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Off-shell effective actions in string theory
Volume 207, number
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LETTERS
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30 June 1988
OFF-SHELL EFFECTIVE ACTIONS IN STRING THEORY T KUBOTA ’ and G VENEZIANO CERN, CH-I21 I Geneva 23, Switzerland Recewed
5 Aprd 1988
A new, “off-shell”, approach to conformal InvarIance, pfunctlons and field equations m first quantized string theory IS presented The method, vahd to all orders m the weak-field expansion and non-perturbatwely m a’, IS Illustrated for a tachyon background at genus zero and one level For a particular choice, we derive a Zamolodchlkov-type equation for the effectwe action
1 Introduction
extension of the theory m a way which, while spolhng conformal invariance, still preserves certain crucial properties It 1s not clear, at the moment, whether such an extension can be derived by giving a meaning to Polyakov’s path integral away from its conformal mvariance points (hence off-shell) In any event, this assumption allows us to define two types of generating functlonals, related to one another by a non-local field redefinition, a modified Legendre transformation In general, both kinds of functlonals depend on rs The condltlons of g-mdependence can be almost trivially studied, however, for one functional and then translated m terms of the other by standard functional derivative techniques In this way we are able to identify what we can define as /?-functions and to relate these to appropriate derivatives of the effective action functional Our results fit nicely with (and extend) those discussed already or partly conlectured in the literature In particular, for one particularly appealing choice of the Legendre transformation, we derive a Zamolodchlkov-type equation [ 11,12 ] for the effective action We shall treat, for slmphclty, the case of a closed bosomc strmg moving m a tachyon background at the sphere and torus level, but to all orders m (Y’ We are thus able to discuss the (finite) mass, couplmg and S-matrix renormahzatlons induced by string loops, confirming and extending previous results [ 13 ] We see no reason why our procedure should not be generahzable to any background field m any string theory and at any (finite) genus This note 1s supposed to give the essential ideas and
A crucial element m the first quantized approach to string theory consists of exploiting the condltlons of conformal mvanance For Polyakov’s path mtegral [ I] describing a string moving m a background [ 21, these condltlons guarantee the decouplmg of the conformal factor (Tof the two-metric gap and are crucial for the eventual consistency (causality and umtanty) of the theory What 1s remarkable 1s that the above equations, which can be expressed as the condltlons for vamshmg &functlons, not only determine, for a string m flat space-time, the dlmenslonahty of the latter but even enforce, m the general case, the equations of motion (and/or mass-shell condltlons) for the background fields m which the string propagates Furthermore, order by order m (Y’ , these equations happen to comclde with the Euler-Lagrange equations derived from an effective action, rconf Fmally, this action also generates the string S-matnx, precisely like an off-shell effective action does m usual field theory Much work has been devoted to understand these welcome, yet somewhat mysterious, comcldences (For a partial list of relevant papers, see refs [ 3-8 ] ) In this note we shall present a new vlewpomt on this relationship It 1s based on the assumption, which 1s largely supported by some old [ 91 as well as by recent results [ lo], that one can construct an off-shell ’ On leave of absence from Department verslty, Toyonaka,
of Physics, Osaka Urn-
Osaka 560, Japan
0370-2693/88/$03 50 0 Elsevler Science Pubhshers ( North-Holland Physics Pubhshmg Dlvlslon )
BV
419
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main results of our approach More details and explicit proofs will be given elsewhere [ 14 ]
2. The basic assumptions Our starting point ~111be the formal expression of Polyakov’s path integral for the (closed, bosomc) string partition function
Z[@LO)l= j DJWz) J&&z) ew
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the condltlons of Weyl invariance by studying the dependence of Z from the choice of Q When the external states are on shell, even after fixing the conformal factor, Mobius (projective) transformations remam as residual symmetry and mduce, at the sphere level, the well-known mfimte SL( 2, C) volume factor .C&,,m T, for n> 3 It appears to us that Mobms mvarlance is a crucial property if we want off-shell finiteness and factonzatlon Indeed, if and only if there 1s Mobius mvanante, are we allowed to factor out Q,., and to fix three mtegratlon variables The number of integration variables is then the correct one for factorlzmg T,, as T,-rT,+l(s-m2)-‘T,-,+,
where So 1s the free euclldean actlon and $i”’ ( Vc,‘)) are the unrenormahzed background fields (vertices), with a labellmg the various backgrounds (tachyon, gravlton, dllaton, ) We would like to give a precise meaning to Z for all fla (fields to be related to @io’ by some appropnate renormahzatlon) at least as a Taylor series Z=
u&z1
(2) where c stands for connected Since the one- and twopoint functions require some special attention, let us consider first T’ correspondmg to eq (2) with the sum starting from IZ= 3 T’ should be, first of all, a finzte functional of its arguments As 1s well known m ordinary field theory, even forgetting ultra-violet divergences, gauge fixmg is needed to obtain a finite result from a gauge theory path integral Reparametrlzatlon invariance can be used to fix the gauge to be of the orthonormal type gnp=&exp[g(z)
1,
g,pdYdXP=exp[o(z)]dzdZ,
(3)
where, for the time being, we limited ourselves to the sphere level Weyl invariance would allow us to fix G as well, and would guarantee that the result 1s independent of the particular choice adopted Vice versa, we can study 420
30 June 1988
,
(4)
since (n-3)=(m+l-3)+(n-m+l-3)+1, where the last + 1 stands for the integration yielding the pole As already mentioned, old [ 9 ] as well as recent [ lo] results show that, indeed, projective invariance can be maintained off shell This allows us to make the followmg educated basic assumptions
Assumptzons There exists a gauge fixed definition of T’ [ @,,a] such that (I ) SL ( 2, C ) invariance holds for all @,, (11) the amplitudes T, factorize properly even off shell, possibly within an enlarged Hllbert space As we have already pointed out, the n-reggeon amplitudes constructed m refs [ 9,101 satisfy expllcltly property (1) As to property (ii), we have verified its validity at the tachyon pole for the tachyomc amphtudes given m these papers at the sphere level Actually, the pole posltlons are unchanged for all levels Both properties use the fact that the dependence upon the conformal factor (T1s only through its value at a fixed, but otherwise arbitrary, point z. The best way to Justify this from the point of view of Polyakov’s path integral 1s to take c to be constant (almost everywhere) At least for the case of tachyons considered here, this restriction 1s not important as will become obvious later, invariance under a global rescalmg of 0 will Imply invariance under any local change The amplitudes T, (n > 3 ) thus obtained are made finite by (a) removal (cancellation) of the infinite SL( 2, @) volume factor &,
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(b) regularlzatlon of products of operators same point through an invariant cut-off AzAz>exp(
-a)
at the
a2
/4)/G@(O)
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30 June 1988
ambiguity m the definition of the quadratic terms m Tand r Indeed, let us add to T’ a quadratlc term and define
(5)
This second step allows us to regularize the functional T’ via a simple renormahzatlon From now on, we shall restrict ourselves to a tachyon background and omit the label a Defining @= (a2)(a
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)
ff=
(a2)-‘”
/4)P*v(o)
one finds /4)Pz@v
(7)
As indicated above, m the process of renormahzatlon, T’ inherits an extra g dependence (besides the one already present m the classical vertex) Such a dependence, which 1s of crucial importance for us, appears already m ref [ 10 1, but was neglected there since it disappears when one goes, eventually, on shell It is now easy to verify that the followmg equation holds aT /aa= Kqi 6T’/6@,
Provided modified
(10)
that I7K-’ + 1 as K+O, one finds that the Legendre transformation defined by
$!kIi-‘6T/&!l,
(lla)
r[$l= T[@l -@‘@
(lib)
)
(6)
@~“)fl(o),[exp(_~)](a
T=T’+~@&!I
(8)
where we have defined
defines a 1TI functional that can be identified with the off-shell effective action m the sense that It generates back T from tree diagrams with Ii’ - ’ propagators Our modlficatlon of the usual Legendre conslstmg of the extra factors transformation, Zi-’ (n) m the defimtlons of q(r), is required by the fact that T generates truncated Green functions, 1 e , amplitudes without external propagators A slmliar observation has been also made m ref [ 7 ] From these defimtlons, it 1s easy to check that i5r/&5= -I@ $ and 6 differ only by interaction finds $=@+n-’
and K(p2) = (1 -a’p2/4) 1s the kinetic operator for the tachyon This equation 1s basically the same as Weinberg’s [ 15 1, who derived the Vlrasoro condltlons on physical vertices by requiring @-mdependence Note at this point that our factorlzatlon assumption (11) contams the non-trivial (bootstrap) condmon that the (tachyon) poles are precisely at the point K= 0 where conformal invariance holds
3. Legendre transformation at the sphere level We would now like to go from the functional T’, which generates connected, truncated amphtudes, to another functional r, which 1s one-tachyon-lrreduclble ( lT1) This procedure has an mtrmslc amblgulty related to the posslblhty of subtractmg some finite part together with the pole terms This amblgulty, which is the analogue of a renormahzatlon ambiguity m the non-linear a-model approach, induces also an
(12) terms
6T’/6+@+0(@2)
Indeed one
(13)
Using eqs ( 10) and ( 11 ), we can also extract the quadratic terms m the T’s r=
-;@zJ+~
) r =0(p)
)
(14)
and verify that 6r’/Z$=l7($5-@)=6T’/?i#
(15)
We finally come to the main purpose of all this exercise g-dependence This does actually depend on the choice of I’I In general, one finds X/so= +
K@I7( q?- $I)
jg(an/aa)
+J(an/a0)
g
(16)
Let us discuss at this pomt three possible choices for 17 They should ultimately be equivalent m the sense of giving the same on-shell theory, but one of them (or maybe a fourth one?) could be most convenient for an off-shell contmuatlon (1) “Mmlmal”subtractlon
We define this by slm421
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ply taking I7= K, I e , a a-independent (14)-( 16), we obtain ar/&s=
ar /aa= - (i?ir/&J) 6T’ /6@
= - (6r/@)
6r’ /@
(17)
(11) “Covanant” subtraction satisfy iXllaa=
17 Using eqs
If we define
m a similar way under the change
(19)
- KqX@= K@ir/Ziq?= -Kg 6T /6@
(111) “‘Intermediate” amusmg happens
subtraction
(20)
Something
if we define I7 to satisfy
a17/aa=m,
(21)
1 e , a-dependence m between choices (1) and (u) this case r satisfies
-~(sr~s~~ m-16rp3$,
In
(22)
which has the same structure as the equation proposed by Zamolodchlkov [ 11,121 for the central charge Since the latter 1s expected to be closely related to the effective action, our result should not be surprising One could ask why It should work only m the intermediate subtraction The answer could be that only in such a case the following simple renormallzatlon group-like equation can be shown to hold
= -P(G)
=K@(6/%0
(WV) ($-$I
(i-6) (23)
Eqs ( 17), (20) and (22) are our main results at the sphere level They show clearly that the free equations of motion for @become, through the modified Legendre transformation, the fully mteractmg equations for $ u-respective of the choice of subtraction This observation makes precise earlier claims by Tseythn [ 16 ] that one set of equations goes mto the other through the renormahzatlon process In our approach, however, the renormalization of UV mfimties was done once and for all at the level of the 422
functional T and, from thereon, we have only conslderedfinzte field redefimtlons Finally, we can re-express our result as follows since stability of the ground state requires 6r/6$= 0, we see that vacuum stability guarantees, through eqs ( 17)) (20) and (22) conformal invariance
17 to
[compare eq (8 ) ] In this case, eq ( 16 ) gives
(aiaa)($-ti)
30 June 1988
(18)
aTjaa=K@ liTIS@
arIaa=
B
4. Extension to the torus
2KLl,
T and T’ transform ofa
arIaa=
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It is relatively simple to extend our method to the genus 1 (torus) level and m principle to higher genera as well We shall have to continue to assume that off-shell amplitudes satisfying factorlzatlon can still be defined Also, by consistency, on- and off-shell amplitudes will be given by integrals over the same region of moduh space (this requirement replaces that of Mobius invariance discussed at the sphere level) Apart from these assumptions, which are presumably realized if string loops are constructed by the procedure of refs [ 9, lo], we have to be careful about the fact that, unlike m field theory, m string theory we cannot resum self-energy diagrams to obtain dressed propagators To each finite order g m the string loop, expansion poles are always at the treelevel position but their order can go up tog (g- 1 for external masses) A related point 1s that we expect the condltlons for conformal invariance to be unchanged by loops as far as the functional T 1s concerned Yet we would like to recover the expected loop effects on couplmgs and scattering masses, amplitudes Let us denote now by the subscripts 0 and 1 ObJects that belong to the sphere and torus level respectively T, contains double- and external-line poles which can be both taken mto account by a term
T,,,~t(6T,/6~)n,‘n,n,’
6T,/6@,
(24)
where 17, 1s the torus two-point function It IS possible to take the a-dependence to be the same for 17, as for I& n;l
aIz,/aa=n,I
an,la0
The double-pole-subtracted
(25) T’,
T; = T, - TDEp can be used to construct
(26) the quantity
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T=T,+T;
(27)
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values for masses, couplmgs and scattermg tudes Expanding m a Taylor series,
Since F factorizes with Just simple poles in the internal lmes, we can use for tt the modified Legendre transform used at the sphere level definmg
amph-
(33) one finds from (30) a mass shift gtven by
l+= ~‘-@I~~ qGI7,’
)
m:=m;+(4/cr’)n,(-m;)
sT/sqkfj+o(qP)
(28)
and obtain, after a straightforward culatlon, the a-dependence
but lengthy
cal-
+~f$(m~/ao-2Kn,)@
+@(2~n,
- ahyatT)
+
(29)
However, the 1TI functtonal which generates full To+,amplitudes is not r but r=p+
the
TD,P I o=r++ril
= f+
f @IT,q+ (higher genus )
- an,/aa)
#+ ~(an,/acb4u7,) 6
$ (31)
defined
prevt-
)
(32a)
minimal ar/aa=-(sr/s~)[K(~-~)+17,~] arla0=K&vp+,
(32b)
intermediate arIaa=-4
1s generated
by r
(plus pole
(36)
The result (34) appears to be m agreement with the prescription obtained by several authors [ 13 ] m rather mdtrect ways As for ( 35 ), this 1s less obvtous since our off-shell amplitudes differ from those of ref [ 13 ] Indeed, we are not sure that the latter can be denved by a Mobius mvanant ObJect and that it obeys factorization
9
In the three subtraction schemes ously, this equation becomes
covariant
(35)
5. Conclusions - anoh)
+ f~(ano/a~-2Kz70) +$(2~n,
~=[1+(4/cu’)17;(-m:,)]“2~~2-“‘~,
(30)
and satisfies ar/afl=$(Kn,
tachyon field becomes
s,, - z~~wrjtip I~~=-_‘,,~
$
+ gi(an,/a0-2Kn,)
Also, the correctly normalized
so that the S-matrix terms) as
KI&-aI&/aa) 4
@/&P$(
(34)
(tirp&)~n,L w/if+
(32~)
This shows that, m all cases, conformal mvartance holds on the strmg-loop corrected equations of motton 6r/6$=0 Furthermore, for the covarlant and intermediate subtractrons, the equatron 1s not changed from the sphere case In particular, Zamolodchtkov’s form keeps its validity up to the g= 1 level Let us discuss how the condttrons of conformal mvariance at the g= 1 level imply the renormahzed
String theory was born as an S-matrix theory Even after twenty years tt seems to remember (suffer from?) this inheritance reflected m its resistance to being extended off shell Yet, we believe that an offshell extension of string theory not only 1s desirable, but could even turn out to be essential to its eventual success Apart from the (purely technical?) difficulty m definmg finite-loop amplitudes while slttmg on-shell all the time, the on-shell theory ts, by definmon, a theory m one vacuum When such a theory shows no dtsease order by order (as seems to be the case for the superstrmg), one may be happy with such a hmttanon, but this means to give up any possrbrllty of (1) dealing with theories with either perturbattve (bosonic string) or non-perturbatrve (superstrmg?) vacuum mstabthttes, and (11) relating different conststent theories (and at present we have too many of them!) as reahzattons of one and the same theory around different vacua and thus being able to select one of them dynamically In this paper we have made an effort to go off-shell 423
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which, so far, seems to work It rests on the basic assumption, which is, however, supported by some explicit results, that an off-shell continuation preserving factorlzatlon and Mobius (respectively modular) mvariance at the sphere (respectively higher genus) level exists Of course, conformal (Weyl) invariance IS lost off-shell and thus the off-shell extension depends on a conformal factor d (on which we do not integrate) The condltlons of o-independence imply the full interacting equations of motion defined by the off-shell extension and vice versa, the equations of motion required by stability of the vacuum imply that no on-shell quantity (1 e , no physics) depends ona The analogy between two-dlmenslonal and spacetime concepts goes even further, at least m the case of the intermediate subtraction defined by eq (2 1) It 1s clear from the space-time action point of view that a stable vacuum corresponds to a mmlmum of r and not Just to a stationary point From the point of view of two-dlmenslonal field theory the same concluslon 1s reached by demanding that the conformal invariant point 1s an ultra-violet fixed point so that a small perturbation does not make the system run away m the limit of infinite (momentum) cut-off (Az-+O or f_+co) It is an easy exercise to solve eq (22) for small fluctuations around a conformal invariant point (see ref [ 14 ] ) and to verify exphcltly that for g+ co one runs away from any pomt except for the true minima of the effective action There 1s even an amusing analogy between eq (22 ) and the equations of potential flow m hydrodynamics which helps understanding the “time” evolution of the system The only puzzle that seems to be left is that, while the action cannot change from a conformal point to another if the two are connected by a path of conforma1 backgrounds, this 1s obviously not the case m general as implied by the existence of saddle points The question then is whether two conformal theories with dzffeerentaction have the same central charge or not On the one hand, one would say that they do since, m string theory, the central charge 1s fixed by conformal invariance, at the same time, this would spoil the claimed [ 11,121 relation between central charge and effective action Havmg not studied yet such a relation, we are not able to say more at present on this point In this paper, we have not dealt with the question 424
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of how the critical dlmenslons originate m our scheme Smce D-26 (or D- 10 for superstrmgs) is known to appear m the dllaton /Cfunctlon, we certainly expect to recover these condltlons as soon as we shall add all the massless backgrounds Slmllarly, the well-known tadpole divergences which are known to occur for the bosomc string at the torus level will presumably induce m r finite linear terms proportional to some combmatlon of massless fields This would be our way of recovering the results of ref [I71
Acknowledgement The authors would like to thank D Amatl, P Di Vecchla, M Martelhm, N Sakal and P West for useful dlscusslons One of us (T K ) wishes to acknowledge the partial support of the Swiss National Science Foundation and of the Japan Society for the Promotion of Science
References A M Polyakov, Phys Lett B 103 ( 1981) 207,21 I [2 M Ademollo et al , Nuovo Clmento 2 I A ( 1974) 77 [3 i C Lovelace, Phys Lett B 135 (1984) 75
[I
E S Fradkm and A A Tseytlm, Phys Lett B 158 ( 1985) 316, B 160 (1985) 69, Nucl Phys B 261 (1985) 1, A A Tseytlm, Intern J Mod Phys A 3 (1988) 365, and references therem 15 1C G Callan, D Fnedan, E J Martmec and M J Perry, Nucl Phys B 262 (1985) 593, C G Callan, I Klebanov and M J Perry, Nucl Phys B 272 (1986) 111, R Nepomechle, Phys Rev D 32 (1985) 3201, G M Shore, Nucl Phys B 286 (1987) 349, G Curcl and C Paffuttl, Nucl Phys B 286 (1987) 399 [6] A Sen, Phys Rev Lett 55 (1985) 1846, Phys Rev D 32 (1985)2102 [7] B E Frledlmg and A Jewckl, Phys Lett B 174 ( 1986) 75, A Jewckl and C Lee, preprint Brown-HET-634 ( 1987) [ 81 R Brustem, D Nemeschansky and S Yanklelowcz, preprint USC-87/004 (1987), H Oogurl and N Sakal, Phys Lett B 197 ( 1987) 109, preprint TIT/HEP-123 (1987) [9] C Lovelace, Phys Lett B 32 ( 1970) 490,703, D Olive, Nuovo Cimento 3A ( 197 1) 399, V Alessandrml, Nuovo Cimento 2A ( 197 1) 32 1, V Alessandrlm and D Amatl, Nuovo Clmento 4A ( 197 1) [4
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793, Proc Intern School of Physics (1971) p 58 (Academlc Press, New York, 1972) [ IO] P DI Vecchla, R Nakayama, J L Petersen, J Sldemus and S Scluto Phys Lett B 182 (1986) 164, Nucl Phys B 287 (1987) 621, P DI Vecchla M Frau A Lerda and S Scluto, Phys Lett B 199 (1987) 49 Nucl Phys B 298 (1988) 526, A Neveu and P West, Phys Lett B 193 (1987) 187, Commun Math Phvs 114 (1988) 613 [ 111 A B Zamolodchlkov, JETP Lett 43 (1986) 730 [P&ma Zh Eksp Teor Flz 43 (1986) 5651 [ 121 A M Polyakov, Gauge fields and strings, Contemporary Concepts In Physics, Vol 3 (Harwood Academic Pubhshers, New York, 1987)
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[ 131 S Weinberg, talk given at the Meetmg of the Dlvlslon of
[ 141 [ 151 [ 161 [ 171
Partxles and Fields of the American Physxal Society (Eugene, OR, August 1985), N Selberg, Phys Lett B 187 (1987) 56, A Sen, preprmt SLAC-PUB-4383 (1987) T Kubota and G Venezlano, m preparation S Wemberg, Phqs Lett B 156 (1985) 309, S P de Alwls, Phys Lett B 168 (1986) 59 A A Tseytlm, preprint Lebedev Physxal Instttute 95 (1986) W Flschler and L Sussklnd, Phys Lett B 171 ( 1986) 383, B 173 (1986) 262, C Lovelace, Nucl Phys B 273 ( 1986) 413
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OFF-SHELL EFFECTIVE ACTIONS IN STRING THEORY T KUBOTA ’ and G VENEZIANO CERN, CH-I21 I Geneva 23, Switzerland Recewed
5 Aprd 1988
A new, “off-shell”, approach to conformal InvarIance, pfunctlons and field equations m first quantized string theory IS presented The method, vahd to all orders m the weak-field expansion and non-perturbatwely m a’, IS Illustrated for a tachyon background at genus zero and one level For a particular choice, we derive a Zamolodchlkov-type equation for the effectwe action
1 Introduction
extension of the theory m a way which, while spolhng conformal invariance, still preserves certain crucial properties It 1s not clear, at the moment, whether such an extension can be derived by giving a meaning to Polyakov’s path integral away from its conformal mvariance points (hence off-shell) In any event, this assumption allows us to define two types of generating functlonals, related to one another by a non-local field redefinition, a modified Legendre transformation In general, both kinds of functlonals depend on rs The condltlons of g-mdependence can be almost trivially studied, however, for one functional and then translated m terms of the other by standard functional derivative techniques In this way we are able to identify what we can define as /?-functions and to relate these to appropriate derivatives of the effective action functional Our results fit nicely with (and extend) those discussed already or partly conlectured in the literature In particular, for one particularly appealing choice of the Legendre transformation, we derive a Zamolodchlkov-type equation [ 11,12 ] for the effective action We shall treat, for slmphclty, the case of a closed bosomc strmg moving m a tachyon background at the sphere and torus level, but to all orders m (Y’ We are thus able to discuss the (finite) mass, couplmg and S-matrix renormahzatlons induced by string loops, confirming and extending previous results [ 13 ] We see no reason why our procedure should not be generahzable to any background field m any string theory and at any (finite) genus This note 1s supposed to give the essential ideas and
A crucial element m the first quantized approach to string theory consists of exploiting the condltlons of conformal mvanance For Polyakov’s path mtegral [ I] describing a string moving m a background [ 21, these condltlons guarantee the decouplmg of the conformal factor (Tof the two-metric gap and are crucial for the eventual consistency (causality and umtanty) of the theory What 1s remarkable 1s that the above equations, which can be expressed as the condltlons for vamshmg &functlons, not only determine, for a string m flat space-time, the dlmenslonahty of the latter but even enforce, m the general case, the equations of motion (and/or mass-shell condltlons) for the background fields m which the string propagates Furthermore, order by order m (Y’ , these equations happen to comclde with the Euler-Lagrange equations derived from an effective action, rconf Fmally, this action also generates the string S-matnx, precisely like an off-shell effective action does m usual field theory Much work has been devoted to understand these welcome, yet somewhat mysterious, comcldences (For a partial list of relevant papers, see refs [ 3-8 ] ) In this note we shall present a new vlewpomt on this relationship It 1s based on the assumption, which 1s largely supported by some old [ 91 as well as by recent results [ lo], that one can construct an off-shell ’ On leave of absence from Department verslty, Toyonaka,
of Physics, Osaka Urn-
Osaka 560, Japan
0370-2693/88/$03 50 0 Elsevler Science Pubhshers ( North-Holland Physics Pubhshmg Dlvlslon )
BV
419
Volume 207, number
PHYSICS
4
main results of our approach More details and explicit proofs will be given elsewhere [ 14 ]
2. The basic assumptions Our starting point ~111be the formal expression of Polyakov’s path integral for the (closed, bosomc) string partition function
Z[@LO)l= j DJWz) J&&z) ew
LETTERS
B
the condltlons of Weyl invariance by studying the dependence of Z from the choice of Q When the external states are on shell, even after fixing the conformal factor, Mobius (projective) transformations remam as residual symmetry and mduce, at the sphere level, the well-known mfimte SL( 2, C) volume factor .C&,,m T, for n> 3 It appears to us that Mobms mvarlance is a crucial property if we want off-shell finiteness and factonzatlon Indeed, if and only if there 1s Mobius mvanante, are we allowed to factor out Q,., and to fix three mtegratlon variables The number of integration variables is then the correct one for factorlzmg T,, as T,-rT,+l(s-m2)-‘T,-,+,
where So 1s the free euclldean actlon and $i”’ ( Vc,‘)) are the unrenormahzed background fields (vertices), with a labellmg the various backgrounds (tachyon, gravlton, dllaton, ) We would like to give a precise meaning to Z for all fla (fields to be related to @io’ by some appropnate renormahzatlon) at least as a Taylor series Z=
u&z1
(2) where c stands for connected Since the one- and twopoint functions require some special attention, let us consider first T’ correspondmg to eq (2) with the sum starting from IZ= 3 T’ should be, first of all, a finzte functional of its arguments As 1s well known m ordinary field theory, even forgetting ultra-violet divergences, gauge fixmg is needed to obtain a finite result from a gauge theory path integral Reparametrlzatlon invariance can be used to fix the gauge to be of the orthonormal type gnp=&exp[g(z)
1,
g,pdYdXP=exp[o(z)]dzdZ,
(3)
where, for the time being, we limited ourselves to the sphere level Weyl invariance would allow us to fix G as well, and would guarantee that the result 1s independent of the particular choice adopted Vice versa, we can study 420
30 June 1988
,
(4)
since (n-3)=(m+l-3)+(n-m+l-3)+1, where the last + 1 stands for the integration yielding the pole As already mentioned, old [ 9 ] as well as recent [ lo] results show that, indeed, projective invariance can be maintained off shell This allows us to make the followmg educated basic assumptions
Assumptzons There exists a gauge fixed definition of T’ [ @,,a] such that (I ) SL ( 2, C ) invariance holds for all @,, (11) the amplitudes T, factorize properly even off shell, possibly within an enlarged Hllbert space As we have already pointed out, the n-reggeon amplitudes constructed m refs [ 9,101 satisfy expllcltly property (1) As to property (ii), we have verified its validity at the tachyon pole for the tachyomc amphtudes given m these papers at the sphere level Actually, the pole posltlons are unchanged for all levels Both properties use the fact that the dependence upon the conformal factor (T1s only through its value at a fixed, but otherwise arbitrary, point z. The best way to Justify this from the point of view of Polyakov’s path integral 1s to take c to be constant (almost everywhere) At least for the case of tachyons considered here, this restriction 1s not important as will become obvious later, invariance under a global rescalmg of 0 will Imply invariance under any local change The amplitudes T, (n > 3 ) thus obtained are made finite by (a) removal (cancellation) of the infinite SL( 2, @) volume factor &,
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(b) regularlzatlon of products of operators same point through an invariant cut-off AzAz>exp(
-a)
at the
a2
/4)/G@(O)
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30 June 1988
ambiguity m the definition of the quadratic terms m Tand r Indeed, let us add to T’ a quadratlc term and define
(5)
This second step allows us to regularize the functional T’ via a simple renormahzatlon From now on, we shall restrict ourselves to a tachyon background and omit the label a Defining @= (a2)(a
LETTERS
)
ff=
(a2)-‘”
/4)P*v(o)
one finds /4)Pz@v
(7)
As indicated above, m the process of renormahzatlon, T’ inherits an extra g dependence (besides the one already present m the classical vertex) Such a dependence, which 1s of crucial importance for us, appears already m ref [ 10 1, but was neglected there since it disappears when one goes, eventually, on shell It is now easy to verify that the followmg equation holds aT /aa= Kqi 6T’/6@,
Provided modified
(10)
that I7K-’ + 1 as K+O, one finds that the Legendre transformation defined by
$!kIi-‘6T/&!l,
(lla)
r[$l= T[@l -@‘@
(lib)
)
(6)
@~“)fl(o),[exp(_~)](a
T=T’+~@&!I
(8)
where we have defined
defines a 1TI functional that can be identified with the off-shell effective action m the sense that It generates back T from tree diagrams with Ii’ - ’ propagators Our modlficatlon of the usual Legendre conslstmg of the extra factors transformation, Zi-’ (n) m the defimtlons of q(r), is required by the fact that T generates truncated Green functions, 1 e , amplitudes without external propagators A slmliar observation has been also made m ref [ 7 ] From these defimtlons, it 1s easy to check that i5r/&5= -I@ $ and 6 differ only by interaction finds $=@+n-’
and K(p2) = (1 -a’p2/4) 1s the kinetic operator for the tachyon This equation 1s basically the same as Weinberg’s [ 15 1, who derived the Vlrasoro condltlons on physical vertices by requiring @-mdependence Note at this point that our factorlzatlon assumption (11) contams the non-trivial (bootstrap) condmon that the (tachyon) poles are precisely at the point K= 0 where conformal invariance holds
3. Legendre transformation at the sphere level We would now like to go from the functional T’, which generates connected, truncated amphtudes, to another functional r, which 1s one-tachyon-lrreduclble ( lT1) This procedure has an mtrmslc amblgulty related to the posslblhty of subtractmg some finite part together with the pole terms This amblgulty, which is the analogue of a renormahzatlon ambiguity m the non-linear a-model approach, induces also an
(12) terms
6T’/6+@+0(@2)
Indeed one
(13)
Using eqs ( 10) and ( 11 ), we can also extract the quadratic terms m the T’s r=
-;@zJ+~
) r =0(p)
)
(14)
and verify that 6r’/Z$=l7($5-@)=6T’/?i#
(15)
We finally come to the main purpose of all this exercise g-dependence This does actually depend on the choice of I’I In general, one finds X/so= +
K@I7( q?- $I)
jg(an/aa)
+J(an/a0)
g
(16)
Let us discuss at this pomt three possible choices for 17 They should ultimately be equivalent m the sense of giving the same on-shell theory, but one of them (or maybe a fourth one?) could be most convenient for an off-shell contmuatlon (1) “Mmlmal”subtractlon
We define this by slm421
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4
ply taking I7= K, I e , a a-independent (14)-( 16), we obtain ar/&s=
ar /aa= - (i?ir/&J) 6T’ /6@
= - (6r/@)
6r’ /@
(17)
(11) “Covanant” subtraction satisfy iXllaa=
17 Using eqs
If we define
m a similar way under the change
(19)
- KqX@= K@ir/Ziq?= -Kg 6T /6@
(111) “‘Intermediate” amusmg happens
subtraction
(20)
Something
if we define I7 to satisfy
a17/aa=m,
(21)
1 e , a-dependence m between choices (1) and (u) this case r satisfies
-~(sr~s~~ m-16rp3$,
In
(22)
which has the same structure as the equation proposed by Zamolodchlkov [ 11,121 for the central charge Since the latter 1s expected to be closely related to the effective action, our result should not be surprising One could ask why It should work only m the intermediate subtraction The answer could be that only in such a case the following simple renormallzatlon group-like equation can be shown to hold
= -P(G)
=K@(6/%0
(WV) ($-$I
(i-6) (23)
Eqs ( 17), (20) and (22) are our main results at the sphere level They show clearly that the free equations of motion for @become, through the modified Legendre transformation, the fully mteractmg equations for $ u-respective of the choice of subtraction This observation makes precise earlier claims by Tseythn [ 16 ] that one set of equations goes mto the other through the renormahzatlon process In our approach, however, the renormalization of UV mfimties was done once and for all at the level of the 422
functional T and, from thereon, we have only conslderedfinzte field redefimtlons Finally, we can re-express our result as follows since stability of the ground state requires 6r/6$= 0, we see that vacuum stability guarantees, through eqs ( 17)) (20) and (22) conformal invariance
17 to
[compare eq (8 ) ] In this case, eq ( 16 ) gives
(aiaa)($-ti)
30 June 1988
(18)
aTjaa=K@ liTIS@
arIaa=
B
4. Extension to the torus
2KLl,
T and T’ transform ofa
arIaa=
LETTERS
It is relatively simple to extend our method to the genus 1 (torus) level and m principle to higher genera as well We shall have to continue to assume that off-shell amplitudes satisfying factorlzatlon can still be defined Also, by consistency, on- and off-shell amplitudes will be given by integrals over the same region of moduh space (this requirement replaces that of Mobius invariance discussed at the sphere level) Apart from these assumptions, which are presumably realized if string loops are constructed by the procedure of refs [ 9, lo], we have to be careful about the fact that, unlike m field theory, m string theory we cannot resum self-energy diagrams to obtain dressed propagators To each finite order g m the string loop, expansion poles are always at the treelevel position but their order can go up tog (g- 1 for external masses) A related point 1s that we expect the condltlons for conformal invariance to be unchanged by loops as far as the functional T 1s concerned Yet we would like to recover the expected loop effects on couplmgs and scattering masses, amplitudes Let us denote now by the subscripts 0 and 1 ObJects that belong to the sphere and torus level respectively T, contains double- and external-line poles which can be both taken mto account by a term
T,,,~t(6T,/6~)n,‘n,n,’
6T,/6@,
(24)
where 17, 1s the torus two-point function It IS possible to take the a-dependence to be the same for 17, as for I& n;l
aIz,/aa=n,I
an,la0
The double-pole-subtracted
(25) T’,
T; = T, - TDEp can be used to construct
(26) the quantity
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T=T,+T;
(27)
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values for masses, couplmgs and scattermg tudes Expanding m a Taylor series,
Since F factorizes with Just simple poles in the internal lmes, we can use for tt the modified Legendre transform used at the sphere level definmg
amph-
(33) one finds from (30) a mass shift gtven by
l+= ~‘-@I~~ qGI7,’
)
m:=m;+(4/cr’)n,(-m;)
sT/sqkfj+o(qP)
(28)
and obtain, after a straightforward culatlon, the a-dependence
but lengthy
cal-
+~f$(m~/ao-2Kn,)@
+@(2~n,
- ahyatT)
+
(29)
However, the 1TI functtonal which generates full To+,amplitudes is not r but r=p+
the
TD,P I o=r++ril
= f+
f @IT,q+ (higher genus )
- an,/aa)
#+ ~(an,/acb4u7,) 6
$ (31)
defined
prevt-
)
(32a)
minimal ar/aa=-(sr/s~)[K(~-~)+17,~] arla0=K&vp+,
(32b)
intermediate arIaa=-4
1s generated
by r
(plus pole
(36)
The result (34) appears to be m agreement with the prescription obtained by several authors [ 13 ] m rather mdtrect ways As for ( 35 ), this 1s less obvtous since our off-shell amplitudes differ from those of ref [ 13 ] Indeed, we are not sure that the latter can be denved by a Mobius mvanant ObJect and that it obeys factorization
9
In the three subtraction schemes ously, this equation becomes
covariant
(35)
5. Conclusions - anoh)
+ f~(ano/a~-2Kz70) +$(2~n,
~=[1+(4/cu’)17;(-m:,)]“2~~2-“‘~,
(30)
and satisfies ar/afl=$(Kn,
tachyon field becomes
s,, - z~~wrjtip I~~=-_‘,,~
$
+ gi(an,/a0-2Kn,)
Also, the correctly normalized
so that the S-matrix terms) as
KI&-aI&/aa) 4
@/&P$(
(34)
(tirp&)~n,L w/if+
(32~)
This shows that, m all cases, conformal mvartance holds on the strmg-loop corrected equations of motton 6r/6$=0 Furthermore, for the covarlant and intermediate subtractrons, the equatron 1s not changed from the sphere case In particular, Zamolodchtkov’s form keeps its validity up to the g= 1 level Let us discuss how the condttrons of conformal mvariance at the g= 1 level imply the renormahzed
String theory was born as an S-matrix theory Even after twenty years tt seems to remember (suffer from?) this inheritance reflected m its resistance to being extended off shell Yet, we believe that an offshell extension of string theory not only 1s desirable, but could even turn out to be essential to its eventual success Apart from the (purely technical?) difficulty m definmg finite-loop amplitudes while slttmg on-shell all the time, the on-shell theory ts, by definmon, a theory m one vacuum When such a theory shows no dtsease order by order (as seems to be the case for the superstrmg), one may be happy with such a hmttanon, but this means to give up any possrbrllty of (1) dealing with theories with either perturbattve (bosonic string) or non-perturbatrve (superstrmg?) vacuum mstabthttes, and (11) relating different conststent theories (and at present we have too many of them!) as reahzattons of one and the same theory around different vacua and thus being able to select one of them dynamically In this paper we have made an effort to go off-shell 423
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PHYSICS
which, so far, seems to work It rests on the basic assumption, which is, however, supported by some explicit results, that an off-shell continuation preserving factorlzatlon and Mobius (respectively modular) mvariance at the sphere (respectively higher genus) level exists Of course, conformal (Weyl) invariance IS lost off-shell and thus the off-shell extension depends on a conformal factor d (on which we do not integrate) The condltlons of o-independence imply the full interacting equations of motion defined by the off-shell extension and vice versa, the equations of motion required by stability of the vacuum imply that no on-shell quantity (1 e , no physics) depends ona The analogy between two-dlmenslonal and spacetime concepts goes even further, at least m the case of the intermediate subtraction defined by eq (2 1) It 1s clear from the space-time action point of view that a stable vacuum corresponds to a mmlmum of r and not Just to a stationary point From the point of view of two-dlmenslonal field theory the same concluslon 1s reached by demanding that the conformal invariant point 1s an ultra-violet fixed point so that a small perturbation does not make the system run away m the limit of infinite (momentum) cut-off (Az-+O or f_+co) It is an easy exercise to solve eq (22) for small fluctuations around a conformal invariant point (see ref [ 14 ] ) and to verify exphcltly that for g+ co one runs away from any pomt except for the true minima of the effective action There 1s even an amusing analogy between eq (22 ) and the equations of potential flow m hydrodynamics which helps understanding the “time” evolution of the system The only puzzle that seems to be left is that, while the action cannot change from a conformal point to another if the two are connected by a path of conforma1 backgrounds, this 1s obviously not the case m general as implied by the existence of saddle points The question then is whether two conformal theories with dzffeerentaction have the same central charge or not On the one hand, one would say that they do since, m string theory, the central charge 1s fixed by conformal invariance, at the same time, this would spoil the claimed [ 11,121 relation between central charge and effective action Havmg not studied yet such a relation, we are not able to say more at present on this point In this paper, we have not dealt with the question 424
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of how the critical dlmenslons originate m our scheme Smce D-26 (or D- 10 for superstrmgs) is known to appear m the dllaton /Cfunctlon, we certainly expect to recover these condltlons as soon as we shall add all the massless backgrounds Slmllarly, the well-known tadpole divergences which are known to occur for the bosomc string at the torus level will presumably induce m r finite linear terms proportional to some combmatlon of massless fields This would be our way of recovering the results of ref [I71
Acknowledgement The authors would like to thank D Amatl, P Di Vecchla, M Martelhm, N Sakal and P West for useful dlscusslons One of us (T K ) wishes to acknowledge the partial support of the Swiss National Science Foundation and of the Japan Society for the Promotion of Science
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[I
E S Fradkm and A A Tseytlm, Phys Lett B 158 ( 1985) 316, B 160 (1985) 69, Nucl Phys B 261 (1985) 1, A A Tseytlm, Intern J Mod Phys A 3 (1988) 365, and references therem 15 1C G Callan, D Fnedan, E J Martmec and M J Perry, Nucl Phys B 262 (1985) 593, C G Callan, I Klebanov and M J Perry, Nucl Phys B 272 (1986) 111, R Nepomechle, Phys Rev D 32 (1985) 3201, G M Shore, Nucl Phys B 286 (1987) 349, G Curcl and C Paffuttl, Nucl Phys B 286 (1987) 399 [6] A Sen, Phys Rev Lett 55 (1985) 1846, Phys Rev D 32 (1985)2102 [7] B E Frledlmg and A Jewckl, Phys Lett B 174 ( 1986) 75, A Jewckl and C Lee, preprint Brown-HET-634 ( 1987) [ 81 R Brustem, D Nemeschansky and S Yanklelowcz, preprint USC-87/004 (1987), H Oogurl and N Sakal, Phys Lett B 197 ( 1987) 109, preprint TIT/HEP-123 (1987) [9] C Lovelace, Phys Lett B 32 ( 1970) 490,703, D Olive, Nuovo Cimento 3A ( 197 1) 399, V Alessandrml, Nuovo Cimento 2A ( 197 1) 32 1, V Alessandrlm and D Amatl, Nuovo Clmento 4A ( 197 1) [4
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793, Proc Intern School of Physics (1971) p 58 (Academlc Press, New York, 1972) [ IO] P DI Vecchla, R Nakayama, J L Petersen, J Sldemus and S Scluto Phys Lett B 182 (1986) 164, Nucl Phys B 287 (1987) 621, P DI Vecchla M Frau A Lerda and S Scluto, Phys Lett B 199 (1987) 49 Nucl Phys B 298 (1988) 526, A Neveu and P West, Phys Lett B 193 (1987) 187, Commun Math Phvs 114 (1988) 613 [ 111 A B Zamolodchlkov, JETP Lett 43 (1986) 730 [P&ma Zh Eksp Teor Flz 43 (1986) 5651 [ 121 A M Polyakov, Gauge fields and strings, Contemporary Concepts In Physics, Vol 3 (Harwood Academic Pubhshers, New York, 1987)
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[ 131 S Weinberg, talk given at the Meetmg of the Dlvlslon of
[ 141 [ 151 [ 161 [ 171
Partxles and Fields of the American Physxal Society (Eugene, OR, August 1985), N Selberg, Phys Lett B 187 (1987) 56, A Sen, preprmt SLAC-PUB-4383 (1987) T Kubota and G Venezlano, m preparation S Wemberg, Phqs Lett B 156 (1985) 309, S P de Alwls, Phys Lett B 168 (1986) 59 A A Tseytlm, preprint Lebedev Physxal Instttute 95 (1986) W Flschler and L Sussklnd, Phys Lett B 171 ( 1986) 383, B 173 (1986) 262, C Lovelace, Nucl Phys B 273 ( 1986) 413
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