Quantum consistency of open string theories

Volume 220, number 1,2 PHYSICS LETTERS B 30 March 1989

Q U A N T U M C O N S I S T E N C Y OF O P E N STRING T H E O R I E S

Jan GOVAERTS CERN, CH-1211 Geneva 23, Switzerland

Received 5 May 1988

We discuss how Virasoro anomalies in open string theories uniquely select the gauge group SO(2 °/2) independently of any regularisation, although the cancellation of these anomalies does not occur in tachyonic theories, and regulators can always be chosen to make these theories (one-loop) finite for any SO(n) and USp(n) gauge group. The discussion is mainly restricted to open bosonic strings. These results open new perspectives for the recent suggestion made by Sagnotti, the generalisations of which allow for the construction of new open string theories in less than ten dimensions.

1. Introduction

The type I open superstring theory is free o f any one-loop gauge anomalies for only one gauge group, namely SO (32) [ 1-5] . For that theory, any amplitude with massless external states can be made finite, provided the contributions from oriented and unoriented world-sheets are appropriately summed [ 5-11 ]. However, divergences in each of those classes of diagrams can be made to cancel against each other for other gauge groups as well, as for example SO(8) [7,8].

These results are not sufficient to establish the SO (32) superstring as a consistent quantum theory. All zero- norm states, and not only the longitudinal gauge bosons [ 1 ] and gravitons [ 12 ], must decouple from amplitudes, and any amplitude with physical external states should be finite (up to the problem of mass renormalisation [ 13 ] ).

The same questions can of course be raised in the context of open bosonic string theories. Using the same summation of oriented and unoriented diagrams which makes type I superstring amplitudes finite for SO (25), it has been shown [ 14 ] that the M-tachyon amplitude in open bosonic strings is finite for SO (213 ), and that the closed string dilaton tadpole vanishes for this gauge group as well [ 15,16 ]. Moreover, it is believed that all zero- norm states decouple from any amplitude [ 17,18 ]. This is indeed to be expected for the massless sector, since the corresponding effective field theory of bosonic fields is free of any gauge and gravitational anomalies. How- ever, this is not so clear for the massive sector. Indeed, decoupling of massive zero-norm states has been discussed in ref. [ 19], with the conclusion that it occurs for the gauge group 50(213) only. In this paper, we show how this group indeed plays a special role, but that decoupling is not realised due to tachyonic states in bosonic string theories. Our discussion differs with that of ref. [19 ] in that we carefully consider and regularise all sources of divergences relevant to this problem, a point neglected in ref. [ 19 ].

The appearance of the gauge groups SO (2 D/2 ) for open string theories must have a more profound origin than that o f merely numerical coincidences; the almost identical contributions from orientable and non-orientable diagrams is indeed very suggestive [20].

The underlying reason for the quantum consistency of open string theories with gauge group SO(2 D/2) is certainly to be found in the recent proposal [ 21 ] that these can be obtained from closed string theories by a Z2- orbifold construction in the wor ld-shee t (and not in the space in which the string degrees of freedom take their values [22 ] ). After all, this is the intuitive idea that an open string is "ha l f" a closed string, with left- and right- moving fields identified.

0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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Indeed, it has been claimed [ 21 ] that such a construction leads uniquely to the gauge group SO (2 D/2) [ 23]. As the concept of modular invariance gains its full applicability again in such an approach to open strings, it is to be expected that quantum consistency (i.e. finiteness and decoupling of zero-norm states) would simply follow from modular invariance, as it does for theories of closed strings [24,25]. In particular, it would determine [ 21 ] how oriented and unoriented diagrams have to be summed, thereby solving the problem of regularisation of these theories. We are thus led to conjecture that all consistent quantum theories of open strings are to be obtained through such a Z2-orbifold construction in the world-sheet.

In this paper, we discuss how quantum consistency of open bosonic strings selects the gauge group SO (2 j3) uniquely and independently of any regularisation. Together with the corresponding results for type I superstrings, this proves the conjecture for all known open string theories.

The approach taken in ref. [21 ] thus opens the way to many new developments. One obvious direction is in the construction of new open string theories (with or without supersymmetry) in space-time dimensions ranging from 10 to 4 (or less) [26 ], with possibly large gauge groups and chiral spectra of fermions. Open string model building becomes possible. Another direction is in the generalization of the approach to other orbifold constructions in the world-sheet (such as Z,). This clearly leads to the conjecture that all consistent quantum string theories can be obtained through such constructions. All string theories would then be theories of closed strings only, characterized by different orbifold structures in their respective world-sheets.

In this paper, the discussion is restricted to open bosonic strings (in 26 dimensions), although our consider- ations also apply to type I superstrings. The question of finiteness of (one-loop) amplitudes (with external open string states only) is first considered, with the conclusion that they can be made finite for any SO (n) or USp (n) gauge group by appropriate regularisations of oriented and unoriented diagrams. Moving on to the question of (one-loop) anomaly cancellation, we discuss how this uniquely selects the gauge group SO (2 ~ 3) [ 19 ] independently of any regularisation ambiguities, although the non-vanishing contributions of the open string tachyon make the theory inconsistent. Details of these results will be presented separately [27 ].

2. Finiteness

From ref. [28 ], we know that for every state

I ~ ; k ) = ~ C,~,...,,,,,a~',, ... a~';,,. 10 ;k ) , (1)

in the Hilbert space of the open bosonic string (we use the notations of ref. [29] with 2 a ' = 1 ), which satisfies the conditions

(Lo=½P2+N) l f ~ ; k ) = d l ~ ; k ) , L, i O ; k ) = 0 , n>~l, (2)

there exists, in one-to-one correspondence, a primary conformal field O(z) of weight A, such that

[ L , , , f ~ ( z ) ] = z " + J O - f ~ ( z ) + A ( n + l ) z ' f ~ ( z ) , [ ~ ; k ) = l i m ¢ ( z ) 1 0 ; 0 ) , (3a, 3b) z ~ 0

and given by

i 0 ( z ) = ~ C~,...,,,,,: f i ( n , - 1 ) !

j = l

where

X ~ ' ( z ) = q ~ ' - i P ~ ' l n z + i ~ a ~ z - ' . n = - - c~;

n v e O

- - O~:JX~'(z) exp [ ikX(z ) ] : , (4)

(5)

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Amplitudes of physical states (i.e. states which satisfy (2) with A = 1 ) are given by the correlation function of the corresponding fields, integrated over the positions of these vertices 0 (z) on the boundary of the appropriate Riemann surface. The factors O~JX~"(z) in (4) can be included into the vertex exp[ikX(z)] as exp [ ik, X u (z) + i~/u 0~' X" (z) ]; by taking the appropriate terms in an expansion in the auxiliary vectors ~/,, the corresponding amplitude is obtained. It is given by the correlation function of the vertices exp [ikX(z) ] multiplied by combinations of various derivatives of the correlation function of the scalar field X"(z) on the Riemann surface

Among physical states, one has zero-norm states, such as L ~ 10; k ) (with k2=0) , L_~e.a~110; k) (with k 2 - - - 2 , 82 - - 1, ak=0) or (L_2 3 2 + ~L _ ~ ) 10; k ) (with k 2 - - - 2 ) . For consistency of the theory, these states must decouple from amplitudes, i.e. amplitudes with external zero-norm states must vanish, and such states (as well as all other unphysical states) cannot contribute to pole and cut singularities of physical amplitudes. As is well known, the properties (3) ensure that this is indeed the case for tree amplitudes with arbitrary external physical states (see ref. [ 24 ] for a discussion in the case of tachyon amplitudes) [ 30 ].

Open string tree amplitudes have cyclicity properties (see ref. [ 24 ] for the case of tachyon amplitudes) [ 30 ], which allow for the introduction of gauge degrees of freedom through Chan-Paton factors [ 31 ]. Factorisation of tree amplitudes restricts the choice of these factors [ 32 ]; for oriented strings, one can only have n X n complex matrices, and for unoriented strings one can only have n × n real matrices or n × n quaternionic matrices. In each case, these Chan-Paton factors 2 , must be anti-hermitian for even mass levels and hermitian for odd mass levels [ 32 ]. These three possibilities lead respectively to gauge symmetries associated with U (n), SO (n) and USp (n). The contribution of these Chan-Paton factors to amplitudes is given by the trace of their product with the same cyclic ordering as that of the corresponding vertices on the boundaries of the world-sheet.

In the case of loop amplitudes, one has to make sure that none of the unphysical (negative-form) nor zero- norm states contributes to singularities due to intermediate states. In the operator approach used here, this can be implemented at the one-loop level through Feynman's tree theorem [ 33 ], and has been done so in the case of planar diagrams for bosonic [ 34 ] and supersymmetric [ 35 ] open string theories. Those results can easily be extended to non-planar and M6bius strip diagrams [ 30 ]; they determine the integration measure over the modular parameter of the world-sheet.

Through (3b), we choose to normalize the vertices 0 (z) for (positive-norm) physical states by requiring that the corresponding states 10; k ) have norm ( + 1 ) (with (k] k ' ) = ~ ( o ) ( k - k ' ) ) ; we also impose that tr 22 = _+ 1. For a process with M external states of momenta k~'(i = 1 ..... M), the S-matrix element is given by

M

S(k, , ..., k i ) = 1-I (2zt)-tD-l)/2(2k°)-'/2A(k~, "", k i ) , (6) i = 1

where A (k~ ..... kM) is the corresponding amplitude. At one-loop, we then have [ 30 ], for orientable (or planar and non-planar) diagrams

I

f z A~o')(k, ..... kM)=--ig M Co I ~ - w J (w) ,=,fi d z ' [ z " [ f i ( z " - z ° )TrR Oi( ~) j , (7) 0 ~ ( w )

and for non-orientable (or M6bius strip ) diagrams, A ~J ) (k~ ..... kM) is given as in ( 7 ) with the factors Co, f 2 (w) and w L° replaced by Cu, f 2 ( -- w) and wE°( -- 1 )N, respectively. We have

f ( w ) = f i ( l - w " ) , g 2 ( w ) = [ - 1 , - w ] w [ w , l l , (8a) n = l

and

Co=l , CN=O for U(n) , (8b)

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C o = ½ , CN=½ f o r S O ( n ) , (8b)

Co =1 , CN = - 1 f o r U S p ( n ) . ( 8 c )

In (7) , g is the open string coupling constant, R stands for the R-ordered product, ~i (zi) are the vertices corresponding to the external physical open string states, the trace is taken over the oscillators a , u, the zero-mode P~' and the C h a n - P a t o n factors (associated to each ~ ( z ) ) , and the factor I za I ~ (za - z °) (with za one of the z~ and z ° an arbitrary point in ~ (w) ) is the gauge-fixing factor for the (real) conformal Killing vector of the cylinder and the M6bius strip.

Eq. (7) represents an integration over the two boundaries [ - 1, - w ] and [w, 1 ] o f a cylinder defined by (w-4< I zl ~< 1, Im z >/0 ) and the equivalence (z.~ wz). The analogous expression for unoriented diagrams represents an integration over the single boundary g7 (w) of a M6bius strip defined by ( w ~ I z I ~ 1, Im z/> 0) and the equivalence (z~ - wz). The factors f 2 ( _+ w) follow from Feynman ' s tree theorem [ 34,30 ]; they lead to the correct part i t ion functions of these theories. The values for Co and CN agree with those given in ref. [ 14 ] in the case of the M-tachyon amplitude; in particular, the relative sign between planar and M6bius diagrams for SO (n) and USp (n) bosonic strings is opposite to that o f the corresponding type I superstrings [24 ]. This is important for our discussion.

The evaluation of the ampli tudes defined above leads to three different types of contributions. The planar one, with all external vertices at tached to the same boundary of the cylinder, is given as a sum over all cyclically inequivalent permuta t ions of the external states with each term of the form

1

igMO(26) (t~=l ki)Co n tr(]-L~(i)) f d~ l 2g i=l

0 R~

X 1-[ (0'(~'~ l',lln(l')lTi")]k'~' , < , < j < M ",, ~,, t,, 7 n - ~ / 2 - - ~ ) ) [N, derivatives in -0/0v,]. (9)

Here, 0~ ( ls lr ) (and 04(lsl~) below) are the Jacobi 0-functions, Ni is the excitation level of the ith external state with ½ k 2 + N, = l, and the integration region R~ is defined by 0 ~< UM ~< UM_ ~ ~< ... ~< U~ ~< 1 (corresponding to the cyclic ordering of the vertices on the boundary) . The gauge-fixing ~(i 'M) in terms of i'M could also be done in terms of any other u~, with the appropriate orderings. Finally, all terms which involve derivatives are combinations of products of derivatives of the corresponding propagator In 01. We also set D = 26.

The non-orientable contr ibution is given as a sum over all cyclically inequivalent permutat ions of the external lines, with each term given as in (9) with the following changes: Co becomes CN, the factor n becomes 213, f - 24 (2) becomes f - 24 ( _ 2 ) and the argument In (2) / 2 ire o f the 0-functions becomes In ( - 2 ) / 2i st.

The non-planar contribution, with the external vertices split into two sets A and B corresponding to the two boundaries of the cylinder, is given as a sum over all such splits, with cyclic permutat ions of A and B or within A or within B not counted as different splits, with each te rm given as in (9) with the following changes: the factor n tr(.~,.~<~N ~ u),~ becomes tr(l-[~A2~ ~)) t r ( l J~B2~ j)), 01 ( u,-- ujl ~) is to be replaced by 01 ( u~- ujl~) or 04( u i - ujl r) depending on whether the vertices ¢~ and ~j are attached to the same boundary or to different boundaries, and the integration region R~ is replaced by the integration region R 2 defined by 0~< uA, ~ ... ~< U A ~< 1 and 0 ~< UBM, ~< ... ~< U~ ~< 1 with U a and u~ corresponding to the vertices in the sets A and B defined above.

In obtaining these results, for oriented diagrams we used the change of variables

2 = e x p ( 4 n 2 / l n w) , u i = l n ( z i ) / l n w, (10a)

and for unoriented diagrams

2 = e x p ( n Z / l n w ) , u ~ = l n ( z , ) / 2 1 n w . (10a)

Let us emphasize that these results are valid for any one-loop ampli tude with any external physical open string

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states. The notorious thing about the planar and non-orientable contributions is that they only differ by a factor ( CN Cff t 2 t 3 ) and the replacement (2--, - ).) in the integrands. The origin of the factor 213 can be traced back to the integration over the bosonic zero modes (or to the transformation relating f ( + w) to f ( + 2) ). In the case of open superstrings, the expressions are essentially the same (the factors 2 - 1 f - 24( + 2) simply do not appear) and the analogous factor 25 (which has been established for amplitudes with M massless gauge bosons only [2,7,9,10] ) can be traced back to the fermionic zero modes. The discussion which follows thus also applies to those theories.

The amplitudes above have divergences of different physical meaning and origin. We are not concerned here with those which arise when some vertices come close to each other; they correspond to singularities due to intermediate states in different channels [ 13,24 ].

A divergence common to all three types of diagrams comes from the region 2--. 1 ( w ~ 0 ) . This divergence is due to the tachyon of the open string, but it does not possess any physical meaning; it is simply a consequence of an inappropriate integral representation of the vacuum loop for such a state [5,19,27 ]. It can be shown [27] that the correct representation is obtained by inserting a factor w ~ in the integration over w, where 7 is an arbitrary parameter which may be different for all three types of diagrams. All integrals must be evaluated with 7 >/1, and only in thefinal result one can take the analytic continuation to 7=0. Such a regularisation does not break any of the symmetries of the theory. All amplitudes are still invariant under world-sheet reparametrisations induced by the Virasoro generators, since the vertex operators are primary conformal fields of conformal weight d = 1, even for 7# 0. Introducing a factor w;' does not amount to a shift in the mass spectrum, since the condition Lo= 1 is still imposed on physical states.

All three types of contributions also have divergent behaviours as 2 ~ 0 ( w ~ 1 ), with a well-known physical meaning; they correspond to closed string intermediate states [24,36]. For the non-planar diagrams, the integrand has the form I

4 n 2 7 f ~ exP( l~)(27r21/8)'ABG(2 ) , ( l l a ) o

where G(2) is an analytic function of 121 < 1, with

G ( 2 ) = ~ G,,2", ( l l b ) n = 0

and SAB = (5~i~ g ki)2= (Y~i~ a k~)2 is the invariant energy exchanged between the two boundaries. Eq. ( 11 ) is to be evaluated by analytic continuation in SAB, with SAB> 8 [24,36 ]. This leads to

G, [16nZT(~Sga+n_l)]1/2Kl([16nZT(~SaB+n_l)l,/2) (12) iim ( 2 n y A" ~Saa+n--1 ;,~0 n = 0

where K~ (z) is a Bessel function with

(½z)2~k+'~ zK,(z)=l+2 L ~(~-(~! [ln½z-½~t(k+l)-½~t(k+2)]. (13) k = 0

Eq. ( 12 ) has poles at SAB = 8 ( n - 1 ) (n = 0, 1 .... ), corresponding to closed string states [ 36 ]. The limit 7- ' 0 can only be taken after the series has been resummed; it cannot be taken in each term separately as the corresponding quantity would diverge. Hence, ( 12 ) has logarithmic branch cuts for ( ~SAa + n-- 1 ) < 0; this is clearly due to the open string tachyon, and is simply a consequence of the non-unitarity of tachyonic quantum theories.

For planar and non-orientable diagrams, the divergences at ,l--.0 follow from tadpoles of closed string states at zero momentum. They can also be regulated by analytic continuation in a fictitious momentum inserted in the tadpole [ 37,14]. For the planar (respectively M6bius strip) diagrams, we introduce a function a (t~) (respectively/3(t2) ) of some parameter t~ (respectively t2) such that

81


l i m a ( t ~ ) = 0 , l i m f l ( t 2 ) = 0 . (14) 11~0 12 ~ 0

The integrands for planar and non-orientable diagrams then take the form, respectively, I 1

exp(4,~--2~l~2"(")F(2), f exp ln2 2 F ( - 2 ) , (15a, 15b) 0 0

where the same analytic function F(2) of 121 < 1 appears in both terms (for a fixed cyclic ordering of the vertices on the boundary), with

F ( 2 ) = ~ F,2". (16) n = 0

In (15), Yl and 72 are the regulators for the open string tachyon, and it is assumed that the parameters t~ and t2 are such that c~ (tj) > 0 and fl( t2 ) > 0. The evaluation of (15a) and (15b) gives, respectively,

o~

lim ~ F, {16uzy,[o~(t~)+n_l]},/ZK~({16u2yl[a(t~)+n_l]}W2) (17a) :,,~o ~=o a(t~) + n - 1 t l ~ 0

( - 1)'F,, lim Z f l ~ ) ~_---~_ 1 {4u2y2[fl(t2) + n - 1]} '/2K, ({4n2y2 [ f l ( t 2 ) + n - 1]}1/2). (lVb) } ' 2 ~ 0 ~ l=O t 2 4 0

Here again, the limits y~ ~0, yz~0 cannot be taken inside the summations and logarithmic branch cuts appear in the contribution of the closed string tachyon (n = 0) due to the open string tachyon.

The quantities (17) are finite as t~ ~0 , t2~0, except for the dilaton tadpole (n = 1 ). However, these divergences can be made to cancel between planar and non-orientable diagrams for unoriented open strings of gauge group SO(n) or USp(n) , if we choose a(t~ ) and fl(tz) such that

lim a(t,)/fl(t2)=(Co/C:c)2-'3n. (18) 11,/2~0

The same clearly holds for type I superstrings; the change in the relativistic sign of Co and CN is compensated by the absence of the divergence due to the closed string tachyon (i.e. we then have f~ d2 2 - IF( + 2) ). By fully using the freedom in regularisation by analytic continuation, any open string theory with gauge group SO (n) or USp(n) can be made (one-loop) finite (up to the problem of mass renormalisation). In particular, choosing O~(tj ) = fl(t2 ) (which seems natural [20] and is supported by the approach of ref. [21 ] ), we have finiteness for 8 0 ( 2 D/2 ) only [6,14], and choosing 4c~(tl ) = f l ( t 2 ) we have finiteness for 8 0 ( 2 D/2-2) only [7,8].

The regularisation which is the most often used in the evaluation of open superstring anomalies is a Pauli- Villars regularisation [ 1,2 ]. Because of the closed string tadpole in (15), this cannot be used for open bosonic strings [27]. However, Pauli-Villars regularisation raises a conceptual problem; in the evaluation [34,35,30] of the integration measure over w through Feynman's tree theorem, it is essential that no additional factors of w'"-' are introduced (and that vertex insertions can coincide). Otherwise, one gets additional surface terms necessary to compensate the unphysical cuts in one-loop diagrams. When factors such as w ~ are introduced (which is necessary only for theories with tachyons), this does not lead to problems since the additional terms are multiplied by 7 and vanish as ~,~0 [27 ]. However, when factors such as w m: are introduced, it is not clear what happens to the unphysical cuts as m2--.~. On the other hand, the regularisations used in the calculations above are applicable to any string theory. They avoid this conceptual problem and follow from a simple physical interpretation in terms of the proper-time representation of propagators [27 ].

In conclusion, by considering the problem of finiteness only, we are still left with many possibly consistent quantum open string theories, with any of the gauge groups SO (n) or USp (n). However, decoupling of zero-

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norm states is still to be implemented; it leads to a restriction on the acceptable gauge groups [ 19 ], as did the requirement of factorisation at tree-level [ 32 ].

3. A n o m a l y c a n c e l l a t i o n

As said before, no restriction is to be expected from the decoupling of massless zero-norm states, namely L_ j I 0; k ) (with k 2 = 0 ). Indeed, the corresponding vertex is Oz: exp [ ikX(z) ] :, and any physical one-loop amplitude with such a state then vanishes, by an appropriate analytic continuation in the variables kflj. Similarly, the massive zero-norm vector state L_lGaLllO;k) (with k2---2, ~2=1, gk=0), with the vertex O- (:iGO-X'(z) exp [ikX(z) ] : ), decouples from any physical one-loop amplitude. We expect the same result to hold for any zero-norm state of the form L_ l lzJ; k ) .

The first non trivial constraint is found for the zero-norm state (L_2 + 3 2 k 2 = ~L l)[0, k ) (with - 2 ) [19 ] . I t s vertex is

O(z) = ~: ik , O~X'(z) exp[ikX(z) ]: + ½ (q~,~ + 3k, k~):iOzXU(z)iO=X~(z) exp[ikX(z) ]: . (19)

Here, we only consider one-loop ampli tudes with one such zero-norm state and ( M - 1 ) arbitrary physical open string states (some of which may be of zero-norm as well). However, we expect [ 27 ] our conclusions to be valid for any zero-norm states since such states can always be expressed as (L_ t [X~ ; k ) +L_2 [Z2 ; k ) ) by using the Virasoro algebra.

From the structure of (19) , it can be shown [ 27 ] that the ampli tude (7) reduces to 1

A(ol)(k, ..... k ~ ) = _ i g i l co½ f dw --W W 0

d z "w L°'~-] X ~ w f 2 ( w ) , = , f i d z ' l z M l ~ ( z M - z ° ) Z r R :exp[ikiX(zi)]: ~=~l--[ 0 , ( i ) , }d ' (20) .Q(w)

where 0~(zj) ( i = l, ..., M - 1 ) are the vertices corresponding to the ( M - 1 ) arbitrary external states, and k 2 = - 2, ZM correspond to the zero-norm state in ( 19 ). For non-orientable diagrams, A ~ ) (kl , ..., kM) is given as in (20) with the factors Co, f 2 ( w ) and w L° replaced by CN, f=(--w) and wL°( - 1) u respectively. Additional surface terms in z~(i= 1 ..... M - 1 ) also appear in the evaluation of (20); they do not contribute by appropriate analytic continuations in kflj.

The ampli tude is thus given by a total surface term in modular space, a result also obtained in ref. [19] following a different approach and to be expected on general grounds [ 17,18 ]. This is also the case for the gauge anomaly in type I superstrings [ 5 ].

Performing the change of variables given in (10) , the ampli tude reduces to sums over different cyclic orderings of the vertices, as discussed above. For planar contributions, one obtains:

I ( M ) f (47[.2~)1) ( 2 4 ) n t r (E i2~ i ) ) d) '2"(") -47r ). _ igM6(26) ~, k,. Co exp

o

[1 f i (Ol(V,_vjlln(2)/2iTO) k'g' X f-24(~.) I~ dP,t~(PM) H O'j(O] ln(2)/2i~z) i= I 1 <~i<j<~M RI

q x (iV,. derivatives in - 0 / 0 u t , i = 1 ..... M - 1 ) J , (21)

with the necessary regularisation factors already included, and the appropriate analytic continuations to be

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performed as explained above. For non-orientable contributions, the factors Co, n, f -24 (2) and the arguments In (2) /2in of the 0-functions are replaced by CN, 2 t3 , f - 24 ( _ 2 ) and In ( - 2 ) /2in respectively, and the regulators o~(t~ ) and 7t by fl(t2) and Y2. For non-planar contributions, the appropriate Chan-Paton factors and the 0- functions 0~ ( v i i ) and 04(vi i ) are to be used as explained before, and only the regulator exp (4n2~/ln 2) is necessary.

For non-planar amplitudes, the integrand has the form 1

cl~ 4n27 2 d 1 f--~-exp( 1 - ~ ) ( _ 4n 2~)(-~(2n21IS)sABG(2)), (22a) 0

with G(2) an analytic function of 121 < 1

G ( 2 ) = ~ G,2" , (22b) n = O

and SAB defined before. If there were no tachyon in the open string channel, this expression would simply be a total divergence and vanish by analytic continuation in SAB (the contribution at 2 = 1 or w = 0 can be shown to vanish in that case). This is the situation in type I superstrings [ 1,24]. Here however, we obtain for (22)

lim (-47r2)(2/r) ;AB ~ G.[167C2y(~SAB+n--1)]I12KI([161r2y(~SAB+n--I)]I/2). (23) 7 ~ 0 n = 0

This contribution is entirely due to the open string tachyon. It has logarithmic branch cuts and does not vanish. Tachyonic string theories are not only inconsistent because of the unphysical character of their amplitudes, but, in addition, zero-norm states do not decouple in such theories!

Considering the planar and non-orientable contributions, we have respectively for the corresponding integrands: l

d 1 d2exp(4n2~Y~2.(',)(_4n22-~)(~F(2)) (24a) f2- k ln,t /-,, \ 0

1 2 d 1 d~exp(~l@~)2~(r2)(-4n22-~)(~F(-2)) , (24b)

0

with F(2) the same analytic function of 12 [ < 1 in both expressions

F(2)= ~ ~,2". (25) 1 l=0

The evaluation of (24a) and (24b) gives, respectively,

4n2F,+ lira (-4n 2) ~ F.[16n2yl(n-1)li/2K,([16n2yl(n-1)] ~/2), Y l ~ O n = O

(26a)

-4n2F,+ lim ( - 4 n 2) ~ (-1)nF.[4n272(n -1)]j12Kl([4nzy2(n -1)]'/2). y 2 ~ O n = 0

(26b)

The second terms in those results are entirely due to the open string tachyon, as in (23). Note their logarithmic branch cut for n = 0. On the other hand, the first terms, proportional to F~, correspond to a genuine anomaly, independent of the existence of tachyons and of any of the regulators. Summing planar and non-orientable diagrams, and ignoring for a moment the contributions from the open string tachyon, the total amplitude with the zero-norm state ( 19 ) is given by

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--igM(~(26)(i~=lki)27~fo(Yl--213-~o)cy~clictr(I~Ii ~(ai))Fi , (27)

where ~cyclic stands for the summation over all cyclically inequivalent permutations of the M external states. The cancellation of this quantity uniquely selects the gauge group SO (213). This conclusion was reached in ref. [ 19], although the relative signs of Co and CN quoted there for SO (n) and USp (n) gauge groups are not the correct ones for bosonic strings. Moreover, the fact that zero-norm states cannot decouple due to the open string tachyon was also overlooked in ref. [ 19 ].

From (20) or (21 ), one sees that the quantity (27) can be interpreted as a tadpole of an unphysical closed string state in the unphysical one-loop amplitude of an off mass-shell open string '"tachyon" (with k~ = - 2 ) and the ( M - 1 ) physical open string states corresponding to the vertices ~i(zi) ( i= 1 ..... M - 1 ).

The same analysis is applicable to type I superstrings. The change in the relative sign of Co and CN is again compensated by the fact that the contribution to the anomaly is now given by Fo, since there is no tachyon in the closed string sector (which would be responsible for a factor 2-~ in the total derivative, as in (24) ). Clearly, the group SO(25) is then uniquely selected, as is well known by considering the one-loop amplitude with six massless vectors [ 1-5 ]. In this case, the same interpretation in terms of a tadpole of an unphysical closed string state has been given for the gauge anomaly [ 5]. However, since the anomaly is related to Fo, it can also be expressed in this case as an unphysical tree-level amplitude on the disk, as is usually done [ 1,24]. The interpretation in terms of a tadpole seems more general and useful.

4. Conclusions

We discussed how one-loop decoupling of zero-norm states in open string theories selects the gauge group SO (2 °/2) uniquely and independently of the regulators, even though decoupling does not occur for tachyonic theories whereas finiteness (of one-loop amplitudes with external open string states only) can be achieved for any SO (n) or USp (n) gauge group by appropriate regularisation. Although finiteness may also result from a dilaton background field [ 38,5,11 ], such a mechanism cannot lead to the cancellation of anomalies since they correspond to tadpoles ofunphysical closed string states [ 5 ]. We also saw how tachyonic string theories not only violate unitarity, but also break the gauge symmetries associated to the Virasoro algebra, i.e. to world-sheet diffeomorphisms, since the corresponding anomalies then do not cancel even for SO (2 t 3 ).

These results show that the recent suggestion [ 21 ] to regard SO (2 °/2) open string theories as closed string theories with a Z2-orbifold construction in the world-sheet actually leads to all the (possibly) consistent quantum theories of open strings which are known (ignoring problems related to tachyonic states). This proposal thus provides the natural setting for studying the full quantum consistency of open string theories, i.e. finiteness (up to the problem of mass renormalisation) and decoupling of all zero-norm states (not only those of the open string sector); those properties should follow from modular invariance as they do in theories of closed strings only [24,25].

Clearly, the ideas of ref. [ 21 ] can be generalised and used in the construction of many new consistent open string theories (with or without supersymmetry) in dimensions ranging from 10 to 4, and characterized by different orbifold structures in their world-sheet. Work in these directions is in progress.

Acknowledgement

It is a pleasure to thank G. Aldazabal and T. Kubota for useful discussions, and P. West for his interest in this work.

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Quantum consistency of open string theories

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