VOLUME 66, NUMBER 14 PHYSICAL REVIEW LETTERS 8 APRIL 1991

Photon and Masses in String Theories

V. Alan Kostelecky ' ' ' and Stuart Samuel ' Department, Indiana University, Bloomington, Indiana 47405 ' ~Physics Department, City College of City University of iVew York, iVew York, lVew York l003l (Received 10 May 1990)

We show that string theories allow interactions potentially leading to and graviton mass terms when one or more scalar fields acquire vacuum expectation values. A general analysis is presented for the case when one of these scalars is the electroweak Higgs-boson doublet. The results provide a new constraint on model building for string theories.

PACS numbers: 11.17.+y, 04.50.+h, 12.10.Gq, 14.80.Am

Tight experimental upper bounds exist for the pho- shell three- coupling. Naively, the second term ton' and graviton masses. In standard theories of is not invariant under the gauge transformation BA„ electrodynamics and general relativity, whether unified =B„k. However, at order g the tachyon gauge transfor- or not, these particles are exactly massless because un- mation includes the term 8p = —(3J3/4)ga'A" ri„k. The broken gauge invariances preclude mass terms. latter makes the tachyon mass term also naively nonin- , which is based on an extended object, variant. However, the two extra pieces cancel when provides a promising and different approach to unifica- combined. tion. Particles correspond to different vibrational modes. The new mass terms arise from the stringy S"T T in- Interactions are spread over a , softening teractions' if one or more scalars S acquires a vacuum forces at short distances and leading to an ultraviolet- expectation value (VEV). For the open bosonic string, it finite theory. In this Letter, we demonstrate that a class has been shown'' that nonperturbative, nonzero scalar of stringy interactions can lead to mass terms for gauge VEVs give rise to a mass term for 4„. In general, the bosons, in particular, for the photon and graviton. The consequences for physics depend on the nature and size appearance of these new terms is a consequence of the of the scalar VEVs. extended nature of strings. Let us define a strongly nonperturbative string vacu- The mass terms of interest arise via collective effects um' as one in which one or more scalar fields have in string interactions. These effects can be examined VEVs of order of the Planck mass Mp. Such vacua are directly in a given string model via covariant field theory. natural to strings since the only fundamental scale is the Such field theories are now available for the open boson- Regge slope a'= I/Mp. However, there are reasons for ic string, the closed bosonic string, and the open super- believing a realistic string theory might not be strongly string. These models may therefore be studied to seek nonperturbative. In the canonical perturbative vacuum, insights into the nature of multistring effects. superstrings generate a wide range of features seen in When each string mode is described by an ordinary nature: a massless graviton, massless gauge bosons cou- particle field, the action contains an infinite number of pling correctly to chiral fermions, and gauge groups kinetic and interaction terms. One striking feature is the large enough to incorporate SU, (3)SSUt. (2) SUy(1). appearance of interaction terms of the form S"T.T, The ingredients are in place for a complete unification of where S is a Lorentz-scalar field and T is a Lorentz ten- the four fundamental forces. In contrast, investiga- sor. Such interactions are truly stringy: Gauge invari- tions" for bosonic strings indicate that the physics in a ances and renormalizability generically exclude them strongly nonperturbative vacuum is unlikely to resemble from the usual particle field theories of gauge and/or that of the canonical vacuum. Since the physics is ap- gravitational interactions. In string theories, the gauge proximately correct in the latter, it seems unnatural to invariance is maintained by cancellations among the ex- search for a radically different ground state. pressions arising from a gauge transformation. Nonetheless, nature cannot be in the canonical pertur- Scalar-tensor-tensor terms of this type do occur in bative vacuum. The SUL(2)-doublet Higgs boson H of some string theories. A simple example is provided by the electroweak model, which is perturbatively a tachy- the tachyon-vector-vector coupling in the open bosonic on, acquires a VEV (H)((Mp. We call weakly nonper- string. The Lagrangian contains the terms turbative any string vacuum in which all scalar VEVs are small compared to Mp. It seems that a realistic — . . string model has a weakly nonperturbative ground state. tr —,y2 gtltA„A" +. 2a' 8 In this Letter, we investigate the possibility that pho- ton and graviton mass terms arise in string theories from where P is the tachyon, A„ is the vector, and g is the on- scalar VEVs smaller than Mp. The issue has import be- 1991 The American Physical Society 1811 VOLUME 66, NUMBER 14 PHYSICAL REVIEW LETTERS 8 APRIL 1991 cause there is at least one scalar field in nature, H, with exclude such interactions if these theories are to be ex- a nonzero VEV. We show that, although (H) is minis- perimentally viable. cule compared to Mp, sizable photon and graviton mass The results apply if the electroweak model is broken terms can be generated. by a technicolor ' condensate (yy), although fewer A source of uncertainty is the identification of H with dangerous terms occur. Possible masses arise from in- a field in string theory: H could be a fundamental sca- teractions such as lar, it could be composite, or it could arise during com- pactification. In certain models, there are other scalars q"a'" "' A A 4 d'x(yy) T with nonzero VEVs. Our main point is that for every ~i scalar in a model one must that the Since (yy) = (600 GeV), m~, = R Mp with R operator string verify ' photon and graviton do not acquire masses above the = 2x 10 . Only p ~ 2 conflicts with experiment. current stringent experimental limits. In certain cases, some interactions can automatically Let us perform a model-independent analysis when be absent, perhaps due to discrete or continuous sym- metries or there is a scalar field 5 with (5) where R((l. . For example, for the =RE, odd-n H"ATA One choice for S is the electroweak SUi, (2)-doublet H, SUL(2)-doublet H the terms cannot ap- ' pear for group-theoretical reasons. Another escape is which has (H)=175 GeV, i.e., R=10 . However, other possibilities exist. First, in some generalizations of seen in the open string with Paton-Chan factors if the the , SU, (3)SSUi.(2)SUi (I) singlets photon generator and the Wilson-line expectation (Al) =(5) are orthogonal in Since open-string acquire VEVs. Second, in compactified string theories S group space. might be a higher-dimensional tensor but a D =4 scalar. amplitudes involve a single trace, no photon mass term is For example, in Wilson-line breaking' ' S is a higher produced. Even when generators overlap, the trace may vanish because comrnutators However, since component of a gauge boson. In contrast, S is probably appear. not the , which is believed to couple only through closed strings and gravity appear at loop level in open derivative interactions. strings, graviton mass terms must still be checked. In the four-dimensional effective theory, the interac- Let us next consider the situation for individual string tions of interest have the form theories. The field theory of the open bosonic string does have interactions I„—S T. T. An explicit example with d4xS g n =1 is provided by the tachyon-vector-vector coupling discussed above. In a stable vacuum, the tachyon must where AT is the photon A„or the graviton h„„and have a nonzero expectation value and a photon mass ' q is the gauge coupling. For the open strings q term is generated. = V 'i J8zcG a' i, where we take the compacti- Next, consider the closed bosonic string. Among the fication volume V= Mp, and only n =1 terms occur ex- cubic interactions in the nonpolynornial string field the- plicitly in the field theory. However, terms with n~ 2 ory one discovers the term (3 /2 )gph„,h"', where p is can be generated by integrating over heavy fields. For the closed-string tachyon, h„, is the graviton, and g is the the heterotic strings, a'q = 16zG. In any case, the in- on-shell coupling between three closed-string . teractions I„produce mass terms m~, = R" Mp, which This term is of the form Ii. When (p)&0 in the true are very small for large n. vacuum, a inass term is produced for the graviton. ' Experimental data place constraints on n. The current A covariant is also known for the limit on the photon mass is' my & 6x 10 GeV. It is a open superstring. Since there is no tachyon, the low- secure bound based on observation of Jupiter's magneto- energy limit can be taken directly. To lowest order, this sphere during the Pioneer-10 mission. A bound as strict limit yields an ordinary supersymmetric gauge field as my & 3 x 10 GeV has been conjectured from theory. One therefore expects no terms I„ for n ~ 2. known and speculated properties of galactic magnetic However, there is no evident reason why terms n ) 3 fields. Other estimates ' range from 3 x 10 to 3 should be absent in general. Such terms would appear as x 10 GeV. For the graviton, one has m& & 10 nonrenorrnalizable corrections to the low-energy theory. GeV from observations of gravitationally bound clusters Their apparent lack of gauge invariance is misleading: of galaxies. They combine with other nonrenormalizable corrections For R = 10 ', these bounds show the necessity of ex- to maintain gauge invariance. Any eA'ort to construct a cluding interactions I„ for n(7. Note that the larger realistic model involving nonzero scalar expectations the VEV (S), the more powers of n must be considered, needs to provide a mechanism whereby gauge mass and therefore the greater the constraints on n. For ex- terms are absent or suppressed. ample, grand-unified theories and typical string models Heterotic strings' are promising candidates for a undergo symmetry breakdown at an intermediate mass realistic phenomenology. ' ' There are two types of scale Mi=10' GeV. This involves a scalar S with D =10 heterotic strings: those without tachyons' ' and (S)=RMp, R) 10, which requires the absence of those with tachyons. The latter theories are still of terms S"ATM for n ~35. Arguments must be found to phenomenological interest because the tachyon p might 1812 VOLUME 66, NUMBER 14 PHYSICAL REVIEW LETTERS 8 APRIL 1991

yield spontaneous Lorentz-symmetry breaking or quartic interactions does not lead to graviton mass terms grand-unified symmetry breaking. either. A direct analysis is hampered by the absence of a Although no terms with n ~ 2 appear in these strings, heterotic-string field theory. For the heterotic strings there is no apparent reason to exclude the nonrenormal- with tachyons, one can use izable corrections I„with n) 3. Identification in the methods to investigate amplitudes that would arise from first-quantized formulation of nonvanishing amplitudes interactions I& in toroidally compactified strings. Here, coming from such terms could, in principle, provide we consider pATA interactions. direct evidence for their existence. To perform the off-shell calculation, ' one needs In summary, we have shown that stringy interactions vertex operators for the states. They are the product of a can allow scalar VEVs to generate photon and graviton left vertex operator VL with a right vertex operator Vg. mass terms. In particular, one scalar, the Higgs field, is For AT, VL""(z) =8X'exp(k X)(z), and VL"(z) =BX' known to have a nonzero VEV. For each candidate xexp(k X)(z) or exp(+1k X )exp(k X)(z). Here, the string theory, it must be explicitly argued that photon X are compactified bosons, k 6 A, where A is a lattice, and graviton mass terms above experimental bounds are and +1k k =2. Equations (4.1b) and (4.1f) of Ref. 24 absent. provide Vg'(z). For the tachyon, Vl=exp. (+1k X ) Under certain circumstances, string theories might xexp(k X)(z) with +1k k =1, while Vg =cexp( —p) produce photon and graviton masses just below the xexp(k X)(z) and Vg =(ck y —y/2)exp(k X)(z) in current experimental limits. For example, the term I7 ' the ( —1) and (0) pictures, respectively. Here, p is the with R = 10 produces a borderline effect that might first-quantized spinor ghost. be revealed by improved experiments. A measurement We find25 with a positive result could thereby provide experimental ~ support for strings. Il (VL (z 1 ) VR (z 1 ) VL (z2) VR (z2) VL (z3) V~ (z3)O) We thank Michael Martin Nieto for discussions on the where 0 X(i ) in the ( —1)-picture formalism and experimental constraints for photon and graviton masses, Duff' This 0= Y-2 in the (0)-picture formalism. Here, X(z) is and Mike for stimulating correspondence. En- the picture-changing operator and Y-2= Y( —i) Y(i) work is supported in part by the U.S. Department of is a ( —2) inverse picture-changing operator, where ergy under Contracts No. DE-AC02-84ER40125 and Y(z) =c ti(exp( —2p). No. DE-AC02-83ER40107. The correlations factorize into a product of left and right sectors. Explicit calculation reveals the right-sector factor is zero due to conservation of ghost and y-fermion ' Electronic address: kostelec iubacs (bituet). numbers. The left-sector expectation is also zero: There b Electronic address: samuel ccnysci (bitnet). is no group invariant involving one vector representation 'L. Davis, Jr., A. S. Goldhaber, and M. M. Nieto, Phys. Rev. with two adjoint representations. Thus, pATA terms Lett. 35, 1402 (1975). are absent. 2J. C. Byrne, Astrophys. Space Sci. 46, 115 (1977). There are three heterotic strings without tachy- 3E. Williams aud D. Park, Phys. Rev. Lett. 26, 1651 (1971). ons:' EsEs, SO(32), and SO(16)SSO(16). Using 4A. S. Goldhaber and M. M. Nieto, Rev. Mod. Phys. 43, 277 the fact that the low-energy limit is an ordinary super- (1971). 5A. S. Goldhaber and M. M. D one expects there are no terms of the Nieto, Phys. Rev. 9, 1119 gravity theory, (1974). form I„with n ~ 2. To verify this, consider Al as a can- SE. Wittenn, Nucl. Phys. B268, 253 (1986). didate for where is a component ~ S, Al of A„with p 4 7T. Kugo and K. Suehiro, Kyoto University Report No. in a theory with Wilson-line breaking for which (Al)&0. KUNS 988 (to be published); T. Kugo, H. Kunimoto, and K. Explicit calculation shows that there is no static n=1 Suehiro, Phys. Lett. B 226, 48 (1989); M. Saadi and B. term. For n=2, consider the four-vector-boson ampli- Zweibach, Ann. Phys. (N.Y.) 192, 213 (1989); M. Kaku, tude given in Eq. (5.17) of Ref. 19. Graphs with mass- Phys. Rev. D 41, 3734 (1990). less particles appearing in propagators must be subtract- C. Preitschopf, C. B. Thorn, and S. Yost, Nucl. Phys. B337, ed. The terms proportional to a single trace of four ma- 639 (1990); 1. Ya. Aref'eva, P. B. Medvedev, and A. V. Zu- trix generators are not problematic because commutators barev, V. A. Steklov Mathematic Institute Report No. SMI- appear. The heterotic string reproduces the quartic vec- 10-1989 (to be published). For the special case n=2, terms of this form can arise in tor-boson coupling tr([A„,A,][A",A'] ) in the low-energy particle field theories in the context of the mechanism. limit, as one can check. Thus, the dangerous terms are Higgs ' Note that the mass terms do not arise from the standard of the form tr( )tr( ). However, these vanish in the stat- = Higgs mechanism, with the possible exception of the case n =2. ic limit s =t u =0. Also, in contrast to the Higgs case, the stringy mechanism does The two-graviton two-vector-boson amplitude can also not necessarily conserve the total number of light degrees of be obtained using the results of Ref. 19. All terms van- freedom. This is possible despite the manifest unitarity of the ish in the static limit, so Wilson-line breaking in the theory because of the presence of an infinite number of states

1813 VOLUME 66, NUMBER 14 PHYSICAL REVIEW LETTERS 8 APRIL 1991 and interactions and because of the stringy modifications to the this paper even more stringent. usual particle gauge transformations. ' D. J. Gross, J. A. Harvey, E. Martinec, and R. Rohm, Phys. ''V. A. Kostelecky and S. Samuel, Phys. Rev. Lett. 64, 2238 Rev. Lett. 54, 502 (1985); Nucl. Phys. B256, 253 (1985). (1990); Nucl. Phys. B336, 263 (1990); Phys. Rev. D 42, 1289 ' D. J. Gross, J. A. Harvey, E. Martinec, and R. Rohm, (1990). Nucl. Phys. B267, 75 (1986). '2The terminology is used because, at the semiclassical level, 2ON. Seiberg and E. Witten, Nucl. Phys. B276, 272 (1986); any nonzero VEVs are necessarily nonperturbative in the string L. Alvarez-Gaume, P. Ginsparg, G. Moore, and C. Vafa, Phys. coupling g. See V. A. Kostelecky and S. Samuel, Phys. Lett. 8 Lett. B 171, 155 (1985); L. J. Dixon and J. A. Harvey, Nucl. 207, 169 (1988). Phys. B274, 93 (1986). '3Y. Hosotani, Phys. Lett. 126B, 303 (1983). ~'R. Arnowitt and P. Nath, Phys. Rev. Lett. 62, 2225 (1989); ' P. Candelas, G. Horowitz, A. Strominger, and E. Witten, Phys. Rev. D 40, 191 (1989), and references therein. Nucl. Phys. B258, 46 (1985). V. A. Kostelecky and S. Samuel, Phys. Rev. D 39, 683 '5S. Weinberg, Phys. Rev. D 13, 974 (1976); 19, 1277 (1989);Phys. Rev. Lett. 63, 224 (1989); Phys. Rev. D 40, 1886 (1979); L. Susskind, Phys. Rev. D 20, 2619 (1979). (1989). ' No general principle such as gauge invariance can guar- 23S. Samuel, Nucl. Phys. B308, 317 (1988); O. Lechtenfeld antee the masslessness of a gauge boson, or the Higgs eAect and S. Samuel, Nucl. Phys. B310, 254 (1988). could not occur. The reader is invited to verify that the exist- 24We follow the notation in Refs. 11 and 23 and in V. A. ence of the term tr(&A„A") is consistent with the on-shell Kostelecky, O. Lechtenfeld, W. Lerche, S. Samuel, and S. first-quantized theory by evaluating the expectation Watamura, Nucl. Phys. B288, 173 (1987).

(c exp(k 1.X)(z 1 )c0X„exp(kz X)(z2)c BX"exp(k3' X)(z3))WO. The proportionality factor involves momentum-dependent This shows that the analysis is independent of the of-shell ex- conformal factors, a coupling constant, and a combinatorial tension. coefficient. ' The graviton terms are unlikely to represent a cosmological O. Lechtenfeld and S. Samuel, Phys. Lett. B 213, 431 constant because the same mechanism generates a photon mass (1988). in the open bosonic string. In any case, the observable conse- D. Friedan, E. Martinec, and S. Shenker, Nucl. Phys. quences of such a cosmological term would make the bounds in B271, 93 (1986).

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