String loop corrections to beta functions

Nuclear Physics B288 (1987) 525-550 North-Holland, Amsterdam

S T R I N G L O O P C O R R E C T I O N S T O BETA F U N C T I O N S C.G. CALLANl, C. LOVELACE2, C.R. NAPPI3 and S.A. YOST3 Joseph Henry Laboratories, Princeton University, Princeton NJ 08540, USA

Received 26 November 1986

We study the problem of finding the beta functions, and the associated spacetime effective action, for interacting open and closed strings propagating in background fields. String loop divergences play a crucial role in this problem. Cancelling them against sigma model divergences gives a consistent set of loop-corrected beta functions, which can be derived from a simple generalization of the string-tree-leveleffective action. This suggests the existence of new string theories which are conformallyinvariant only after all world sheets have been summed. 1. Introduction The close connection between conformally invariant two-dimensional sigma models and classical string physics is by now well understood: The coupling constant functions of the sigma model may be identified with spacetime expectation values of the massless string modes; the conformal invariance condition (the condition that all the renormalization group beta functions vanish) amounts to a set of spacetime equations of motion for these modes [1-5]; all of these equations may be derived from a single spacetime effective action; and, finally, this action is the one-particleirreducible generating functional for the massless particle string tree S-matrix [6, 7]. One virtue of this approach is that it allows us to translate information about two-dimensional field theory, gained by standard methods, into information about classical string theory. It is natural to ask whether this method can be generalized to deal with the effects of string loop (i.e. quantum) corrections. There is a well-defined string-loop expansion parameter, e ~ (where ~ is the background dilaton field) and it is plausible b o t h that the background field equations of motion should have a power series expansion in e -~ and that they be derivable from a spacetime effective action, itself having a power series expansion in e ~. What is not clear is whether such loop-corrected equations of motion could be interpreted as the conformal invariance conditions of some appropriately generalized two-dimensional field theory. A posi1Supported in part by DOE grant DE-AC02-76ER-03072. 2 On leave from Rutgers University. Supported in part by NSF grant PHY84-15534. 3 Supported in part by NSF grant PHY-80-19754. 0550-3213/87/$03.500 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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C.G. Callan et al. / String loop corrections to S-functions

tive answer would shed useful light on the deeper symmetry structure of string field theory, and would probably suggest new ways to look for string-compatible vacua. Lovelace [8] and Fischler and Susskind [9] have suggested a way of generalizing the renormalization group beta functions of sigma models to include string loop effects. The idea is that, if the loop expansion is defined by the Polyakov path integral sum over world sheets of different topology, nonstandard sigma model counterterms are required to remove the divergences in the modular parameter integrations over inequivalent world sheets of the same topology. Letting the renormalization group flow be defined by the sum of the standard and nonstandard counterterms defines loop-corrected beta functions whose vanishing can be taken as the conformal invariance conditions of a generalized sigma model. Fischler and Susskind showed that, by using this strategy in a very simple contest, it was possible to see the cosmological constant of closed bosonic string theory appearing as a one-loop correction to the beta function for the background metric field. In this paper we extend this very promising idea to more elaborate contexts where it is possible to apply nontrivial internal consistency checks. We regard this as quite important, since the regulation and renormalization procedure used to define the new counterterms is rather ad hoc and not guaranteed to maintain the spacetime gauge symmetries of string theory. Specifically, we determine the first corrections to closed string beta functions due to open string loops, with the effects of open and closed string background fields included. With this knowledge of background field dependence, we can make a nontrivial check of the mutual consistency of the loop-corrected beta function equations. We do find consistency and find furthermore that the full equations may be derived from a loop-corrected spacetime action of very reasonable form. It is probable that this action generates the appropriate loop-corrected S-matrix elements of string theory. In short, the entire set of relationships between sigma models, beta functions and the string theory S-matrix seems to survive the passage from tree to loop amplitudes. The precise nature of the conformal invariance of the underlying sigma model, apart from the observation that it involves cancellation of conformal invariance violations between different world sheets, remains only partially understood. The paper is organized as follows: In sects. 2 and 3 we determine the tree-level beta functions for open and closed strings in general backgrounds. This actually allows us to infer what the open string loop corrections to the closed string beta functions must be. In sect. 4 we review, and slightly improve, the Fischler and Susskind computation of the first closed string loop corrections to beta functions. In sect. 5 we find the counterterms needed to renormalize open string loop divergences in general background fields and derive the associated loop corrections to closed string beta functions. In sect. 6 we discuss the meaning and internal consistency of our results and draw conclusions. In an appendix, we present an alternative derivation of our key results by more conventional operator methods (as opposed to the sigma model methods used in the body of the paper).

C G. Callan et al. / String loop corrections to B-functions

527

2. Background field sigma model beta functions The sigma model describing the coupling of a closed string to the massless backgrounds g~, b~, and ~ is 1 Sc = - f d2z [!/y'gabg~v ( X ) CgaX~Obx v 4 ~rot'

+eabb,,,(X) OaX"ObX"-½a'gr~R(2)q~(X)],

(2.1)

where X" is the spacetime coordinate of the string, 3'ab is the world sheet metric, and R (2) is its curvature. Equations of motion for the backgrounds are obtained by imposing conformal invariance or, equivalently, demanding that the renormalization group beta function for each field vanish. Among other things, this makes the physics independent of conformal transformations on the world sheet metric. At string tree level and to leading order in a', the beta functions are [4] j~gv = Ritv

1 2 V~V/p, - an;p-

f i b = trvX H

, 1

fl~' = - R + 1 H 2 + 2V'2q~ + (V't~) 2,

(2.2)

where Hx~ . = 3Vixb,, ]. Remarkably, the equations obtained by setting these beta functions to zero are equivalent to the equations of motion for the spacetime action

Sg~gsed=f dDxgrge*[R-~H2-(V',)2-av2eO]=- f dDxv~eq'flq',

(2.3)

where D is the spacetime dimension. The relation between the variational equations for this action and the above beta functions is 3

sclosed

eel

-

¢~-e*fl*,

3 3b~,,

8 / ,closed =

+ ½gp,~/"eft

grg-e*flgp •

(2.4)

A systematic loop expansion of the two-dimensional field theory leads to an expansion of the beta functions and the effective action in powers of a', or

CG. Caftan et al. / String loop corrections to fl-functions

528

derivatives of the background fields [6,10, 7]. This whole procedure generates the conditions on closed string background fields arising from tree-level, or classical string physics. Our main concern in this paper will be the way in which the background field equations are modified by non-classical, or string-loop, effects. As mentioned in the introduction, we have found it very useful to approach this problem indirectly, by studying tree-level open strings in general backgrounds. This gives us very useful information about string loop physics because of the well-known fact that the basic interaction between open and closed strings is really a loop phenomenon. The problem of an open string coupled to open string backgrounds has been considered by several authors [11-14]. The appropriate sigma model is

So=

1

f d 2z ll~yabOaX ~*ObXJ*-~dsAt,(

0

X)-~s X ~ ,

(2.5)

where ds is the length element around the boundary induced by "lab- In this case, the beta function was found to be ,8~ = 2rro~'V~F~ [1 - (2rra'F)2] 21 .

(2.6)

This result is valid to all orders in a' and to lowest order in derivatives of F. Setting /3A= 0 gives an equation of motion equivalent to that which follows from the Born-Infeld action Sef pen= f

f dDx ¢det(1 + 27ra'F)

(2 7)

The variation of this action is in fact

.~,eff = ~A ~ vopen

-¢det(1 + 2~ra'F)[1

_

-1 u (2~ra r F ) 2 ],v fl]-

(2.8)

This is compatible with the observation that the effective action and the beta functions are not expected to be identical, but rather to satisfy the relation 8seff/~A~

=

v

A

for a nonsingular matrix X,, [6,15]. For our purposes, it will be necessary to know the beta functions for the open string in a general background of both closed and open strings. The relevant results and representative calculations are presented in the next section.

C G. Callan et al. / String loop corrections to fl-functions

529

3. Open strings in general backgrounds The coupling of an open string to all massless backgrounds, arising from both closed and open strings, is described by a sigma model of the form

S = m 1 f d 2 z [ v ~ 3 , . b g . . ( X ) O~X.ObX ~ 4 ~ra'

--ieabb.~( X) O~X g ObX ~ -

1

+ 2ira'

[

o

ds iA.(X)--XU-os

½a'Cy-R(2)~ ( X ) ]

]

½a'keo(X) '

(3.1)

where k is the extrinsic curvature of the boundary and A t has been rescaled to include a factor of 1/2~ra'. Both the world sheet and spacetime metrics are taken to be euclidean. The coupling of the dilaton to the boundary curvature is needed because e e is the coupling constant of the string theory, and therefore must multiply the entire Euler density. If we work in a conformal gauge 3'ab = eZ°~ab and let r denote the coordinate along the boundary, then v~-R (2) = 2 [] p and k ds = - Onp dr, where O, is the exterior normal derivative. Then the world sheet action may be rewritten as

1

S [ X ] = ~ a , f d2z[g.~(X) OaX"O.X ~

--ie~bb..( X) O~X" ObX ~ + a' O~pV/oO~X"] i + 2rra---7~d'rA~,(X) O.~X~'.

(3.2)

We will expand (3.2) about a classical background X, using a normal coordinate defined about X. Since the dilaton action is of a higher order than the rest of the action, we take .~ to satisfy the variational equations of the ~-independent part of the action, i.e.

( gff Oa -1-l'l~vXOa~Tt nt- ~11-...txvx,Eab 11 Ob2X ) Oa2v = O, 0n ~u - i (b + F ) "~ O, 2 ~ = 0

on the boundary.

(3.3)

C.G. Callan et al. / String loop corrections to fl-functions

530

The normal coordinate expansion of (3.2) is [14,16] 1

-47fdazOoo[~v~G¢(x) Gx"+ V . O ( X ) D ~ " +

812, (1--s[xl +

v

O(~2)]

1 + ~

f d2z [g..(.Y) N ~ " N a ( ~ + (3~b--e ~b)

i + 47r-a'~dr

[(b(X')

+ F( ~-'))..~" D~"

+~x~°v~(b + F ) x . 0 . 2 " + 0(~3)],

(3.4)

1 where ~ is the generalized curvature for the connection F + IH, namely

~ . x p . = ~,xo. + ½iv~H.xo -

- xH,,.o H ~x +

and we define

D o e = (g". aa + r ~ a a ~ ) ~ ~, ~"

= D ~ ~ + 2lH

u)~eab O b X ~

v.

Comparing with [13], we notice that on the boundary, F.. is replaced by F~. + b~.. Along the lines of [13], one can use the terms g.. O~" 0 ~ ~ and (b + F)..~" 8.~" in (3.4) to define the propagator. At open string tree level, we may take the world sheet to be the upper half-plane, with z = r + io. Then we obtain the propagator G . . ( z , z') =

-

1[

~a

G.lnlz

+

g

-

-

z'l 2

~7]~

i n ( z - ~') +

~-b-~

.~

l n ( ~ - z')

•

(3.5)

In the limit where g, b, and F are constant, this propagator is exact. Using (3.5) to compute the tadpole graph of fig. la gives a contribution to flf analogous to the previous result (2.6), namely ~7"(b+ F ) ~ [ g - ( b +

F)2] -1

As shown in [14], there is also an H-dependent contribution to flA, which comes

C.G.Callanet aL / Stringloopcorrectionsto fl-functions

531

HSrX

(a)~

(b)~Or~ ¢Va~ ¢

(F+b)OrX

F+b

Fig. 1. Graphs contributingto fla. The line . . . . indicatesthe propagator (3.5). from the graph fig. lb. If we use the exact propagator to compute this graph and use the equations of motion (3.3), we find an addition

~x [ b+F ½(b+ F)u,H °[g_-~+F)2]xp to /37. There is also a dilaton contribution to flA, which, to lowest order in derivatives of @, is determined by a sigma-model tree-level calculation. The tree level contribution to the trace of the stress-energy tensor is the variation of the dilaton action with respect to the conformal factor of the world sheet metric, i.e. 6

2

1

1

1

g f d z 4= Oap v.epOoX" = - -VV.p/OoY"OoE"-4=

1

+ ~8(o(z))

V,, 0,2".

The last term is localized on the boundary and may be rewritten as i

4= 8( o( z ) ) gr/p( b + F)", 0,2" using the equations of motion (3.3). It obviously contributes a term ½V"q~(b+ F)~, to fla. The complete fla is the suha of the above contributions and is found to be flA = V " ( V + b)uX[g - ( F + b)2]• 1

+½(F+B),,H~XO[

F+b ] g_~++b)2jxo+½V~ep(r+b),~.

(3.6)

The noncovariant F-dependent terms in (3.4) do not contribute to flA when 2 satisfies (3.3). The spacetime effective action associated with the equation of motion (3.6) is the generalized Born-Infeld action

Sa, °n = f

dDxe ~'/2 (det(g + b + V ) .

(3.7)

C.G. Callan et a L / String loop corrections to fl-functions

532

The variation of this effective action with respect to A ~ is - e~'/Z(det(g + b + F ) [ g -

(b + F)Z],~lfl~

with flA as defined in (3.6). Therefore, the vanishing of this variation is equivalent to/3 A = 0 . Although the open-string and closed-string beta functions are logically independent of one another (after all, they arise from different sigma models evaluated on different world sheets), it is quite natural to guess that the proper way to describe interacting open and closed strings is simply to add the associated actions, (3.7) and (2.3), together (with an as-yet-undetermined relative coefficient x). The proposed full effective action is therefore

=f

d~x [ 7 r g e * ( R - ~2H 2 - (Vq~) 2 - 2V2q~)+ t¢ e¢/Z~det(g + b + F ) ] . (3.8)

Since the original terms in the effective action are of second order in derivatives of the fields, while the added terms are dimensionless, we note that K will be of order a'-1. Note that (3.8) is invariant under the combined transformation b,, ~ b~, + 2VI~P~I,

A~

(3.9)

A ~ - ~'~.

This implies that A, can be totally absorbed in b,~ by a Higgs mechanism of a type familiar in supergravity theories [17]. This effective action exactly reproduces the open string beta function equations, but not those of the closed string. There are additional terms in the variational equations for g,,, bu~, and q5 which amount to gauge-field source terms for the closed-string fields. Specifically, we find 8 g

S eft t°t~l --- -v/ge*fl ~' + ½x e 't'/2 ~/det(g + b + F )

b+F ]~

8b~"8 oeffct°t~= 7r~e./3b + ½~C,/2~det(g + b + F ) [ g _ ~ + F ) 2

8 --+2g#v~-~

"eft

(b+F)2]~ ' (3.10)

where fig, fib, and fl* are given by (2.2). The new terms are certainly reasonable (after all, gauge fields must act as a source for gravity), but are not visible in the standard treatment of closed string beta functions: As noted in refs. [13] and [14],

C.G. Callan et al. / String loop corrections to S-functions

533

the presence of a boundary does not change the beta functions of the closed string massless fields. When the relevant graphs are computed, it is found that the only possible changes arise from the boundary terms in the propagator (3.5). However, these terms do not contribute to the divergences which give the closed string beta functions. In particular, the boundary contribution to G~,(z, z') is finite as z ~ z' in the interior of the world sheet, so the logarithmic tadpole divergences responsible for the one-loop contributions to/3g and/3 ° are unchanged. It is straightforward to check that the other contributions to (2.2) are unchanged as well. The only difference is that/3 ~ is now the coefficient of the entire Euler density, rather than R ~2), in the trace of the sigma model stress-energy tensor. Therefore, if the new terms in the closed string beta functions are correct, they must arise from as-yetunaccounted-for physics. An essential clue is that the new terms contain a factor of e ~/2 relative to the original closed-string beta functions and must therefore be regarded as string-loop corrections. In the next section, we will discuss a general proposal for computing such corrections to beta functions and, in the rest of the paper, we will use it to show that all the equations of motion derived from (3.8) are interpretable as loop-corrected beta functions.

4. Loop corrections to beta functions: general strategy That string loops must modify the sigma-model beta functions is obvious. Fairly concrete suggestions about how to calculate this modification have been made by one of us [8], and independently by Fischler and Susskind [9]. The idea is that the divergences of string loop perturbation theory can be eliminated by new counterterms in the sigma model action, over and above those needed to renormalize the usual perturbative divergences, and that the new counterterms can be thought of as generating corrections to the usual renormalization group beta functions. In particular, for the closed bosonic string, Fischler and Susskind showed that, in a fiat spacetime background, the renormalization of one-string-loop diagrams adds a cosmological constant term to the beta function for the metric tensor. In the rest of this paper, we will adapt these ideas to the case of nontrivial spacetime backgrounds and show that it reproduces the loop-corrected closed-string beta functions derived heuristically above. In this section we will rederive, from our point of view, the results of Fischler and Susskind, both to review the essentials of the method and to introduce, in a simple context, some arguments which will be important later on. Consider the closed bosonic string in general metric and dilaton background fields, g ~ ( X ) and ~(X). It is described by the nonlinear sigma model action Sc given in (2.3) with b~ = 0. In the Polyakov path integral approach, the partition function of this theory on the two-sphere (toms, etc.) gives the physics of string no-loop (one-loop, etc.) amplitudes. Since the two-dimensional nonlinear sigma model is renormalizable, not finite, a counterterm action built out of dimension-two

534

C.G. Callan et al. / String loop corrections to B-functions

operators must be added to Sc to yield finite results. The general form of that action is 8Sc - l ° g A

2¢r

fd2z[f~V~Sg..(X) OoX"ObX"-½a'v/-~R'2'Seo(X)] (4.1)

where A is a cutoff parameter and 8g, 8~ are functions of the background fields. They define the renormalization group beta functions, whose vanishing is the condition for sigma model conformal invariance. In performing the sum over two-dimensional world sheets to get the string loop expansion, the conformal invariance of the sigma model makes it only necessary to sum over conformally inequivalent surfaces. For a given genus world sheet, this leaves a finite-dimensional parameter space, known as Teichmiiller space, to integrate over. The crucial points for our purposes are that the Teichmiiller parameter integrations in general do not converge; that the divergences come from boundaries of the parameter space where topological fixtures (such as handles) shrink to zero size; that the divergence coming from shrinking a single fixture away is proportional to the insertion of a simple local operator on the lower-genus world sheet obtained by removing the shrunken fixture altogether [18]. For the sigma model corresponding to flat empty spacetime, the divergence associated with shrinking away a single handle is equivalent to the insertion of the operator log A

2~r C~I~,~OoX/zO~Xv,

(4.2)

where A is some convenient cutoff on the Teichmiiller integration (which we identify with the cutoff used in the counterterm action (4.1)), C is the closed string cosmological constant (as determined by a calculation of the closed string partition function on the torus [19], for example) and ~,, is the Minkowski metric of flat spacetime. Not coincidentally, this is the vertex operator for the emission of a zero-momentum dilaton, and the divergence has to do with the existence of an amplitude for emitting such a dilaton into the vacuum. Now consider doing a closed string calculation to one-loop order. This is given by the joint contribution of the genus zero and genus one surfaces (sphere and toms). The observations of the previous paragraph indicate that the divergences of the Teichmiiller parameter integrations for the torus can be eliminated (and the string loop calculation renormalized) by adding a new counterterm lagrangian 8S~ °°p

A . = log2--~fd2zC~,~OaX OaX

(4.3)

to the sigma model action. Since this is a dimension-two operator, it is a perfectly legal counterterm from the point of view of two-dimensional field theory. When

C.G. Callan et al. / String loop corrections to r-functions

535

evaluated on the sphere, it gives a divergent contribution which precisely cancels the divergence of the torus. Since it is already a one-loop effect, it would, to this order, be neglected in calculating the torus contribution. Thus 8Scl°°p renormalizes the string-loop divergences in very much the same way that 8S~ renormalizes the ordinary field theory divergences. We must make two adjustments to the above counterterm: In a general coordinatization of flat spacetime, the Minkowski metric */,~ must be replaced by the general metric g,~ and, in the presence of a constant dilaton field, we must include a factor e ~ to account for the well-known topologically-determined dependence of the path integral on the dilaton zero mode (the torus behaves as constant while the sphere behaves as e~'; thus the counterterm must behave as e -~ in order that, when evaluated on the sphere, it reproduce the divergence of the torus!). The result of these corrections is 8S2oop :

log____AA f d2z

aox.aox

2~r

(4.4)

We will actually use this counterterm in non-flat spacetimes with nonconstant dilaton backgrounds, in which case it must be regarded as the first term in an expansion of the true counterterm in powers of derivatives and curvatures. The proposal of Fischler and Susskind is that the string-loop-corrected renormalization group beta functions and background field equations be read off in the obvious way from the combined counterterm /iSc + 8S~l°°p. For the background metric this gives fife = R ~ - V',~7,¢ - C e - ~ g ~ , ( X ) = 0,

(4.5)

an equation which resembles the Einstein equations for nonzero cosmological constant. It differs from the similar equation written down in [9] mainly in that it includes explicit dependence on the dilaton field. This dependence is, of course, crucial to the consistency of the equations and it is not permissible to ignore it, as also noticed in [20]. There is also a string-loop-corrected dilaton equation of motion, but it is not easy to evaluate the counterterm which generates it. We will determine what it must be, and at the same time learn something important about the consistency of this method, by an indirect procedure. The point is that the equation fl~, = 0 is not guaranteed to be a consistent equation: taking its divergence, and using the equation itself plus the Bianchi identities, it is possible to show that o = v

v

g

= v.(

1

-

-

) 2)

(4.6)

which means that r * = ½R - V'2q, - ½(V'd?)2 = const

(4.7)

536

C.G. Callan et al. / String loop corrections to fl-functions

is the consistency condition for fig = 0! An equivalent system of equations (with const = 0) is obtained by varying the spacetime action

L= f dDXv~[e~'( R - (V'ep)2-

2~y2q~)+ 2C1.

(4.8)

This action is the obvious one-loop, cosmological constant generalization of the tree-level effective action for closed strings (the relative powers of e * distinguish terms arising at different string-loop orders). Exactly the same construction has been used in the standard sigma model approach to string tree physics [4] where the existence of a covariantly constant r * is guaranteed by the existence of a c-number conformal anomaly in conformal sigma models. We know of no principle which guarantees that the same construction will work in the current context. The upshot of all this is that the loop-corrected beta function for the metric implies a loop-corrected beta function for the dilaton field and the mutual consistency of the two equations is guaranteed by the fact that both can be derived from a single spacetime effective action. It was by no means obvious a priori that this construction would continue to work for the loop-corrected beta functions, and we take the fact that it does as an encouraging indication of the consistency of this string loop renormalization scheme. In the rest of this paper, we will show that the above construction may be carried out for the much more demanding case of interacting open and closed strings in nontrivial gauge field backgrounds.

5. Open string loop corrections to beta functions In the previous section, we studied the corrections to closed string beta functions arising from closed string loop divergences. In this section we will consider the effect of open string loop divergences on the same beta functions. Our interest in open strings, we emphasize, stems from the ease with which they may be coupled to nontrivial background fields, and the opportunity they provide for serious tests of the internal consistency of string loop renormalization schemes. In contrast to the closed-string case, where the divergences come from handles shrinking to zero size, the open string divergences come from the shrinking holes in the world sheet to zero radius. This divergence is proportional to the effect of replacing the small hole by the insertion of a certain local operator on a lower-genus world sheet. We will identify that operator by studying the behavior of the annulus partition function in the limit of small inner radius. We will add some instructive complication to the problem by coupling all world sheet boundaries to a constant background gauge field strength. The end result of this computation will be an identification of the background field dependence of the open-string-loop corrections to closed string beta functions. Consider an annulus with inner radius a and outer radius one. The propagator on the annulus in the presence of general constant open and closed string background

537

C.G. Callan et al. / String loop corrections to fl-functions

fields, and with equal and opposite U(1) gauge charges assigned to the inner and outer boundaries, can be shown, using the techniques of [13] to be

G~(~,~ , )_- - ~1. , -[%(z,z , ) +

(5.1)

where

a2nz)( a2nz l

G~,,(z z ' ) = g ~ , l o g ( z - z') + g~, ~__llOg 1 '

-

,

+

1 - --

z

z'

J]

~ , 5 1 lOg 1 - - ~ - ) ( 1 - a 2n- 2zz,') .

g+ b+

(5.2)

We choose equal and opposite boundary charges in order that the states propagating around the loop be electrically neutral. If we consider only orientable world sheets (i.e. if we do not include such surfaces as the projective plane or the Mrbius strip in our world sheet summation), it is always possible to choose the boundary charges such that all internal lines in loop diagrams are neutral. The other possible choice, of equal charges on both boundaries, has been considered in [11]. It gives different results, corresponds to different physics and will not be considered here. If the background fields are not constant, the propagator is the same, but nontrivial vertices, involving derivatives of background fields, appear in the expansion of the sigma model action. We will ignore the effects of such terms, and our results should be thought of as the first term in a systematic expansion of powers of spacetime derivatives of background fields. The expansion of the propagator in the limit as a approaches zero is g+ b+

-a

[(z

2 gjzv

7

+ --

z

-}-

log(1 - zU) ,uv

g+b

~

( 1)1 ZZ'+

--

z,T'

+O(a4).

(5.3)

Substituting the first two terms in (5.3) into (5.1) gives the propagator on the unit disk, with the boundary condition O,G,,= -¢x'g,~ at ]z] = 1. A straightforward calculation shows that the a 2 terms are equivalent to the insertion at z = 0 on the unit disc of the operator (see fig. 2)

2a2(g-b-F) a' g + b + F

~ ~vOzX O~X.

(5.4)

This means that a scattering amplitude on the annulus of inner radius a may be

538

C.G. Callan et al. / String loop corrections to fl-functions

Fig. 2. A small hole in the annulus is mimicked by an operator insertion on the disk.

computed as a power series in a 2 whose term of O(a °) is the same scattering amplitude on the unit disc, whose term of O(a 2) is the same scattering amplitude on the disc with one insertion of the operator (5.4), and whose higher terms correspond to the insertion of higher-dimension operators we have not troubled to determine. This is a particular case of known general features of the behavior of amplitudes on higher-genus world sheets at the boundaries of moduli space [18]. Since the inner radius of the annulus is a Teichmfiller parameter, it must be integrated over, with a measure given by certain functional determinants, in order to obtain the total contribution of the annulus to any amplitude. Since the measure turns out to behave as a-3 as a ~ 0, quadratic and logarithmic small-a divergences will come from the zeroth and first terms, respectively, of the operator insertion expansion described in the previous paragraph. In old-fashioned language, the quadratic divergence arises from a tachyon, and the logarithmic divergence from a dilaton or graviton, disappearing into the vacuum. The logarithmic divergence will be proportional to the insertion on the disc of the operator (5.4), while the quadratic divergence will be proportional to the insertion of the unit operator. Note that the divergences are associated with insertions of operators of dimension zero and two, the dimensions allowed by renormalizability in the sigma model. The constants of proportionality come from the measure and will be important to us because they depend on the background fields! To fix them, it suffices to examine the divergences of the vacuum-to-vacuum amplitude on the annulus. The vacuum-to-vacuum amplitudes on the disk and annulus in the presence of open string backgrounds (but in the absence of closed string backgrounds) and expanded to lowest order in derivatives of F, have been obtained in refs. [11] and [13]. The result for the disc is [11]

Z disk= f d°X~/det[1 + F( X)] Zoaisk ,

(5.5)

where Z disk is the vacuum amplitude on the disk without a gauge field. For the annulus, with the choice of equal and opposite boundary charges the result is [13]

Z ~ nuluS= f dDXdet [1 + F( X)] Z ~ u'us ,

(5.6)

c.G. Callanet al. / Stringloopcorrectionsto fl-functions

539

where Z~nul.s = [1 da oo Jo ---T a I-I (1 - a 2n)-(°-z) n=l

=

[lda Jo -a-T[1 + ( D - 2 ) a 2 + O ( a 4 ) ]

(5.7)

is the zero-field partition function for the annulus, including the ghost determinant. Note that (5.5) is claimed to be proportional to the spacetime effective action for the background gauge field! The background field dependence of any partition function is in fact very divergent and this simple finite result is obtained by a careful use of zeta-function regularization. It is in accord with the general arguments of refs. [11] and [8] that the Polyakov path integral for a background field sigma model can be defined to be the spacetime effective action. The expansion of the integrand of (5.7) in powers of a 2 should, according to the arguments given above, correspond to an expansion in insertions on the disk of operators of increasing dimension. With a little algebra we can show that (5.7) is reproduced by z~mnulus = [1 d a

Jo a 3 ( ( 1 - 2a2)¢det(1 + F ) )disk

+~

---LS-,Cdet(1 + F)

O~X"O~X ~ /~1,

+ finite. (5.8) disk

Note that the expectation value ((9)disk=f.~Xe-S(9(X) of an operator (9 is proportional to the determinant (5.5), so each expectation value in (5.8) contains an implicit factor of Cdet(1 + F ) . The explicit factor of Cdet(1 + r ) inside the brackets in (5.8) then serves to reproduce the overall factor of det(1 + F) in (5.6). This is the background-field-dependent constant of proportionality we wanted to fix. The first term in (5.8) reproduces the quadratic divergence, as well as the " - 2 " part of the logarithmic divergence (attributable to the ghosts), in the expansion (5.7). The second term reproduces the remainder of the logarithmic divergence, coming from the D fields X ". If we cut off the a-integration at a short distance A, we see that the counterterm action needed to compensate for all these divergences is

[2

8S~°°P= f d2z ~-log ACdet(1 + F)

~

~,OzX~'O~X~

+ ( A 2 - 21ogA)¢det(1 + F ) ].

(5.9)

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C.G. Callan et al. / String loop corrections to fl-functions

The first term is a counterterm for the standard dimension-two operators in the sigma-model action and will lead, via the Fischler-Susskind type of argument, to loop contributions to the beta functions of the massless closed string fields. The second term is a counterterm for dimension-zero operators, which are used in the sigma-model action to describe the interaction with background tachyon fields [21]. This term should therefore generate loop corrections to the tachyon beta function! Since the tachyon is not present in realistic string theories, we will not concern ourselves further with this type of term. Our final task is to include the effect of closed string backgrounds on these counterterms. If we work to lowest order in derivatives of background fields, the vacuum amplitudes (5.5) and (5.6) become Z disk=

Z annulus =

f dDXe~/Zfdet(g + b + F) Z disk ,

f dDXvfgdet( 1 + g - l ( b +

F))Z~

(5.10)

nnulus .

As usual, the coefficient of ~ is determined by the Euler character of the relevant world sheet. The full structure of the logarithmically divergent, dimension-two counterterm insertion is then given by

2-71°g Afd2z~det(l+g_l(b+F))e_~,/2(g-b-F g+b+F ) 8zX~O~X~" (5"11t Following the approach of ref. [9], we treat it as a new counterterm for the sigma model couplings to the metric and antisymmetric tensor fields and read off from it (by separating the symmetric and antisymmetric parts) the following open string loop corrections to the beta functions fig and fib:

)

~/~Aflg = ---~7-e-~'/2(det(g+ b+ F) g + (b+ F ) 2 g - ( b + F ) 2 ~' 1/~Aff~ -

b+F

47re-*/2~det(g+b+ F)

_ _ =-

g_

1

(5.121

:71 /

Comparing with (3.10), we see that these loop corrections to the closed string beta functions are precisely those implied by the conjectured action (3.8) if we take the undetermined coefficient ~ to have the value - 8 7 r / a ' . Note that since we have not kept track of the world sheet curvature, or world sheet ghost field, dependence of the loop counterterms, we cannot directly identify the loop correction to f * . We

541

C G. Callan et al. / String loop corrections to fl-functions

can, however, infer what it must be by a consistency argument of the type which led to (4.7) and will necessarily get the same equation as found by varying (3.8). Before turning to a discussion of these results, we would like to present a brief description (details are in the appendix) of another way to derive the crucial background field dependence of (5.9). The divergences of open string loops can be interpreted as due to zero mass states of the closed string disappearing into the vacuum. In the appendix, we give a general technique for calculating the transition amplitude, T, for this process: One first builds a Fock-space representation of the operator V(A*, F~), made of the closed string creation operators A*, that sews a boundary, with boundary conditions appropriate to the presence of background gauge field strength F,., onto a closed string world sheet. We find that

V(A t, F..) = [det(1 +

F)]'/Zexp

E

~

~,,u=l

A~*,. ,

(5.13)

/ t~v

times a BRST ghost factor (A.26). For F = 0, this reduces to a result of Ademollo et al. [22]. The transition amplitudes of interest to us are the expectation value of this operator between the massless closed string states and the vacuum. We see that for an annulus in an external electromagnetic field both the graviton and the antisymmetric tensor field have vacuum transition amplitudes that are respectively the symmetric and antisymmetric piece of

T(g~,b~ ->vac)=[det(l+ F)]l/2(l-F) tzv.

(5.14)

This reproduces the graviton and antisymmetric tensor part of (5.9). In the appendix, we will show how, by including ghost operators in the above considerations, it is possible to reproduce the dilaton piece as well. It is reassuring that the potentially ill-defined background field dependence of the overall normalization of this quantity can be derived in two completely independent ways. Since overall vacuum stability of the closed string requires that loop amplitudes cancel against the tree vacuum amplitudes, which are precisely the original sigma model beta functions [8], the full stability condition will be precisely the loop-corrected beta functions derived above.

6. Discussion and conclusions Let us now try to summarize what we have learned. Perhaps the best way to appreciate the physical content of Aflg and A/3 b is to expand both of them in

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C.G. Callan et al. / String loop corrections to r-functions

powers of F in flat space and for vanishing b: 27t

Aflg=-a7 ( g~ - ~ g ~ F 2 + 2F2 + . . . ) , 4~ -

7 G +

.-

..

(6.1)

The first term in Zlflg obviously corresponds to a finite cosmological constant addition to the equation of motion for gravity. Whether the contribution of open string loops to the cosmological constant of a bosonic string theory should be finite or not is a question concerning which there has historically been some confusion. According to our method, the answer is unambiguous. The second term in Aflg corresponds to the classical contribution of the gauge field to the energy-momentum tensor of matter. The origin, within the context of the sigma model, of this absolutely crucial term has always been somewhat mysterious (though its possible emergence from some kind of anomalous effect of small holes [23,4] had been guessed at). Our method shows that it comes, in effect, from divergences not accounted for in the standard field theory approach to conformal anomalies. This is reminiscent of a recent discussion [24] of tachyon background fields, which showed that nonperturbative divergences, due to sums of infinite subsets of sigma model Feynman diagrams, generate the tachyon contribution to the energy-momentum tensor of matter. Similar remarks could be made about the effects of world sheet instantons: nonconvergence of the instanton scale size integration generates divergences which must be eliminated by extra sigma model counterterms. This leads, in turn, to new nonperturbative contributions to the beta functions and, in some circumstances, qualitatively new physics [25,26]. Finally, the leading term in Aft b obviously expresses the well-known linear mixing of an abelian gauge field with the antisymmetric tensor field [17]. For a nonabelian gauge field we would expect to recover the more subtle Chern-Simons mixing first discovered in supergravity theories [27]. In other words, the qualitative physics incorporated in the loop-corrected beta functions we have found is just right. We expect, but have not checked, that the effective action (3.8) which generates the new beta functions also generates the appropriately loop-corrected string S-matrix. It should perhaps be said that, at the level we are working, the term "loop correction" is a bit of a misnomer. Although we used the annulus to evaluate our counterterms, the divergence we have emphasized arises first at the level of the disc! The point is that, although the disc gives finite contributions to purely open string S-matrix elements, it gives divergent contributions to mixed open and closed string amplitudes. Because of the relative Euler character of the sphere and the disc, the needed counterterms will be of order e -4/2 relative to the perturbative counterterms. They could have been evaluated directly on the disc, but we found it

C.G. Callan et al. / String loop corrections to r-functions

543

convenient to identify them as a subdivergence on the annulus. In the sense of powers of string loop coupling constant, the disc looks like a loop contribution, but its contribution to the S-matrix will have the analytic structure of a tree amplitude (poles, but no cuts). This, in fact, is why we were able to correctly guess the structure of the full effective action from a tree-level (i.e. disc) open string calculation. The first true loop correction should come from the limit of the annulus in which the inner and outer boundaries collide. We believe that the associated counterterm is a boundary integral corresponding to a correction to flA, the gauge field beta function. A major defect of this treatment of string loop renormalization (and most others as well) is the lack of a systematic regulation and renormalization procedure. The internal consistency of our results indicates that our cavalier treatment of such matters has not led us astray, but we can hardly expect such good luck to persist to higher loop orders. Further progress in this area may well depend on the development of a systematic regulation scheme which is compatible with the symmetries of string field theory. A very interesting question concerns the sense in which the loop-corrected sigma model is conformally invariant. The sigma model on the disc has fl 4:0 and thus cannot be conformally invariant by itself. Mansfield and Martinec [28] recently pointed out that string loop divergences invalidate the formal arguments for both BRST invariance and the decoupling of Virasoro states from the physical sector. Since we have arranged the divergences and the vacuum instabilities to cancel between the disc and the annulus, presumably these conformal anomalies also cancel each other and the complete theory is free of negative metric states. We hope to check this explicitly in future work. Our model suggests the existence of a big new class of string theories, consistent only on all world sheets at once. They may well be incarnations of the abstract structures on moduli space proposed by Friedan and Shenker [29].

We wish to thank G. Moore, E. Witten and L. Yaffe for useful remarks.

Appendix OPERATOR FOR ATTACHING A BOUNDARY

In this appendix we derive the operator which imposes an arbitrary boundary condition on the world sheet. It generates the ancient pomeron factorization [22, 30] and is related to a modern proposal for the offshell extension of the string propagator [31]. Consider first a single harmonic oscillator with frequency co and

544

C.G. Callan et al. / String loop corrections to fl-functions

unit mass. Let

u,(q),

n = 0,1, 2,... be its normalized position space wave functions { ~

u.(q)=[~)

/1/4

2-"/2(n?)-Z/2H.(~l/2q)e ~q2/2,

(A.1)

where H,(x) are Hermite polynomials. We are interested in the functional integral over a finite euclidean time 0 ~< • ~< t,

exp{-jo'd"[( )'+..o.

1.:

},

where SO and St are extra actions on the boundaries which serve to impose the desired boundary conditions. Inserting

f dqi~(~(0)-qi)f dqf6( @(t) -

qf)

we get by a variety of standard methods

It= ~_, e-"~'f ~

dqie

s°cq~l..(qi) f ~ dqfun(qf)e -s'[qf].

(A.3)

(In this expression, the zero point energy has been subtracted.) Inserting (A.1) and doing the sum gives a gaussian in qi, qf, whose coefficient matrix is the inverse of the Neumann function restricted to the boundaries [11, 31]. Eq. (A.3) exhibits the factorization of the two boundaries, but we would like to express it in terms of the creation and annihilation operators A* and A, rather than Hermite polynomials. Therefore we look for an operator V(At, S) satisfying

£ for any given function

dqu.tq)"' e -s[ql S[q]. Using

~

1

(OIA"V(A*,S)[O)

(1.4)

the Hermite polynomial generating function

,z £ ~nV. ' u,(q)z"=--) ("'~"exp{-Jo~qZ+(2wl'/2qz -"I

n=0

(A.~

turns the right-hand side of (1.4) into (01e"V(A*, S)[0) =

V(z, S).

(A.6)

Therefore

V(A*,S)= (4.~)-V4f~dQ exp{-S[(2o.,)I/'Q] - ¼Q2 + QA*-

~(A*)2}, (A.7)

C.G. Callan et al. / String loop corrections to fl-functions

545

Fig. 3. The annulus is mapped into a rectangle. The inner boundary becomes the top.

where Q -= (2~)1/2q. In terms of this operator, (A.3) then becomes

I t = COl V(A, So)e ~'A*AV(A*, S,)IO),

(A.8)

displaying the factorization of the two boundaries in a convenient form. Notice that the position variable inside the boundary action S[q] is scaled by (2w) -1/2 in the integral (A.7) for V(A*, S). We now extend these formulae to the string by multiplying the separate oscillator pieces. The conformal transformation (shown in fig. 3)

r + io = - l o g z maps the annulus a ~ tz[ ~< 1 into a rectangle 0~/2 ]

o

m=l [ d r

d'r

(A.IO)

Writing ~ - ~-(~

+ iX~m)

(A.11)

gives 2D correctly normalized real oscillators for each frequency oa = m = 1, 2, 3. . . . .

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C.G. Callan et al. / String loop corrections to fl-functions

Now we consider the boundary action. The two relevant boundaries are r = 0, r = t. If we describe them clockwise, they will have opposite charges. Let us consider the ~--- t boundary alone, with a positive charge so o runs from 0 to 2rr. Then by (3.4) with ¢ ~ o, the boundary action for a constant field-strength is -i

S,=--/0

r2~r

4~ra'

v

0

doF~ (o,r ) ~o ~'( °' ,r).

(A.12)

Substituting (A.9) gives D

St= E F,~ ~ m~(t)fm(t ).

(A.13)

(,r = t is fixed on the boundary, so the argument could be dropped.) We are now ready to construct the boundary operator (A.7). First we notice that S t contained q scaled by (2~0)-1/2. Here oa = m, so rescaling each coordinate in (A.13) gives D i,, v = l m = l

Eq. (A.1) and therefore also (A.7) require that the coordinate Q be real. We therefore first associate creation operators Bm ~* and C~* with the coordinates ~ and X~m in (A.11), and then combine them by

A~,, = f~ ( B f -T-iC~*) .

(A.14)

The last term of (A.7) then becomes D

--½~.~ E

o0 [(Bm~t)2W(cm~t)2]=-E

m = l p.=l

D

~A~tA~-*m m = l /.t=l

-

(ArIA-t)

(1.15t

in vector notation. The complete operator (the product of (A.7) over all modes) in this notation is [,nI~I=1 (4¢rm)- n/2] f ~ b ~

e x p ( - ½ ( t~[F, ~b)-½( dT[~ )

+(Atl~)+(~IXt)-(AtlAt)}. (A.16) Since only creation operators are present, everything commutes. We do the gaussian

547

C G. Callan et al. / String loop corrections to fl-functions

functional integral, and evaluate the determinant by zeta-function regularization log

mI-J1

= limc

s o ds

1

(xm-D:) s

, 0) = ~ ( 0 ) l o g x + 51 D~(

=-½1ogx+....

(A.17)

There is one factor [det(1 + F)] -1 for each m, so we finally get

( D ~=1 ~, A~,(1-F] / (A.18) v(a*'F~")=[det(l + F)]'/2exp~,~=l£ rn~ 1+Flay A~* -mj" This displays how the left and right moving modes A~,, are coupled together by boundary reflections. For F~ = 0 it reduces to a formula of [22]. This fixes the numerical constant dropped in (A.17), since they derived their formula from open string unitarity. To see the physical significance of this operator, consider an annulus e - ' ~< Iz [ ~