On a nonperturbative vacuum for the open bosonic string

Nuclear Physics B336 (1990) 263-296 North-Holland

ON A NONPERTURBATIVE VACUUM FOR THE OPEN BOSONIC STRING V . Alan KOSTELECKY* Physics Department, Indiana University, Bloomington, IN 47405, USA Stuart SAMUEL** Physics Department, The City College of New York, New York, NY 10031, USA Received 3 November 1989

In the context of covariant string field theory, we investigate the existence and properties of nonperturbative vacua for the open bosonic string. The study is performed using successive orders in a level-truncation method . We find a candidate nonperturbative vacuum and obtain numerical values for nonzero expectation values of all fields to level two. Perturbation theory about this vacuum is shown to be well defined. The occurrence and location of poles in the propagators of all fields to level two are established. We observe that string theories necessarily have running couplings at tree level and that the open bosonic string is asymptotically free . The nontrivial momentum dependence means that certain poles seen in the canonical vacuum are absent in the nonperturbative vacuum . This result follows from the nature of the string as an extended object and is likely to hold for general string theories .

1. Introduction Superstring theory [1] has many features rendering it attractive as a framework for a fundamental description of our universe. In particular, it incorporates a quantum theory of gravity that is likely to be finite . Furthermore, the lowest excitations include states exhibiting similarities to known particles, and string interactions reproduce couplings among the states that are those of gauge theories and gravity. The gauge groups are anomaly-free [2] and are large enough to contain SU(3) X SU(2) X U(1) . Despite these attractive features, superstrings or heterotic strings [3] in canonical ten-dimensional vacua are unsuitable as explicit models of the universe. The dimensionality of space-time is too large, the perturbative spectrum is not realistic, * Bitnet address : KOSTELEC at IUBACS. ** Bitnet address : SAMUEL at CCNYSCI . 0550-3213/90/$03 .50CElsevier Science Publishers B .V . (North-Holland)

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and the gauge groups are too large. To make contact with nature, the large gauge group must be broken and six of the ten dimensions must compactify into a miniscule internal manifold, perhaps a Calabi-Yau manifold [4]. It is natural to ask whether the canonical vacuum is unstable, in which case the string theory might exist in a different vacuum. If this is so, both the gauge-group breaking and the compactification could be spontaneous. One might hope that the new vacuum retains the attractive features of the canonical vacuum while eliminating the undesirable ones . Evidently, it is important to understand vacuum structure in string theories. In particle physics, a useful approach to the vacuum structure of a model is through field theory and functional-integral methods. These often efficiently determine relevant candidate vacua through the effective potential. The true vacuum is found as an appropriate extremum of the effective potential. In particle theories, other vacua may decay by barrier penetration into the true vacuum [5,6]. A covariant field theory for the open bosonic string was achieved by Witten [7] and developed and studied in refs. [8-35]. The reformulation of the theory in a Fock-space representation, i .e. in terms of particle fields, is now known. The perturbative analysis of the theory about the canonical vacuum is well established. It reproduces the Koba-Nielsen tree-level amplitudes and provides their off-shell extensions. Similarly, the standard N-point one-loop amplitudes have been obtained from the theory, along with their off-shell extensions. Some results also exist for higher-loop amplitudes . In this paper, we use covariant string field theory to explore the vacuum structure of the open bosonic string. This string theory is a good candidate for such an endeavor not only because there exists a covariant field theory with calculable couplings, but also because the canonical 26-dimensional vacuum of this theory is unstable. The spectrum includes a tachyon . These particles have negative mass squared, so it is energetically favorable to pair produce them. Such situations are not uncommon in particle physics; for example, the electroweak model in the naive vacuum has scalar fields with negative mass squared. There, the instability is resolved by the observation that there exists another vacuum, which is stable and in which the scalars acquire vacuum expectation values . Similarly, the presence of the tachyon in the open bosonic string may merely signal an incorrect choice of vacuum . The string interactions may stabilize the theory in a different vacuum, in which the tachyon field 0 would acquire an expectation value. In an effort to address this possibility, the lowest-order nontrivial interaction in the static tachyon potential was calculated [34] to see if any signal of stability could be discerned. This calculation yields the coefficient of the 04 term exactly at tree level, i.e. the effects of all mass levels are included . The sign and magnitude of the coefficient do not suggest stability. However, the same work proved that all higher-power 0 interactions were as important as the 04 one. Thus, the question of stability is inherently nonperturbative .

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One of our goals here is to obtain nonperturbative information about the static tachyon potential in an effort to determine whether there exists a nonperturbative stable vacuum in the theory. We use a level-truncation scheme to obtain the dominant tree-level contributions to the static tachyon potential at all orders . The calculation involves truncating the Fock-space expansion of the string field ~P to the lowest-lying states . Intuitively, one expects light states to dominate most physical processes. This is not always true ; for example, Regge behavior in the Veneziano amplitude [36] is lost in Fock-space truncation. One important point in favor of the validity of our procedure is that the couplings among three states decrease exponentially with the masses of the states . Details of arguments concerning convergence and regions of validity are discussed in subsect. 2.2. Some notation and the necessary background for the calculations are presented in subsect. 2.1. In subsect . 2.2 we establish the methodology of level truncation. We also test our scheme against the exact tree-level result for the coefficient of ~° in the static tachyon potential expanded about the canonical vacuum [34]. We find that states to mass-level two account for 72% of the exact result . Including two more levels generates 84% of the total. In sect . 3, the structure of the effective static tachyon potential is investigated in varying orders in the level-truncation method . With only the tachyon field included, we find a locally stable vacuum with (0) = 0.91/ga', where a' is the Regge slope and g is the on-shell three-tachyon coupling at tree level. The candidate vacuum remains as more levels are included, with the value of (,p) increasing first to 1 .083/ga' and then to 1 .088/ga' . In this nonperturbative vacuum, other scalar fields acquire expectation values, which we determine. Sect. 4 analyzes some physics in the nonperturbative vacuum. The possibility of tensor-induced spontaneous Lorentz-symmetry breaking [35,37,381 is examined . We show that perturbation theory is well defined and we consider the question of stability. Level mixing is discussed and treated, and the spectrum of the lowest-lying states is computed for different truncation orders . We find that the spectrum changes only by small amounts when higher-order corrections are incorporated. The momentum dispersion relation, as given by the propagator, is graphed for several states . We observe that, as a consequence of the extended nature of strings, the string coupling runs even at tree level. We find that momentum-dependent couplings can have dramatic effects on the propagators and on the spectrum when compared to the free theory . Many propagating states of the perturbative vacuum are removed. Sect. 5 provides a summary and a discussion. In appendix A we present our conventions for a Fock-space expansion of the bosonic string field. Appendix B contains the full lagrangian density up to terms trilinear in level-two fields . The decomposition of level-one and level-two fields into irreducible multiplets of the Lorentz group is described in appendix C. Appendix D contains the inverse propagator for the scalar fields in the nonperturbative vacuum.

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We follow the notational conventions of ref . [39]. In particular, the metric qu, in Minkowski space is the matrix with diagonal entries (-1,1, . . . ,1), and the representation of the momentum operator pu as a differential operator is -i d/dx". 2. The level-truncation method 2.1 . BACKGROUND

Our analysis is performed in the framework of Witten's covariant string field theory [7] . This may be viewed in terms of an analogy with differential forms. The string field * is like a matrix-valued one-form . The BRST operator [40] Q is the analog of the exterior derivative . The other ingredients in the theory are the star product * and the string integral f . They correspond respectively to the wedge product of forms and to the integral of forms over a manifold times the trace in matrix space. The action is analogous to a Chern-Simons form and is S

1 ( ( 2a,JP*Q'Y+9J~*~ *q .

The normalization of the action (2.1) is chosen to yield conventionally normalized kinetic-energy terms. The interaction in eq. (2.1) is characterized as follows. Divide the string, parametrized by a, a E [0, it], into two at the midpoint, a = 7T/2. Let the left half be a E [0, 7/2] and the right half be a E [7r/2, v]. The interaction involves three strings. The right half of the first string is joined to the left half of the second, the right half of the second string is joined to the left half of the third, and the right half of the third string is joined to the left half of the first . The vibrational modes of the string are the particle states . The lowest vibrational modes represent the lightest particles . The field T has an expansion in this basis known as the Fock-space representation : T is a linear combination of ordinary particle fields whose coefficients are the solutions of the first-quantized theory [10-17] . Our conventions for the first few terms in the expansion of T are given in appendix A. When this representation of ~P is substituted into eq. (2.1), the theory in terms of particle fields is obtained. For any comparisons with nature, a particlefield interpretation of string field theory is needed since experimentally we observe particles, not strings. The action (2.1) possesses gauge invariances of the form

2a'

g( ~P *A-A * f) .

(2 .2)

Gauge fixing is necessary to render the functional integral well defined . We choose

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the Siegel-Feynman gauge [41] : bo~P = 0 .

(2 .3)

Quantization in this gauge has been developed in refs . [26,27]. At tree level, the string field theory contains two basic types of particle fields . First, there are the physical states of the first-quantized approach. Second, there are auxiliary fields introduced to render the theory covariant and local [42-46] . Some of the latter are ghosts in the sense that their free kinetic-energy terms, i.e. terms in the action quadratic in the fields, have the opposite sign to physical fields *. These ghosts serve to cancel unphysical time components of tensor fields in tree amplitudes. The presence of ghosts makes it important to draw physical conclusions only

from physical quantities . The gauge condition (2 .3) sets to zero terms in the expansion of * containing c o . In this gauge, the quadratic piece of eq. (2 .1) in terms of particle fields to level two is

2a'

Q*

=z (d26XIduodfLo+d11A,dAA " +dlB d"B'+d ;, BAvd~Bt"' -d~ißt d~/3 t

+

1

at

( - 02+B,,B"+B,~,B"° -#i)I .

(2 .4)

The cubic piece of eq . (2 .1) in terms of particle fields to level two is given in appendix B. Note that particle fields f in the interaction lagrangian Yi .t = Enyt"> enter everywhere as fields smeared over a distance a'

f= exp[ a' ln(3 V-3/4) d. d AI

f.

(2 .5)

2 .2 . METHODOLOGY

As mentioned in sect. 1, the basic idea is to truncate the Fock-space expansion of the string field ~P . There are several possible types of level truncation : one can truncate the expansion in ~P at a certain point and use all lagrangian terms involving the remaining fields, or one can truncate in the level number n of the interaction lagrangian Pint = E"Y(") . In this work we do a combination of both . We first truncate in * and then consider successive truncations in ylnt . There is a reason why level truncation might be a good approach . It is possible to 21 ( n ) are proportional to a factor of (413r3)"= show that the couplings in _ * The word ghost is used in three different contexts : there are the Faddeev-Popov ghosts b(a,T) and c(a, T) of the first-quantized formalism, there are the ghost auxiliary fields that we are now discussing, and there are Faddeev-Popov ghosts of the second-quantized formalism that enter in loops .

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268

exp(-0 .26 n), i.e . they decrease exponentially as n increases. If 4/3C = 0 .77 were a small number then level truncation would be an excellent approximation . Given the actual value = 0.77, the validity of the truncation is harder to judge but it can be discerned by examining higher-order corrections to lower-order results. Numerically, we find good convergence properties . As a test of the level-truncation method we calculate in successive orders the tree-level contribution to the static tachyon potential and compare to the known

e

exact result. The idea of approximating a four-point amplitude by saturating it with low-lying intermediate states dates to the days of dual resonance models : Mandelstam used it to check his Neumann-function methods [47] . It has been used repeatedly since. Ref. [48] showed that saturation with a few low-lying states is particularly good in covariant string field theory . The exact contribution in order $4 to the static tachyon potential is [34] Vstatic - -

1 2,ar

2

9/2 3 04 , + + 4~ 26 3!3 g

(2 .6)

where X = iâg 2 (-11 .2 ± 0.1) .

(2 .7)

This exact result involves the complete sum over intermediate states . In the four-point amplitude, the contributions of odd-level fields cancel between s- and t-channel diagrams so that only even levels in the truncation need be considered . Furthermore, as we seek the static tachyon potential we can disregard the momentum dependence in the interaction lagrangian . Using the interaction terms given in appendix B to incorporate fields up to level two in saturating the intermediate line yields 72% of the exact result . To include interactions involving fields up to level four, we use the additional quadratic mass and cubic interaction terms determined from eq. (2 .1) as 'a =

,(Dlg v D1v+D2PP D2° +DWvPa D PvP' -28183 - 2 - m3Pm3) 2a 25 13 52 33F P w P~ + g Dlw + 34~ D2w + 2 . 34~ Dwv 2~ 3 3 ( +

245 34

bl +

19

245 5-11 8 2 + 35 s3 + ,~P2 . 34 35 m 3PP l

(2 .8)

Other level-four fields do not contribute to the static four-point tachyon amplitude . Saturating the intermediate line produces an additional 12% of the exact result .

V.A . Kostelecky and S. Samuel / Open hosonic string

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TABLE 1

The contribution of the level-two and level-four fields to the four-point static tachyon amplitude. Level

Field

Contribution

% of Exact result

Level 2

B,,,, ßi

-6.43 a 'g2 2.39 a'g2

Level 4

Dl, D2~,

-0.72 a,g 2 -0.18 a'g 2 -0.51 a'g 2 -0.12 a'g 2 0.46 a'g 2 0.03 ag2 0.36 a'g 2

114.8% -42.7% Total level 2 : 72 .0%

S_= 8+ -

Dw, n ,

(si + 83)/r (si - S3)/~F2 82 m3pp

12.9% 3 .2% 9.2% 2.1% -8.3% -0.5% -6.4% Total level 4 : 12 .2% Total levels 2 and 4 : 84.2%

Table 1 displays the contributions of each field to A/(a'g 2) in absolute terms and as a percentage of eq . (2 .7). The total contribution to the static four-point tachyon amplitude to level-four fields is thus 840. We consider this as evidence that the level-truncation method is converging. For many quantities computed in sects. 3 and 4, numerical results indicate even better convergence.

3. The nonperturbative vacuum The sign in eq . (2 .7) means that the theory, if truncated to 04 , has no stable vacuum . There are no local or global minima of the static effective tachyon potential (see fig. 3b of ref. [34]). However, it can be shown [34) that 0" terms for arbitrary n are equally important. This suggests that one should try to include n-point contributions. Presently, this is too difficult to do exactly. Instead, we find these terms in the level-truncation method . The approach is systematic ; by including more and more levels a better picture of the potential is achieved . The effective tachyon potential is generated by integrating over all fields except in the functional integral. This produces a complicated nonlocal action for 0. For vacuum structure one is interested in translationally invariant states, i.e . states with zero momentum . Equivalently, one sets derivatives of 0 equal to zero . Although a multiscalar effective potential can be defined, that is, an effective potential function of all the scalars, it is useful to focus on the tachyon since it is the source of the perturbative vacuum instability . In addition, any minimum of the multiscalar potential necessarily must be a minimum of the tachyon potential. Fields integrated out of the functional integral become functions of 0. If 0 achieves

27 0

V.A . Kostelecky and S. Samuel / Open bosonic string

a nonzero vacuum expectation value (4)), the other fields gain vacuum expectation values determined by substituting (0) for -p in the equations of motion. 3 .1 . TRUNCATION TO LEVEL-ZERO FIELDS

The lowest-order truncation restricts the expansion of * to the first term, containing only 0. Here, we consider the effective potential for 0 in this limit, which we call the level-zero truncation. This is useful for an understanding of general features as well as for comparison with the higher-level results below. From the level-zero terms in the quadratic and cubic parts of the lagrangian density given in appendix B and setting ~ = 0 for the static situation, one gets Veff(0)-

where 4K

2 0 +gK~3 + . . ., 2a'

1 3F 3! ( ) = 0 .365 .

(3.2)

The perturbative solution at 0 = 0 is a local maximum and hence unstable . It is therefore an unsuitable choice of vacuum. A local minimum in Veff occurs at (P) = 1/3Kga' = 0.91/ga' .

(3 .3)

Eq. (3.3) shows that ((P) is of order 1/g and hence is nonperturbative . It can be shown that this result holds to all orders at tree level [34]. Note also that this vacuum is inaccessible in the commonly used zero-slope limit a' - 0. A graph of eq. (3.3) is given in fig. 1. The value of Veff at the minimum is a contribution to the cosmological constant:

n=

1 - 2 . 33K 2g2a t3

-0.14 g2a~3

(3 .4)

3 .2 . TRUNCATION TO LEVEL-TWO FIELDS IN THE COMB APPROXIMATION

Let us incorporate the next-order effect by including the level-two fields in the expansion of ~ in yc2> and _rp (4) . These interactions are of the form (p-(p-B and O-B-B, where B is a generic level-two field . Integrating over level-two fields in the functional integral generates the comb approximation truncated to level-two fields . In general, comb contributions are diagrams with the structure given in fig. 2, where 0 are the external states and higher-level fields appear in the internal lines. In this subsection, we consider level-two fields propagating along internal lines. We call the truncation to level-two fields in the comb approximation the order-four truncation

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271

q ô a x

ga'~

Fig. 1 . The effective potential for ¢ near the local minimum . The order-zero truncation, the comb approximation, and the order-six truncation are shown . The latter two are largely indistinguishable but are beginning to separate for 4> > 2/ga'.

because it includes effects up to P(4) . The advantage of the comb approximation is that the analysis can be performed analytically because the propagating fields enter quadratically in the relevant kinetic and interaction terms . In the static case only ß1 and B contribute, where B is defined as the trace part of Bu,: 1 (3 .5) B :=-71' Bl" . 26 The factor of 26 in eq. (3.5) ensures that B has the standard normalization in the kinetic-energy part of the action . The comb approximation generates 1

2

Veff ( ) ?,Cl r0

+

33~ ,~,3 27

g`Y

131v/3 ( 3417 2 4( 530 1 + g -1 + 213 g 223 217 '~ 2432

g4

97 2 + 6 2 33 g 2)

Fig . 2. An example of a comb diagram for the eight-point amplitude .

(3 .6)

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V.A . Kostelecky and S. Samuel / Open bosonic string 10

-20

-30 -40 C, -40

I l

-30

I

I

-20

-10

II

0

1

10

Fig . 3 . The effective potential V((p) for 0 in the comb approximation. The dash-dot lines are asymptotes . In the region 0 < -28 the potential V(O) is indicated by a dotted line and has been linearly scaled by a factor of 1/6000 . Thus, for example, the local maximum displayed at ¢_ -36 has V(-P) = -124,000.

A plot of eq . (3 .6) is shown in fig. 3. The region around eq . (3 .3) is plotted in fig. 1. A second local minimum appears at (0) = -6 .98/ga'. Since Iga'(0) I is large

there, shifting 0 by its expectation value produces sizeable terms in the action, so that this minimum might disappear when higher-level contributions are taken into account. Throughout our work we are using semiclassical techniques . They work best when vacuum expectation values are not large. We show in subsect. 3 .3 that when y(6) is included Veff no longer has a minimum at negative 0. Nevertheless, fig. 3 is indicative of the complicated structure that can be generated. The local minimum in the order-zero truncation in eq . (3 .3) is increased by about 20% to (0) = 1 .083/ga' and the value of

Veff

(3 .7)

at the minimum becomes A = -0 .192/g 2a' 3 .

The corresponding expectations for B and

(B,~,>

1

-

26

n~v(B)

°

ßl

(3 .8)

are

(B) = 0.37/ga',

(ßt)

- 0 .35/ga' .

(3 .9)

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273

3 .3 . TRUNCATION TO LEVEL-TWO FIELDS AT ORDER SIX

Incorporating the next-order effect involves all 0 and B terms in Y ., up to and ; including P(6). We refer to this as the order-six truncation. Due to the presence of trilinear B-terms it is no longer practical to proceed analytically. Instead, we take a numerical approach . The static equations of motion for the truncated system involving 0, B and ßl are 1 34 2 3-11 3-5 13 . 4P B~ ßl$ 26 / F3 8a + 27 26 19

+ 1

3-5 13

B

73=)

2'3 ßi +

+

5-11 13 2~32 2

27

7 . 83 Bl1 + 2732 BZ = 0 ,

5-11 13 2632

(3 .l0)

7 . 83 ßl~ - 2632 Bo

7-41 5-19 13 7-11 -83 -73 Z B ß1 + 2734 + 2635 Bß1 + 2'3 4 13 _- 0' 1 73==ga'

ßl 1

(3 .11)

3- 11 2_ _ 19 5-11 13 tP Bip ßl 263 2632 + 27

+ 27 ßi +

5- 19 13 2634 Bß1

+

7-11 -83 --p3-5 B2 = 0 .

(3 .12)

We have used several methods to solve these equations . One simple approach is to solve the quadratic equation (3.12) for ßl in terms of 0 and B, substitute the solution into eq. (3 .11), and manipulate it to give a quartic equation for B with -0-dependent coefficients. Next, (p is fixed at a numerical value The quartic equation for B is solved to give a value B°, and ß° is obtained from eq. (3.12). The static potential is evaluated at 0 B = B° and ß1 = ß°, thereby yielding Veff(0). The procedure is repeated for many values of 4° so that a picture of Veff is obtained. The known solutions of a quartic equation [49] permit us to determine exactly certain key points of Veff, although we present only two decimal places below. Note that this method incorporates into Veff(q)) all extrema of the full static potential, not merely stable ones. The usual restriction to minima of the potential is inappropriate here because of the presence of ghosts, which have opposite-sign quadratic terms. The question of stability is addressed to in subsect . 4.6.

= e,

e.

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Fig. 4. The effective potential for 4) at order six, showing the branch structure in the region of the local minimum.

For a given value of 0° , the maximum number of solutions is four. There are two possible solutions for ß° and two for B ° *. Fig. 4 displays the region of interest. Outside this region, higher-level fields may significantly change the results . For -0.35 < 0 < 0.53 there are four solutions, three of which are displayed in fig . 4. The fourth is located at Veff < -4000 ; it corresponds to large expectation values of the fields and is not trustworthy . There are four solutions also for 6.7 < (P < 145, but again one cannot trust the results here. With the exception of these two regions there are two solutions . The existence of the various solutions suggests a structure for the effective potential that may become more complicated as more levels are included . The unstable perturbative solution with ((P> = 0 is part of a branch of the effective potential denoted as branch 1 in fig. 4. This branch terminates at (P = - 0.35, where it coincides with the terminal point of branch 2. Branches 2 and 3 merge at 0.53 . The termination of two branches at the same point is easily understood in terms of a model. Consider the potential V(S) = a(A)S 3/3 + b(X)S 2/2 + c(A)S + d(X ), where X is a varying parameter and S is a real scalar field. The extrema of V(S) occur when a(a)S 2 + b(a)S + c(X) = 0. When the discriminant (62 - 4ac) is positive, there are two solutions, one a local maximum and one a local minimum . When the discriminant vanishes the two solutions merge. Then, for (b2 - 4ac) < 0, no real * As the solution for B° arises from a quartic equation one might expect four solutions . However, reversing the procedure and solving first for B involves a quadratic equation, so there are at most two.

V.A . Kostelecky and S. Samuel / Open bosonic string

275

solutions exist . This is what locally happens in fig. 4 with 0 playing the role of A and B,,81 playing the role of S: two real solutions merge at the common endpoint of two branches. Beyond this point the solutions are complex. We find no minimum of Veff(~) other than the one with ~~) = 1 .09. This suggests the candidate vacuum at (~) = -6 .98/ga' seen in the comb approximation is untrustworthy . The solution in eq. (3.9) persists : (0) -= 1 .088/ga',

(B) = 0.404/ga',

(ß,) = - 0.380/ga',

n ,= -0.194/g 2a'3 .

(3 .13)

The small change between eqs. (3.9) and (3.13) suggests that our level-truncation scheme is good . A comparison of Veff in the different truncation schemes considered in subsects . 3.1-3 .3 is presented in fig. 1. The minimum in fig. 1 around (4)) = 1 .09/ga' is shallow . This is not merely a figment of the scales used. With our normalization conventions, numerical values for various quantities are expected to be of the order of one in units with a' = g = 1 . The value A = -0.19/g 2a' 3 in eq. (3.13) is thus fairly small. Nonetheless, the agreement in fig. 1 between the different truncation schemes suggests that the local minimum persists at higher orders. 4. Physics in the nonperturbative vacuum

In this section, we analyze physics in the candidate vacuum by shifting fields by their vacuum expectation values and considering small perturbations. The idea is to use this vacuum as a laboratory for nonperturbative calculations . We explore the possibility of tensor-induced spontaneous Lorentz-symmetry breaking, we obtain the spectrum in several truncation orders, and we discuss the vacuum stability. 4 .1 . TENSOR-INDUCED SPONTANEOUS LORENTZ-SYMMETRY BREAKING

It is interesting to determine whether tensor-induced spontaneous Lorentz-symmetry breaking takes place. Ref. [35] revealed that string theory has a natural mechanism by which Lorentz-symmetry breaking can occur. Such breaking is desirable given that the Lorentz group of our (3 + 1)-dimensional world is 0(3, 1), much smaller than the Lorentz groups 0(9,1) and 0(25,1) of the perturbative superstring and the bosonic string . The point is that couplings of the form STyTM exist in string field theory, where S denotes a generic scalar and Tm denotes a generic tensor. If scalars acquire vacuum expectation values of the appropriate sign then a negative mass squared might be generated for a tensor field. Then, this tensor field would play the role that a Higgs field plays in a gauge theory : it would acquire a vacuum expectation value and spontaneous Lorentz-symmetry breakdown would ensue.

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Y.A . Kostelecky and S. Samuel / Open bosonic string

As an example, note that the term OA,,A ,` is present in y(2). Here, a negative value of <0) generates a negative mass squared for A~', since the latter has no mass contribution from the quadratic part of the lagrangian density. An expectation value for Au breaks 0(25,1) to 0(24,1). In contrast, a positive value of (0) favors no Lorentz-symmetry breaking . To the order-six truncation considered in sect. 3, the shift of 0, B,' , and ßl by their expectation values in the candidate vacuum yields positive mass-squared terms for A~, B,, and B,. . No Lorentz-symmetry breaking occurs in the model to this order.

4 .2 . THE SPECTRUM : GENERAL CONSIDERATIONS

In general, the spectrum is obtained by examining the small fluctuations about the shifted fields . In the action no terms linear in the fields appear because we have a solution to the equations of motion . The quadratic terms define a quadratic form whose inverse provides the propagator. In momentum space, the quadratic form may vanish at various values of p2 = -pot . The corresponding pole in the propagator represents the mass squared of a state. Whether this is the mass of a physical particle depends on whether it enters in physical processes. Some poles may correspond to ghost fields and/or may be gauge artifacts. The evaluation of the quadratic form simplifies when a Lorentz decomposition of states is used. Since states transforming under different representations do not mix, the quadratic form is block-diagonalized . Appendix C contains the details of the Lorentz decomposition of all states to level two. To this level, the irreducible multiplets include one two-tensor, bTT, three vectors, AT, BT and bTL, and six scalars 0, A L, BL , B, bLL , and ßl. String theory has asymptotic freedom incorporated at tree level. The couplings in momentum space contain the factor exp[-3a'p2 ln(3V3/4)]. This may be seen directly from eq. (2.5) and the form of the interaction lagrangian in appendix B. For short spatial distances or large space-like momenta the effective couplings are small. The momentum scale is set by 1/ a' , which is expected to be the order of the Planck scale. At distances smaller than this, string theory in any vacuum is perturbative. In establishing the spectrum, one is interested in large time-like distances. Equivalently, one sets P2= -pot and looks for poles in the propagator . A pole at po = m means there is a state of mass m. Note, however, that masses large on the scale of the truncation involve large values of po, for which the effective coupling g(po ) = g exp[3a'pô ln(3vr3 /4)] is large. This means that perturbation theory cannot be trusted. Any poles found at scales beyond the truncation scale are unreliable . In fact, it is a general result that the spectrum as determined by the free theory cannot be trusted at high mass levels as soon as interactions are introduced .

V.A . Kostelecky and S. Samuel / Open bosonic string

277

4 .3 . THE SPECTRUM IN THE ORDER-TWO TRUNCATION

In this subsection, we consider the simplest interesting case, the order-two truncation . The string field ~k is truncated to -0 and AN, fields, and Ti,,, up to Y(2) is used . This case illustrates the phenomena that can arise in a nonperturbative vacuum . It also serves as a base from which one can judge the size of higher-level corrections . At order two, ((P) is given in eq . (3 .3). Shifting 0 by ~0) and examining small fluctuations reveals that there is no mixing between .0 and AL . The lagrangian to quadratic terms is

2

ô,,4) ô"0 +

2

ôwA°â "A' _

1

1 24 23 a,$2+ 3 3 a,A N Al'+ a,~Ava vA``, _2a'02+ 33

(4 .1)

where the meaning of the tilde is given in eq . (2 .5). The quadratic form associated with 0 in euclidean momentum space is p2-

1

a' +

2 -exp [ -2a'p21n(3~/4)] .

(4 .2)

Setting p2 = -pot, a pole in the propagator occurs if there is a real root of 1 2 -P2 - ; + ~exp [ 2afp2 ln( 3~/ 4 ) ] = 0 . a a

(4 .3)

For all values of p2 , the right-hand side of eq. (4 .3) is positive . There is no pole associated with 0. This is a consequence of the momentum dependence in the coupling stemming from exponentials of p2 : the equation determining the zeros of the Minkowski-space quadratic form is transcendental. If 0 instead of ~ appeared in eq . (4 .1), the mass squared of 0 would be 1/a'. The effect of the momentum dependence in the coupling has a dramatic effect on the spectrum . The quadratic forms associated with AT and AL are respectively p2

p

25 + 3Ta' exp[ -2a'p2 1n(3r/4) ], ~

+2 2 + ~ 33a' 25

33 4P2

exp [ -2a'p21n(3F/4)] .

(4 .4) (4 .5)

When p 2 = - M2, eq . (4.4) never vanishes as a function of M2. There is no pole for the transverse vector field. The scalar AL has a pole at

278

V.A . Kosteleckj and S. Samuel / Open bosonic string 1.2

t+0 ~, 0 w

a

0 .6

0.4 0.2 0.0

à 0 bD 0

a 0 w a

po t a,

2

Fig. 5. (a) The propagator for AT at order zero, in the comb approximation, and at order six. The latter two are indistinguishable in the region shown. (b) The propagator for AL at order zero, in the comb approximation, and at order six. The latter two are indistinguishable in the region shown.

It is unclear whether this pole is physical : it does not enter the 04 scattering process and one must look at other processes to ascertain whether it enters physical amplitudes . A plot of the transverse and longitudinal Au propagators is provided in fig. 5 . It is interesting to contrast the above with what would have happened if the factor exp[-a'p 2 1n(3v~3 /4)] associated with each field were absent . The pole in AL would be at p 2 = -25/43a'= -0.74/x' instead of at -1/a' and a mass pole in AT would occur at - 25/3 3a' = -1 .19/a' . The exponential momentum dependence in the couplings has removed a potential state from the spectrum .

V.A . Kostelecky and S . Samuel / Open bosonic string

279

4 .4 . THE SPECTRUM AT ORDER FOUR IN THE COMB APPROXIMATION

In the comb approximation, the quadratic forms associated with AT and AL are respectively

247z 3 5 13 2 24 zP 2 +icgexp [ -2a'p2 1n(3 Vr3 /4)](3y(0)3

1

+ 32C$)a P2-

2 1 i 335 (#,)a P 2-

31 35

3

(B)

2s + 3 5 13

i

(B)a P2 ~'

(4 .7)

411

24 72 (B) ~ßi) - 3s s 3 13

( B)a~Pz+

7

3 5 2 13

( B)a/2pa~ > (4 .8)

where the expectation values in eq. (3 .9) are to be used . It turns out that the A fields do not mix with the others . The quadratic form for AT is found to vanish as a function of p 2 ' where P2= Pô, at a value m2 = 16 .81/«' . However, this value cannot be trusted due to higher-level effects as explained in sect. 2. The propagator in the region of validity is plotted in fig. 5a . Going from order two to order four produces only a slight change . The pole for A L found in order two persists in order four . It is shifted from m=1/ a' to MAL = 1 .17/Fa ,

(4 .9)

a 17% change. There is also an unreliable pole outside the region of validity, at m 2 = 10 .82/«' . Fig. 5b depicts the propagator versus po. The quadratic form for the tensor field, bT, T, is 1 28gic zP2+ 2a' + 35 ($)exp [ -2a'p21n(3~/4)] .

(4 .l0)

Setting p 2 = -p2' one finds that this equation never vanishes. The propagators in

V.A . Kostelecky and S. Samuel / Open bosonic string

280

ô

0.2

0.0

Fig. 6. (a) The free propagator for b,T. (b) The propagator for hTT in the comb approximation and at order six.

the canonical and nonperturbative vacua are compared in fig. 6. The pole in the canonical vacuum at p2 = 1/a, is removed. Again, a significant difference in the physics occurs. The vectors BST and b'TL mix and generate the following quadratic form when P2= -Pô: Mu

Mt2

M12

M22

V.A . Koste%cky and S. Samuel / Open bosonic string

281

with Mil - - 2Pô +

- -

1 2a'

27gK

+ 3a

($)exP [ 2a~P0 ln(3~/4) ]~

23 KK a' P0($)exp[2at p2 ln(3F/4)] ,

I

t a Po ($)exp [2a'Pô ln(3F/4)] , (4 .l2)

M22 = - iPô + 2a' + gK

where Mll, M12 and M22 are the BST-BST, BST-b~L and bTL-bTL entries, respectively. Converting the matrix (4 .11) to euclidean space and numerically examining the determinant reveals that the determinant never vanishes . The propagator in Minkowski space exhibits a pole at m,c =

.451a-' 1 .

(4 .13)

One state is associated with the BT-b. L system . The scalars 0, ßl, B, b LL and B L mix to produce a five by five matrix for the quadratic form . Its explicit form is provided in appendix D. Taking the determinant of this form and carrying out a numerical evaluation, zeroes at po = 1.14/ rd, po = 1 .44/ a' and po = 3.97/ a' are found. They correspond to M2= 1.30/a', m 2 = 2.08/a', and M2= 15 .76/a' . The last value cannot be trusted due to higherlevel effects. Hence, we have two scalar states at ml = 1 .14/ a' , (4 M2 'e 1 .44/ a' .

.l4), (4 .l5)

It is easy to see from the form of the interaction lagrangian given in appendix B that all the scalars couple to $2 and hence enter the $a amplitude. The states associated with eqs . (4 .14) and (4 .15) should manifest themselves in a physical process. 4.5 . THE SPECTRUM IN THE ORDER-SIX TRUNCATION

The length of the order-six interaction lagrangian -W(6 ) makes the determination of order-six corrections for all the states considered in subsect. 4.2 laborious. Here, we focus on a few specific cases. The A,, fields do not mix. The quadratic forms given in eqs. (4 .4) and (4.5) are valid at order six if the order-six expectation values in eq. (3 .13) are used . The pole in AL shifts by less than 1% from M A L = 1 .17/ a' to M A L = 1-18147 .

(4 .16)

282

V. A . Kostelecky and S. Samuel / Open bosonic string

An unreliable pole at m 2 = 9.56/«' also occurs. The AT propagator contains only an unreliable pole at m2 = 15 .63/«'. The changes in the A L and AT propagators between orders four and six are indiscernible in the region of validity . This again suggests that higher-level corrections do not significantly change our lower-level results. The quadratic form for bûT at order six is

zP 2 +

1 2a'

+ 8K

28

3s

2 811

p

2811

13

(B)

(4 xexp [ -2a'p 2 1n(3 vf3 /4)] .

+

212 , 2 iii= (B)a P

1 .17)

Setting p 2 = -p 2 ' one finds only an unreliable pole at m 2 = 16 .63/«' . A comparison of order-four and order-six results for the propagator is provided in fig. 6.

4 .6 . THE STABILITY OF THE NONPERTURBATIVE VACUUM

Summarizing, in the new vacuum we find three scalar poles and one vector pole with masses within the range of validity . Empirically, we find high-level corrections do not significantly change lower-level results. One can ask whether the new vacuum is perturbatively stable at the quantum mechanical level. The functional-integral representation is useful here . If the quadratic form of the small fluctuations is positive definite in euclidean space then the functional integral is mathematically well defined when interactions are ne-

glected. The presence of ghosts in the string field theory and other theories complicates the situation, however, since ghosts have wrong-sign kinetic-energy terms. Certain pieces of tensor fields involving time components also act like ghosts in Minkowski space. In such theories the quadratic form is not positive definite . Instead, the functional integral should be defined by analytic continuation . A ghost integration variable is made complex and analytically continued to purely imaginary values . One then performs the functional integral and analytically continues back . This is 2 like defining f dx exp(ax /2) to be - ~r/a for all complex values a :* 0*. Hence, the relevant question is not whether the quadratic form is positive definite but whether a sensible perturbation theory can be defined. Perturbation theory can be performed if the propagators in euclidean space never diverge. This means the quadratic form is invertible . We therefore should require the nonvanishing of the determinant of the quadratic form rather than its positive definiteness . We have numerically analyzed these determinants for all particle fields * A branch for the square root must be selected.

V.A . Kostelecky and S. Samuel / Open bosonic string

28 3

in each of the truncation orders in sect. 4 and found that they do not vanish in euclidean space for any value of p2 . In contrast, the canonical bosonic-string vacuum is found to be unstable, as expected. The propagator for the tachyon in euclidean space diverges at p2 = 1/a'. Another question is whether the vacuum state is nonperturbatively stable, that is, whether it can tunnel through the potential-energy barrier. The analysis of the barrier-penetration problem in string theory has not been performed . Generally speaking, the more degrees of freedom in a theory the harder it is to tunnel . Roughly, this is because a suppression factor is generated for each degree of freedom. In quantum mechanics, which is based on a finite number of variables, penetration of a potential-energy barrier of finite height can always take place. The ease with which tunnelling occurs accounts for the absence of spontaneous symmetry breaking in quantum mechanics. In field theory, where the number of degrees of freedom is infinite, barrier penetration occurs when one vacuum has a lower energy than the other [5,6]. When vacua have the same energy, tunnelling may or may not exist depending on the situation. Spontaneous symmetry breaking occurs when a broken vacuum state is unable to tunnel into partner vacua. String field theory is like a particle field theory with an infinite number of fields. For this reason tunnelling processes, if they occur, should proceed with more difficulty . A local vacuum in the effective potential may be nonperturbatively stable . 5. Discussion and conclusions We have presented a systematic nonperturbative string calculation via the leveltruncation scheme using functional-integral methods and covariant string field theory . We have obtained the effective potential in successive orders and uncovered a candidate nonperturbative vacuum . The physics about this vacuum was analyzed . Perturbation theory was shown to be well defined. We determined the perturbative spectrum of states in the new vacuum as well as the contribution to the cosmological constant produced at tree level. In principle, our results are subject to higher-level corrections. Numerically, however, we find such corrections are not substantial. In any string theory, the determination of the vacuum state is of considerable importance. In general, the effective potential provides us with a means of finding candidate vacua. The plots we have presented in figs . 3 and 4 suggest that string effective potentials might be quite complicated. There could be a rich structure of branches and many vacua. For the open bosonic string, part of the interesting structure we have found occurs in a region where higher-level corrections are important. These corrections may wash out local minima or may instead generate further complicated functional behavior of the potential with more minima . We have found that the mass spectrum is quite different in the new vacuum. The massive level-one symmetric two-tensor seen in perturbation theory about the canonical vacuum appears absent in the nonperturbative vacuum. A massive vector

28 4

V.A . Kostelecky and S. Samuel / Open bosonic string

and several massive scalars are observed. They involve linear combinations of fields at different levels . When higher-levels are included more mixing will occur. Since the couplings decrease exponentially with level number, such mixing should diminish . The size of the mixing matrix will increase but the magnitude of the new off-diagonal entries will decrease. Results of this type are likely to be true in any string theory when developed about a nonperturbative vacuum. The spectrum should be dramatically changed and level mixing should take place, particularly among lower-lying states . We have also uncovered a mechanism by which propagator poles are removed. The origin of this is the factors involving the exponential of p2 that are present in the couplings. This means that the tree-level string field theory in the nonperturbative vacuum has many features found only at the loop level in a particle field theory. Poles in propagators are renormalized and disappear in some cases. Wave-function renormalization is needed in any scattering process. The string coupling runs already at tree level, producing tree-level asymptotic freedom. Among the consequences are perturbatively calculable short-distance effects and difficulties in the analysis of the spectrum of heavy states beyond a certain mass level due to strong coupling. Tree-level asymptotic freedom should occur in any string theory. Physically, it results from the extended nature of the string, which smears interactions at short distances and provides good ultraviolet behavior . In a realistic model, if certain fields acquire expectation values, then factors of exp(-ca'p 2) appear in the quadratic form, where c is a constant. The zeros of the quadratic form in Minkowski space are then determined by a transcendental equation in pot instead of by a polynomial . This can cause the propagator poles of many states to disappear. The number of states in a nonperturbative vacuum of an interacting realistic model could be significantly less than the number indicated by naive perturbation theory . This would have substantial consequences for any attempts at a phenomenological interpretation of strings. In commonly used particle field theories, the number of degrees of freedom remains unchanged when some fields acquire vacuum expectation values . For instance, in the context of the Higgs mechanism a massless gauge vector absorbs a scalar but becomes massive, thereby maintaining the total number of degrees of freedom. Strings can evade this by shifting certain poles away from the real axis in the complex energy plane* . The point is that the nonperturbative vacuum is full of string . Certain modes may not sustain propagation beyond the string scale a' . For example, in the truncation to level-zero fields performed in subsect. 4.3, the 0 propagator has no real pole. However, a complex pole exists at pô = (0.24 ± 1.9i)/a' . This corresponds to an enhancement in the S-matrix or to tiny violations of causality at Planck time scales, as might be expected in a nonlocal theory. From this * Loop contributions in a particle field theory can generate similar effects.

V.A . Kostelecky and S. Samuel / Open bosonic string

285

viewpoint, the appearance of the extra tachyonic degree of freedom in the canonical vacuum merely signals an inappropriate choice of vacuum*. It is likely that the spectrum in the nonperturbative vacuum contains only massive states, at least at tree level. A string theory without tachyons or massless states is finite because there are neither infrared nor ultraviolet divergences. There is therefore the possibility that the open bosonic string in our new vacuum is a finite 26-dimensional theory. One-loop effects are important to this issue. At one loop, closed bosonic strings must be taken into account. In perturbation theory about the canonical vacuum, the closed-string tachyon produces the leading divergence in the one-loop amplitude. To date, it is unclear how to handle closed strings in the field theory of the open bosonic string. If a field theory for both were available, one would need to seek an expectation value of the closed-string tachyon field and expand about the combined nonperturbative vacuum. If the spectrum of the combined theory is also massive then this bosonic string is a finite 26-dimensional theory . An intriguing question arises from conformal considerations . It is generally believed that to each vacuum state of string theory corresponds a conformal field theory . This statement is based on the first-quantized world-sheet approach . The world-sheet action is a conformally invariant two-dimensional theory . It is unclear that this folklore applies to the nonperturbative vacuum but, if it does, one can ask what is the corresponding conformal field theory . Furthermore, as conformal methods are among the most powerful for analyzing string theory, the extension of off-shell conformal field theory [30] to nonperturbative vacua is likely to yield a useful calculational tool. The incorporation of higher orders in our truncation scheme would provide more evidence as to whether the nonperturbative vacuum persists . It would be interesting to examine the resulting effective potential for additional structure and candidate vacua. However, there are substantial physics questions that could already be addressed at the present truncation order. These include the determination of scattering amplitudes and other physical processes, and the investigation of gaugesymmetry breaking [51]. The trilinear couplings in the nonperturbative vacuum are identical to those of the canonical vacuum so that only the change in the propagator need be incorporated . It would also be pleasant to have a simple criterion for determining whether or not propagator poles are physical . In conclusion, we have identified a candidate nonperturbative vacuum in the open bosonic string . We have found dramatic changes in the spectrum associated with this new vacuum. These results originate in the nature of strings as extended objects and provide evidence of the importance of nonperturbative effects in string theories. ` In

an investigation of quantum effects in the presence of black holes, 't Hooft [50] has found a string-like spectrum of states with complex masses .

286

V.A . Kostelecky and S. Samuel / Open bosonic string

We thank R. Bluhm, P. Carruthers, J. Challifour, A. Hendry, G. 't Hooft, D. Lichtenberg, and R. Newton for discussions . This work was supported in part by the United States Department of Energy under contacts DE-AC02-84ER40125 and DE-AC02-83ER40107 . Appendix A. The string-field expansion The expansion for ~P begins as = ( ¢ + A Wa_ 1

v + iab_ 1 c0 + 1iB a_ 2 + 1Bpv a_ 1 a_ W 1 +a NO b_2c0 + Nalb_1c

+ikt, b 1 coa1~ 1 + C

+iyl b_ 2c_ 1

+

1

al` 3 + '

iy2 b_ lc_ 2

+ 131, b_ c_ 10W 1 l

+

+

1

i7-6

h

1

+ 76~ i"n

À v CX ~a 1aW la 1

T Dl w vaa 3a°_ 1 +

1

+

1

+ r ilu vb_lcoat` 1a°

1 +

iyab_3co

12t.b_2coal=1

2F D2Wvau 2a° 2

+ 73== im 1,,b_ 1 coa1`3 +

+ &1b-3c

1

+ 82 b

2 c_ 2

1 2im 2t b_ 2 coa4 2

I c 1 au 2 + im41,b_ 3 co a1` 1 + im 5N b_ 2c_ la' 1 + im b1 ,b l c_ 2a1` 1

1 1 + 2ml,wb Icoa1'2 0' 1 + 2m21L b_2coa1` 1 1

1

+

D~Pa al_ l a" la? 1aQ 1 + So b_ 4ca

+83b-lc-3 + 84 b_ l b 2 c_ 1 c o

+ r im 3,~ b

iCgal_2aY 1

1 F ub lcoa~ 2

+ i iD~a 1_ 4 +

+ ziDxl, a~ 2aw 1ae 1

1

1

-

-

-

1

+ F m3Ab_ lc_ lat' l aL 1

---)

Here, 4p is the tachyon, A. is the massless vector, B,,v is a symmetric two tensor at the first massive level, etc ., these descriptions applying in the canonical vacuum . The fields a, ßo, 81, k,,, etc . are auxiliary fields . The first-quantized string vacuum is I 0) = c1 IQ), where I fl) is the SL(2,R)-invariant vacuum [52]. The states created by applying al` , n > 0, c , n > 0, and b_ n , n > 0 to 10), together with 10) itself, are solutions to the first-quantized theory in a harmonic-oscillator representation [1]. All fields are real; the factors of i are needed to ensure the reality condition

V.A . Kostelecky and S. Samuel / Open bosonic string

287

`pt = *t [7] and are readily found using appendix C of ref. [21]. The numerical factors ensure that the kinetic-energy terms in the action have the standard normalization. The level of a state is defined as its mass squared in units of 1/a' as measured above the tachyon-mass squared. The tachyon 0 is at level 0, A, is at level 1, etc. We use the letters A, B, C . to denote fields at level 1, level 2, level 3. . . . . Note that in the unoriented string the odd-level fields are absent .

Appendix B. The lagrangian density at order six In this appendix, we present in Siegel-Feynman gauge the full interaction lagrangian Yjt involving fields up to and including level two. Using the expansion of * to level-two fields given in appendix A and the Fock-space representation of the three-vertex given in refs. [10-13], the interaction lagrangian in terms of particle fields is ) +y (4) yint=y(0 +2(2)

+y(6)

~ ~

(B .1)

where the superscript is referred to as the order and indicates the sum of the levels of the fields . We find (B .2) where g is the on-shell three-tachyon coupling at tree level, K is a constant defined in eq. (3.2), and the meaning of the tilde is defined in eq . (2.5). For the order-two terms, we have `~ (2) = Kg ~

-

23 32

32 2

Blw$z -

Kga'(2 al$ Â

Il 32

Qi(p2

2a

+

3iAlAMiP

ô °Al` - ô~A v d',!" - dl ô,,~Â1`Â")

- 3 Kg a~ $ gwBw - 3 2

a

Kga'( $ dl dÀB~`° - a~ a v BuY)

(B .3)

For the order-four terms, we find ~(a) = rpôa) +yia) +yza) +~3 a) +y(a) ,

(B .4)

288

V.A . Kostelecky and S. Samuel / Open bosonic string

where the subscript counts the number of explicit derivatives . The terms are 2 52 8 ~ -5-11 _ 19 27 ~
r

2 28 411 , 23 r2- - 5 '+  A"A"B (B B" - 35 A"A"/3i l , 35 3s A'A'

=

+

f' 2 35

Kg a'(2

2~2 -5

~ca) - +

B"v-2d"~ÈB"v+ ~-2 Â"d"Â BV- ~-2 Ä Ä ô"B° ) 25

Kg a'~d"B"B"- 3Kg â Ä"A"dB°+

34

( 24 5 ~B d 1L d_ v BPP + d" d v~ B"vBPP ) - Kga' d"~ B"P dvBPP 35 Kga~1 3s

35

2' r235

-5

Kga'( d" A dvA" BPP + A" A d " d °BPP - 24" d"A v dPBPP

Kga'(2 d"Â~ dXÂ"B")` + 2Â" - d"Â

+

2' 35

ap i?, d"B°~ +Â" dv dXÂ"B° X

B" )` - 241, d,Â,\ PB"Y - 2Ä" d" d,Äx b°-k)

Kga'(2 dj BX d °B")` - d" dj B" ~'hx° - ~ d" B,\ d °B"x

2 311 35 Kga(2A"d4y dvßl -A"A v d"Ffi -d"4 v d°A"fi ) 2 3 F-11 35 Kga'(2 d"~ B"° d, f,

+

22 -11 34 Kg a' M d"B", (B .6)

23 5

22

-

.5)

22 33

KgA5 d"B"d B°,

- ¢B" d" d P#1 -

d" d~ B ll )

(B .7)

V.A . Kostelecky and S. Samuel / Open bosonic string y(4)=

24

34 -

289

Kga' a'(2Â,,dPÂv ava~ÈX -ÂPÂ aPavadPÂavÂ~dABX) r2 4 V2 34

Kga' a' (2 a,~ a, dNB~Bu° -

P av a 11B,, 13, x - au av~a x h ~l

25 + 35 Kg a'2( P a.BP° aP a°Bwv+ 2 aP a~~hp° aP a°BPv+ ap a, aP a°~ hP`âP° a -22 a,, av ap~ B" a° BvP - 22 aP~ av hp . a P a °BPv + 22dp av~ a P B°° a°BPP ) +

26 v~2 35

Kgâ

2(ÂP aP aP a°dv avBP° + 14

aP a° â u a°BP° -`4P aP a°ÂY aP a~âP°

- aPÂ,, av aP a°ÂP BP° + a,Â,, a,° a° a°Bup + aP a.ÄP aP a° Âv h- 2 a~,Â, av a PÄ° a°B"P) .

(B .9)

For the order-six terms, we find (6) - ~(6) + yl(6) +?(6)

+y(6)

(6) + 275(6) + Y6(6)

+

.

(B .10)

These order-six terms are I/ ~ôb> _ + Kg/ -

26~ - 5 ., BP BPBvv 3'

+

211 ~ 38

2711 BPBv BPv - 37 #1BPBP

7 B~ah ,h, - 2 ~ 5 B"BPv hP v 2~-3 3 53

9

+

213F2 3

B~ P Bv~BPv

5'11 2811 5-19 1 P B'Bvv BI, BI" l3iB - -ßi 34 2-38 ß1 38 ßl P v )

1

8 - 28 5 6) =- 36 Kg ya aP BPBy B° Kg a'(aP B BP° BP P- BP B,Pa ,BP P) 3' 213 - 38

Kg

a' ( aP Bv BPPBP' BP BYP a'BPP ) -

2"F - 11 3x 29

- y Kg Va

Kg

â

2 -19 36

(aPBvBPßI-BPBPvavßl)-

do, BPB~ph~p-

52 y

Kg

ja7 aP

2 ~ - 5 - 11 3-,

BPl;i Kg

v« aP B~`Bvvl31

Kg a' dABPBvvBPP(B .12)

V.A . Kostelecky and S. Samuel / Open bosonic string

290

10

~

a"BPÉP" - B a"aP BPB"P) Kga,( am BP P

3

9 2 211 2 C BP B §'P - B a" Ë" BPP) Kga' aP BP a" B "ßl a" P K g a ( aP P aP 3 3 -

+

2C-5 23 r2-19 a Kga , a,BPa,B"Bp P aP a "Nl - BP " a Pßl a"äl) 36 Kga,(BP"Rl 3' 2 811 38

Kga'(2BP" d"b 'p aPßl - BP "BPP a' a pt3 1 - aPB"p a"BPPßl)

2 3 5 . 11 Kga'(2BP , aPBpp a"al - BPPB"P a" d pßl - BP  aP a"Bp p fil) + ,8 3 211C2

Kga'(2

38

,, aPBP° aPB" ° + BP "Bp° aP a°BP" - 2f3,, aP BP° a°B 'P

"° - BP , aPBP° a"BP °) -2h,YBp° aP apä +

+

-

2' vr2-

-5

38

Kga'(2BP," aPB"p aphQ -

22 C2 . Sz 38 211

Kga'( äP" aPBP ° a"B.o -

ICgâ «' (2 a,,ä" ap

38

BPP

a"Bp° aPB "° - BP  BPp a" aPB°° )

BPPB"P

°) ,

(B .13)

a" aPB°

B`a°B'P - 2 aP a"BP B°p a°B"P + aP B" Bp° ap a°BP"

+ aP a" aP B° B P"B p° - BP a"Bp° a P a °B P" - aP a"BP B°P a °B P" ) -

+

23

s~

Kg« ,

2 4 vIr2 37

-

11

aP BP a"ä" aP BP /~ Kga ' «' (2 aP d PBPB"PapNl - aP BPB" P a " a

pßl - aP a" aP BP B "p,Bl)

_ _ _ _ _ _ 29 + 3Kga' a' (2 aP a,,BPBP° aPB "° - aP a" aP BP B°"Bp° - a,BP a"Bp° a °B p " ) 24 5

B

~ " a PB°°- a a" aP B P ä"P B°°- aP +B K g« r,/«r (2 a a" P Bp P

3

P

BP É, a " a P °° )

V.A . Kostelecky and S. Samuel / Open bosonic string

2(6) °

-

25~

291

K g a ,2 (ap Bp a ap a a B v B pc - ap a v B p ap aQ B p BYa )

3°

+ 24C-5 Kgaa(22 v a ffB?, apaph07 a° a'Bpp+ 22 Bpo apBp,, a ap 38 -22

aQBT, - 2Bp ap aaBpT avhp,, ap apBp avhp,, ap

-BpP

+

2 C 3

ar apBoT

aa aTBpp

aBTT) - Bpvhpa ap av ap a

K g « 2 ( - 2 apBp aaBpT av a 'Bpa + apBp aok, a p a 'B po +Bpv ap apBoT av a -B pr +Bp, a"b"' a0 ao a'Bpp + ap B p

- Bpv 25 + 38

11

rha' a a aTBpp - B" ap

ao bi-, a , a

hp.

aafip*) apBoT ap av ?

j~

Kg«l2(22Bpv apBp,, av ap a"N, + 22 a"B'p av a pBp. a '#, -22 de B p a "Bpa ap aaßl - 2Bp,, ap,, ,Bpa ap aaß,

-Bpv Bpa dg av ap a-ß, - ap aTBpa ap a°BP'R,) , e(6) - -

6

3~

(B .15)

Kg« ,2 «- (22 a aY a TBpa - 22 ap a, ap aTBpa apBp ,,,,BT* a,,Bpà,, là

a'Bpv + 2 ap a, ap Bpha, aa a'BYP + ap a v ap da d,BpkPBar -22 ap a,Bpaphar ao +ap Bp ,,' ap har aa a 'kp ) , (6) -

,0

+ 2

+

IC

39

2 8vfj 38

g«,3 ap

(B .16)

a n BTT aT Bp,, a p aT Bpn a °

icg« ' 3 (2 ap B p da aTBpa a" ap anB°T + Bp ap ap aa B,a F a' aTBpa +ap Bp aa àpn a , ap aT aTBpa)

28V2 39

K g « ,3 (3Bpv ap aaBpn a , a v a , a"b" + ap a~Bpa aT a n Bpv ap al?")

,o 2 v~2 Kg« ,2 ap a' aTBpa . a,,BTp a' a,,BTn 3

(B.17)

292

V.A . Kostelecky and S. Samuel / Open bosonic string

Appendix C. Lorentz decomposition of states to level one The evaluation of the quadratic form simplifies when a Lorentz decomposition of states is used. Since states transforming under different representations do not mix, the quadratic form is block-diagonalized . From a vector field such as A P or BW, a scalar object can be formed by applying a derivative dA. This is the longitudinal piece of the vector . The piece left over is the transverse part and carries spin. Define BP = B, + BIL,

where

(C .1)

BT = B - a P a ~Ba BL BL = ` dB w w a a2

,

=

ri -à-2-il

,

(C .2)

and where d2= aA aA is -p 2 in momentum space. The normalization for BL is at one's disposal but with the above convention the kinetic-energy term is standard. This follows from fd26XBtBP= fd26X(BTBTP+BLBLt`)

= fd26X(BTBTI`+BLBL) .

(C .3)

Note that d,,BI` = dIIBLw = I - d 2 l BL, which is Ip I BL in momentum space. The symmetric tensor B., contains a spin-two representation, a spin-one representation and two scalars . One of the scalars, B, involves the trace of BP, : 1 1 1 BP."

bp~ gPvB, B = + 26

bu"

,nPo BPu, 26 q,``

= BP"

26

,qP°BP° . (C .4)

The traceless part of BPv is bPv . Again, we have normalized B so that BPP BI'v = bu~P' + BB . The field b L . can be further decomposed: bl,,= bTT +bTL +bLL ,

where

d dg a ~'b a , TT bPP =bPYa2 bTL = a,,bVL + 21-a21 bLL PL

v

aW a ° bLL a2

bTL = u bLL =

_

d, a ~'be,\ a2 avb~L

21-a21

a2

g dY a P a bpQ, 2 a a2

,

,

( a abPx dPa°bPo

+

(C .5)

a,, bLL ) , (C .6)

V.A . Kostelecky and S. Samuel / Open bosonic string

293

These irreducible multiplets satisfy

ff

26x bTTgv bTTgp + b'TLbTLgY + b,LLbLLux,)

d26x bgvbgY =

= f d26X (bTTbTTgP + gY

b~LbTLA g

b LL b LL ) ,

+

TT= dgd " bTL= dgbgY gY 0,

dgbg ,= dgb~+dgbeL

=

I

- ~I

b

+d bLL

a2bLL . d ,a d PbgY =

(C .7)

Appendix D. The scalar inverse propagator at order six In this appendix, we provide the quadratic form for the scalars in the comb approximation for the order-six truncation . It is a hermitian five-by-five matrix for the ordered scalar fields (0, B L , B, bLL, ßl) with the form m11 -im 12 exp [ - 2a'P21n(3F/4)

m13 m 14 m15

1 m12

m13 'M 23

m14 im 24

m22 - im 23

'M 25

m33

m34

m35

- im24

m34

m44

m45

- im25

m35

m45

m55

m15

The entries are as follows .

roll - ( 2P2 -

« 1 e2 pzln(3~/4) + 31cg
5 13 11 24 Kg
21 7

- 19

35 13

KS~B) +

22 1 1 26 Vif 3a Kg~ß1) - 3a 13 ~gCB) p 2 1l

(D .2)

W Pl , (D .3)

V.A. Kostelecky and S. Samuel / Open bosonic string

294

M13 =

1

2

~-

2-5-11 13 2-5 13 27-83 K9W + g(B) + Kg(ßl) 2 s K 3 3 3s 3

3 33139

+ 32 2 13 3

-

mla

= 21

~

+

Kg~B)atP2

7

Kg~ßl)aP2+ 313Kg ( B)aaPaJ 3s 13

(D .4)

23 V - 43 23V2 Kg(B) 3 2 Kg~~) - 34 13 23

Vr2- . 11 35

2' r2 Kg~B)a,pz a,P2 I > Kg(ßl) + 3s 13

(D .5)

2511 2-5-11 13 2-19 1 2-11 , 2 Kg(B) + Kg<ßl) - 3s ug + Kg(op) + 33 Kg($)a P2 m22 = 34

(D .7)

24 1 22 - 7 - 37 m23- 2[13 Kg~~)~ a~Pl + 34 13 Kg~$)a~P 2~ a~P~J , 3s

(D .8)

24~

M24

1 ~ 28C2 ( t = (D 2 - s Kg(~)~ a~ P + 34 Kg $)a p2I

M25

= 2f -

3

M33 =

1

(2

W jj P ,

2z11 l a~P~' , a Kg~~)I 3

.9)

(D .10)

4 3 2 7 383 P2 + 2 2 â ) e2ap 2 ln(3V/4) + -Kg(o) tp Kg~$)a - 3513 + 3513 Kg(4,)a,2P4 (D .11)

34 -

1 ~ z

-

25> 2 23 F . 11 l z KgWa,zp4 ~$)aP + 3s -i'13 Kg 13

(D .12)

V.A . Kostelecky and S. Samuel / Open bosonic string

2-5- 11 13 2311 _11 2 Kg(o) _ -Kg(o) a~p2 35 35 13_

m3s maa

295

r1 2 = 1 2P +

2a'

l ez«pz ln(3f1a) +

8

3s

Kg+

8

3s

(D.13) Kg 4,)a ,p2 +

5 3s

Kg(,p)a,2P4 (D.14)

mas

-

1

2~

-

23~ . 11

mss-(-2P2-

35

Kg(9p)a tp 2

l

(D .15)

J

2a')e2«p2tn(3f/a)+

39Kg($) .

(D .16)

References [1]

[2] [3] [4] [51 [6] [71 [8] [91 [10] [11] [12] [13] [141 [15] [161 [17] [181 [19] [20] [21] [22] [231 [241 [25] [26] [27] [28] [29] [30]

J .H . Schwarz, ed., Superstrings, Vols. I and II, (World Scientific, Singapore, 1985) ; M.B . Green and D.J . Gross, eds., Unified String Theories, (World Scientific, Singapore, 1986) ; M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory, Vols . I and II, (Cambridge University Press, 1987) M.B. Green and J .H. Schwarz, Phys. Lett. B149 (1984) 117; B151 (1985) 21 ; Nucl . Phys. B225 (1985) 93 D .J . Gross, J .A . Harvey, E.J . Martinec and R. Rohm, Nucl . Phys. B256 (1985) 253 ; B267 (1986) 75 P. Candelas, G .T. Horowitz, A . Strominger and E . Witten, Nucl. Phys . B258 (1985) 46 P. Frampton, Phys. Rev. D 10 (1977) 2922 S . Coleman, Phys. Rev. D 10 (1977) 2929 E . Witten, Nucl. Phys . B268 (1986) 253 S . Giddings, Nucl. Phys . B278 (1986) 242 S . Giddings and E. Martinec, Nucl . Phys. B278 (1986) 91 D . Gross and A. Jevicki, Nucl. Phys. B283 (1987) 1 E . Cremmer, A. Schwimmer and C . Thom, Phys . Lett . B179 (1986) 57 S . Samuel, Phys. Lett . B181 (1986) 255 N . Ohta, Phys . Rev . D34 (1986) 3785 D . Gross and A . Jevicki, Nucl. Phys. B287 (1987) 225 C . Thom, in Proc . XXIIIrd Int . Conf. High-energy physics, Berkeley, 1986 E . Cremmer, in Proc . Paris-Meudon Colloquium, 1986 S . Samuel, in Proc. XVIII Int. GIFT Conf. : Strings and superstrings, El Escorial, Spain, 1987 S . Samuel, Phys. Lett . B181 (1986) 249 D . Gross and A . Jevicki, Nucl. Phys . B293 (1988) 29 K . Suchiro, Nucl . Phys. B296 (1988) 333 S. Samuel, Nucl . Phys. B296 (1988) 187 K . Itoh, K . Ogawa and K. Suehiro, Nucl . Phys. B289 (1987) 127 H . Kunitomo and K. Suehiro, Nucl. Phys. B289 (1987) 157 S. Giddings, E. Martinec and E. Witten, Phys. Lett . B176 (1986) 362 O . Lechtenfeld and S . Samuel, Nucl . Phys . B308 (1988) 361 C . Thom, Nucl. Phys . B287 (1987) 61 M. Bochicchio, Phys . Lett. B188 (1987) 330 ; B193 (1987) 31 J.H . Sloan, Nucl. Phys. B302 (1988) 349 S. Samuel, Nucl. Phys . B308 (1988) 285 S. Samuel, Nucl. Phys. B308 (1988) 317

296 [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52]

V.A . Kostelecky and S. Samuel / Open bosonic string D. Freedman, S . Giddings, J. Shapiro and C. Thom, Nucl . Phys . B298 (1988) 253 R . Bluhm and S . Samuel, Nucl . Phys. B323 (1989) 337 R . Bluhm and S . Samuel, Nucl . Phys. B325 (1989) 275 V.A. Kostelecky and S. Samuel, Phys . Lett . B207 (1988) 169 V.A. Kostelecky and S. Samuel, Phys . Rev. D 39 (1989) 683 G. Veneziano, Nuovo Cimento 57A (1968) 190 V.A. Kostelecky and S. Samuel, Phys . Rev. Lett . 63 (1989) 224 V.A. Kostelecky and S. Samuel, Phys . Rev. D 40 (1989) 1886 V.A. Kostelecky, O . Lechtenfeld, W. Lerche, S . Samuel and S. Watamura, Nucl . Phys . B288 (1987) 173 M . Kato and K. Ogawa, Nucl . Phys . B212 (1983) 443 ; S . Hwang, Phys. Rev. D28 (1983) 2614 W. Siegel, Phys. Lett . B142 (1984) 276 ; B151 (1985) 391, 396 A . Neveu, H . Nicolai and P .C. West, Phys. Lett . B167 (1986) 307 ; Nucl . Phys. B264 (1988) 573 T. Banks and M . Peskin, Nucl. Phys . B264 (1986) 513 T. Banks, M .E . Peskin, C.R. Preitschopf, D. Friedan and E. Martinec, Nucl . Phys . B274 (1986) 71 W. Siegel and B . Zwiebach, Nucl . Phys. B263 (1986) 105 H. Hata, K. Rob, T. Kugo, H . Kunitomo and K. Ogawa, Phys . Lett . B172 (1986) 186, 195 ; Phys . Rev. D34 (1986) 2360 S . Mandelstam, Nucl . Phys. B64 (1973) 205 ; B69 (1974) 77 A .R . Bogojevic, A. Jevicki and G-W. Meng, Brown University preprint HET-672 (Sept. 1988) L . Ferrari, as described by G . Cardano, in Ars Magna, siva de regulis algebraicis (published 1545), translated and edited by T .R. Witmer, The Great Art, or the rules of algebra (MIT, Cambridge, 1968) G. 't Hooft, private communication V .A. Kostelecky and S. Samuel, preprint IUHET 174 (October 1989) D. Friedan, E. Martinec and S . Shenker, Nucl . Phys . B271 (1986) 93