)

(15)

(16)

Expression (15) is not Jt'd-covarmnt We have verified that, by adding the divergent parts of the corresponding graphs w~th one and two connectmns, one gets the covanant expressmn

1 12 f d4z(~'Tkt")(~jTktm)f)~'D~l~

(4zt) 272

(17)

AS explained m ref [ 6 ], m order to recognize the two-loop fl-functlons from eqs (13), (14), (16) and (17), one has to take into account a possible wave function renormahzatmn 175

Volume 186, number 2

PHYSICS LETTERS B

Q

(3

a

b

5 March 1987

Fig 2 Contribution of the wave function renormahzatlon to the fl-functlons

q' ~ q' + Z'j( ~)q J

(18)

Z' s can

be determined [ 11 ] by putting the lagranglan (2) m normal form The relevant term to

at one loop one loop is 1

12i

f d4z(~l,ko+T,nkT,,,o) ~q,~qj (qkqt),

which leads to

Z'j = (1/4n)( I/6 )R'j

(19)

The wave function renormalizatlon (18), (19) contributes at two loops through the graphs in fig 2 graph 2a+graph 2 b =

1_

(4/~)2

I 6

f d4zRgt~,kO[)CI)'(1+ ~ 3 ) D ~ b J

(20)

The term (20) cancels exactly the/-part of expression (13) and at two loops one is left only with I2-corrections to the metric Therefore there is no two-loop contribution to fl~ and l/h, which according to (12) are given by

fl,j-

(1/47t)~tv),

fl~-

(l/41t)~tv I

(21,22)

This is in agreement with the result of ref [ 8 ] Let us consider in what follows the model (1) in external gravitational background In the superconformal gauge the classtcal action does not depend on the background and consequently the normal coordinate expansion is sltll given by eq (2) This does not mean of course that the gravitational background can be ignored on the quantum level In fact, in the regulartzed theory the coupling with the superconformal background does not vanish because of the presence of regularizing massive fields (of the Pauh-Villars regulanzatlon) To see how it works, let us consider first the purely bosonic case The action of a free massive scalar field ~ in a gravitational background 7~ (x) is

Sm(~o)=l f d2xx/~(,u,,Ou~Ov~o-m2tp 2)

(23)

In the conformal gauge

yu~(X)= exp [ 2tr(x) ] qua, Sm takes the form 1 Sm(~) =2

d2x{Ou~(x)o'u~O(X)-m2exp[2a(x)]~o2(x)}

(24)

This expression can be easily generalized to the ease of an N = 1 real superfield ~ ( z ) in a superconformal background as

Sm(~) =~1 I d4z{f)q~(z)D~(z)+ 2m exp[tr(z)] q~2(z) }, 176

(25)

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where a(z) is now a real superfield, (26)

a(z) = a ( x ) + OX(x) + ½0Of(x) From eq (25) one gets for the massive superpropagator

G(z', z, rn exp(a)) = ( z ' [ {½I)D- m e x p [ a ( Z ) ] } - ' [z),

(27)

where

a(Z) Iz) =a(z)Iz),

(28)

(z' Iz) =~(z'-z)-~(x'-x)~(O'-0)

Consequently, the Pauli-Vlllars regularized propagator is

G( z', z, a) = Go(z', z) - G ( z', z, m exp(a))

(29)

In addition to the substitution (29) in all supergraphs, one has to regularize also the determinant Det ~--d/E)(I)D/2) In conformity with (29), we define the regularized determinant by Det ~- d/E) {(I)D/2) ((I)D/2) - m exp [ a(z)] } - t }

(30)

From eqs (29), (30) it follows that the regularized effective action F depends on a(z) Exactly as in the bosomc case, the conformal properties of the model are characterized by the a-dependence of F More precisely, the superconformal mvarlance IS controlled by the central charge 2 [ 12 ], which is given in our approach (see e g ref [ 13 ]) by 8 2 F ( ~ , a)/Sa(z')Sa(z) Ipg=ph=o= (1/21)2I)D~(z' - z ) +

,

(31)

where the dots stand for less singular terms In the [z' - z I - ' 0 hmlt When ;t = 0, a(z) does not acquire a kinetic term in F ( ~ , a) Consequently in this case a(z) is not a true dynamical degree of freedom and the superconformal mvanance is not anomalous [ 14] In the rest of the paper we will be concerned with the computation of 2 For this purpose we consider the expansion

a(Z)=a(z)+(Z-z)SOsa(Z)+½(Z-z)s(Z-z)tOsOta(z)+ s - ( l t , a),

t - ( v , fl),

,

Os=-OlOz~,

(32)

which inserted in eq (27) gives

G(z', z, m exp(a)) = ( z' [M[z) + m(z)O~o(z) ( z' IM( Z - z ) ~ M I z ) + ½m(z)[O.Ota(z) +Osa(z)Ota(z)] ( z ' I M ( Z - z ) ~ ( Z - z ) t M [ z ) +m2(z)O~a(z)Ota(z)(z'lM(Z-z)~M(Z-z)tMIz) +

,

(33)

with M-

[½I)D-m(z)]-~, re(z) = m

exp[a(z)]

(34)

The dots in eq (33) stand for terms which are irrelevant in the m ~ o o limit, because the model has at most linear divergences in superspaee The a-dependence of F is governed by the degree of divergence of the underlying supergraphs Being mindependent, finite supergraphs are also a-independent For logarithmically divergent supergraphs, in the m ~ o o limit the a-dependence survives only through the first term in the expansion (33) Consequently, in the corresponding expressions for flat space one has to make the replacement 177

Volume 186, number 2

a

PHYSICS LETTERS B

b

5 March 1987

c

Fig 3 Graphs relevant for 2

Fig 4 Contribution of the wave function renormahzatlon to 2

I ~ I + 2tr(z)

(35)

The hnearly divergent supergraphs get contributions from all the terms explicitly written m (33) From this consideration it follows that the new (with respect to the fiat case) supergraphs to be analysed are those which are hnearly divergent At two loops these are the graphs presented m fig 3 Apart from a mass renormahzatlon which is unessentml for the central charge 2 at th~s order, one obtains graph 3a=0,

(36)

1 graph 3 b : ~(4n) graph 3c=

1 f d4zRDaDtr '

12

1 2 12 If (41t)

d4z T,~kT'sk~)aD~+

(37) (38)

,

where the dots m eq (38) stand for terms resulting from eq (15) after the replacement (35) (In order to avoid the tedious 0-algebra in computing (38), one may consider the particular background tr(z) = ½00fwlth f = const and afterwards supersymmetrlze the result ) The wave function renormahzatlon (19) contributes through the supergraph in fig 4, which gives (4n) 2 12

(39)

d4zR(I+2~)E)De

As for the determinant (3) one finds ( ~ / ~ ) D e t (-a/2~ { (I5D/2) [(I)D/2) - m ( z ) l - ' } = ( 1 / 4 n ) ( d / 4 ) f ) D a ( z )

(40)

Summing the contributions from eq (12) with the replacement ( 35 ) and from eqs (36) - (40) one obtains 8 F / ~ a ( z ) = (l/21)[fl~I)~'Dqt~ +fl~D~'73Dt~ j + c ( ~ ) I ) D a +

],

(41)

wherefl g andfl h are given by eqs (21) and (22) respectively, c ( ~ ) = (1/4zt)(d/2) - (1/4n)2 ( R - 13T2),

(42)

and T2-- T,jkT 'sk The dots m (41) represent terms which do not contribute to the leading singularity m (31) For the central charge 2, from eqs ( 31 ) and (41 ) one gets 2 = c ( ~ ) [p,=ph=o

(43)

By means of the Bronchi identity one easdy verifies that fig =fib = 0 imply that 2 is a constant on JCd [ 12 ] From eqs (21 ), (22), (42) and (43) it follows that in general 2 ~ 0 even if the theory is finite (#g =fib = 0) [ 15 ] This is for example the case of parallehzable manifolds ( ~,jkt = O) A comparison of the central charges of the bosomc and N = 1 supersymmetrlc models shows that the fermlons contribute at one loop with one half of the bosomc contribution On the contrary, fermlons do not contribute at two loops 178

Volume 186, number 2

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A n o n t n v m l check of our c o m p u t a t i o n is provided by the expression of 2 o b t a m e d by m e a n s of current algebra methods for the particular case when ~¢/d lS a group manifold [ 9 ] There is a complete agreement with eqs (42), (43) Finally we c o m m e n t on the effect of the dflaton term [4,5 ] SF(~) =~

~ d4z D D t r F ( ~ ) ,

where the dllaton F is a scalar field on ./¢a Following exactly the procedure adopted in ref [ 6 ] for the bosonlc case, one finds

6F/Str(z) =

(1/2)(1/4n) { I ) ~ ' D t / ~ ( ~ , j + 2V , V j F ) + I ) ~ ' 7 3 D t / ~ [ ~,j + 2 ( V

- f ) D a [ ( d / 2 ) - (1/4n) ] ( 3 R - T 2 - 1 2 V , F V

'F+ 12V ,V ' F ) ]

+

kF) Tk,j]

}

S u m m a n s m g , we have presented a m e t h o d for the c o m p u t a t i o n of the renormaltzatlon group fl-functlons a n d the central charge 2 of N = 1 supersymmetrlc a-models in a gravitational background The exphclt expresston of the central charge confirms the c o m m o n b e h e f t h a t for N = 1 supersymmetry, the fermlons do not contmbute to 2 at the two-loop level The above m e t h o d can also be apphed to more general cases which are currently u n d e r consideration

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