The pure gauge sector of compactified superstrings

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PHYSICS LETTERS B

30 July 1987

THE PURE GAUGE SECTOR OF COMPACTIFIED SUPERSTRINGS K. E N Q V I S T 1, R. G A N D H I and D.V. N A N O P O U L O S Physics Department, University of Wisconsin, Madison, W153 706, USA Received 26 January 1987

We study the classical field equations of the pure gauge sector of superstrings compactified to four dimensions. Instantons no longer turn out to be solutions of the field equations. We also determine the dynamical behavior of the gauge coupling g in the background of a monopole field, with the result that g and the strong CP-violating angle 0 tend to a constant far from the monopole. The energy of such a field configuration is finite.

equations o f such a system, and show that for S = constant, the field equation for S alone is sufficient to determine the gauge structure of the vacuum. In particular, the self-dual instanton is no longer a solution to the classical field equations and we will discuss some o f the possible physical consequences o f this. We also find static finite energy solutions in a monopole background. Far away from the monopole the gauge coupling tends to a constant. This provides a possible dynamical mechanism for the determination of the value of g. In the effective d = 10, N = 1 supergravity theory the bosonic part of the pure gauge interactions is given by [ 7 ]

Anomaly-free superstring theories [1] in ten dimensions are serious candidates for superunified theories o f all interactions, including gravity. The low-energy field-theory limit o f the heterotic string [ 2 ], which appears to be the only string theory leading to sensible low-energy phenomenology, is d = 10, N = l supergravity coupled to Yang-Mills fields with the gauge group E8 ×E~. U p o n compactification to d = 4 it is expected to yield an effective theory o f gravity and matter which still has one supersymmetry. Such compactifications were studied in ref. [3], where it was found that solutions do exist, provided the six-dimensional compact space is a Calabi-Yau space, whence E8 x E~ is broken down to E6xE~. Considerable effort [4,5] has gone into finding the phenomenological consequences of these superstring-inspired effective supergravity theories, which include new particles and new gauge interactions at low energies. A simple truncation [4] of the d = 10 lagrangian results in an SU (n, l ) no-scale model [ 6 ]. In the present paper we will focus on the pure gauge sector of the effective d = 4 theory, which is modified from the more conventional one by the addition o f gauge singlet scalar fields coupled to the gauge field strength Fu~. The effect of this coupling is that the gauge coupling strength becomes a dynamical quantity with ( R e S ) = 1/g 2. We study the classical field

5a(l°) = - ]e0-1 Tr F ~ N •

On leave of absence from the Department of Theoretical Physics, University of Helsinki, Helsinki, Finland. 54

(1)

Here ~ is the dilaton field belonging to the gravity supermultiplet together with the graviton gMu and an antisymmetric tensor field BMN. In addition to (1), there will appear an infinite series o f higher-order operators. The full effective action has certain classical symmetries, such as scale invariance ~--,2-10, gMu--'2gMu. This is in fact a tree-level property o f the superstring [4 ] and therefore is not destroyed at treelevel by the inclusion o f the higher-order corrections to (1), involving e.g. powers Of FMN. There is also an axion-like symmetry BMN-"BMN+ alUg2U I. The classical symmetries fix uniquely [ 8 ] the form of the pure gauge part o f the d = 4 , N = 1 lagrangian, which is the one found by simply truncating [4] d = 10, N = 1

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action down to d = 4. The euclidean lagrangian for the gauge kinetic terms reads e - I ~ . t ' ( 4 ) _ --

a sa,,s* (S+S,)2

+ ~ i ( S - S * ) Tr F,,~i~u~,

TrF2

U

V=S+ S. Ic+h[ 1 +3(S+S*)/2bo] × e x p ( - 3 S / 2 b o ) Iz ,

(2)

where flu. is the dual of F~,~ (and real), and S is defined in terms of the ten-dimensional degrees of freedom as

S=O -I e 3 ~ - 3 i v / 2 D ,

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(3)

where e ~ is the conformal factor of the metric, and D is related to the field strength HMNPof BMNthrough Hu~p=e-t~2e-6"eu~p,~OPD. Fermionic terms are related to (2) by supersymmetry. In the notation of Cremmer et al. [ 9 ] f,p = J,~pS. Classical symmetries may be broken at the quantum level. It may also turn out that not all the symmetries of the effective d = 1 0 field theory are symmetries in the full tree level string theory. Indeed, it has been proposed [ 10] that the Wess-Zumino terms introduced by Green and Schwarz to cancel anomalies will modify (2) by allowing also other fields to couple directly to Fu~.On the other hand, it has also been argued [ 11 ] that Wess-Zumino_terms are not physical and should not change (2). This latter position appears to be vindicated by a recent calculation [ 12] of the five-point amplitude in the heterotic string, which involves both gravitons and gauge bosons. It shows that the Wess-Zumino term is absent. Therefore we can with some confidence take (2) to represent the correct tree-level expression in the heterotic string theory. In addition to the kinetic terms (2) there also appears a potential

(5)

where bo is the coefficient of the one-loop beta function of the hidden sector, and c and h are constants. Still V~ 0 as S ~ oo although the actual cosmological evolution of S may in fact be such that S is driven towards small values [ 14 ]. The potential (5) would be created by gauginos condensing in E~ (which needs to be broken in order that this mechanism works) at the scale #o
(6)

The field equations for S* and S are given by

O~,O~S=2 (OuS)2 - ~(F~,~Fu~+ iFu~fiuv)(S+S*) 2 (S+S*) --0, S*)2 O~OuS*- 2(0u (S+S*)

=0.

(7) ~(Fv~FV~_iF,,~ffu~)(S+S,)2 (8)

If we set S = c o n s t . = 1/g2+iO, these admit only

FU"Fa.=FU"ffa. = 0 ,

(9)

i.e., the vacuum solution (6) or a plane wave with

U

(4)

BZE, independently of the Yang-Mills field equa-

where U does not depend on S (but depends on the other scalar fields). Vmay receive modifications from the higher-order terms, and the case for eq. (4) does not appear as solid as the case for the kinetic terms. Moreover, the potential (4) shows a minimum at S--.oo or g ~ O , a situation which is physically unacceptable. It has been suggested that certain instanton-like configurations [ 13 ] of the d = 4 theory give corrections to (4) resulting in the following potential:

Fu.= +ff.. do not satisfy the classical field equa-

V=S+S.,

tion. Note that dual or self-dual instanton solutions tions. Nevertheless, they still are field configurations leading to a finite action. What is not obvious is that there are no other field configurations with a small action, and therefore the non-perturbative form of the potential (5) could pick up corrections proportional to/lo/Mc. These modifications could be of great importance for the transmission of the supersymmetry breaking into the observable sector. Let us first 55

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set SI = Im S = constant and consider the equation of motion for SR = Re S alone. From (7) and (8) we find that

F,,~P ~'~= 0

(10)

and O,,O"¢= ½e° FuvF "~ ,

(11 )

where we have introduced the notation O=ln SR.

(12)

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Therefore, far away from the monopole the magnetic field is given by

B= Bor/r 3 .

(17)

The field can be expected to be modified near r~-Mo but these changes are going to be smooth and should not lead to any dramatic consequences. With the approximation (17), (15) becomes the Liouville equation and can be solved to give g(~) = e - ~/2

The Yang-Mills field equation in this case reads

D~,F~"+O~F~=O,

(13)

= ½a e x p ( - x / ~ ) -

exp

),

(18)

where D~, is the covariant derivative, which shows that solutions vanishing by virtue of the Bianchi identity, d F = 0, such as conventional instantons, are not solutions of the classical Yang-Mills field equations. F~,~= 0 is still a solution, but there will be other finite energy solutions. Let us observe that in the presence of a static monopole, for which

where ~ = l/r, and C and a are constants of integration. The energy of the system is given by

OoAu=O, Ao=0, Fo=aokBk,

where prime denotes differentiation with respect to ~, and we have an energy cutoff at ~=Mc, where corrections to (17) become important and the monopole itself dissolves as the world becomes tendimensional. Energy is always positive and finite for g * = g > 0. If we keep the value of the coupling constant at Me fixed, as would be natural ifg(Mc) would be determined in the ten-dimensional effective theory, inspection shows that the solutions with lowest energy are those with C= 0. These are given by

(14)

where Bk is the magnetic field, eq. (10) is satisfied. Therefore we will now study solutions of the field equations in the background of a monopole field. Monopoles have in fact been shown [ 15 ] to arise when superstrings are compactified to four dimensions. This may sound surprising because there are no Higgs fields to provide the source for the gauge fields a la 't Hooft-Polyakov [ 16 ]. However, the role of the Higgs fields is played by the gauge field component mp, p > 4 . In a multiply connected internal manifold some of these gauge fields cannot be gauged to zero despite their vanishing field strength, a fact which in a sense provides them with a "vacuum expectation value". The magnetic charge of such a monopole is [ 15 ] an integer multiple of 2n/g. For such a static monopole configuration the field equation reads V20=e ° B 2 , V # X B + V XB=j,

(15) (16)

where we have introduced an external source j, representing the monopoles arising from compactification. Note that in the case of spherical symmetry (16) is just the usual monopole field equation. 56

1

E--~ f d3x [(V0) 2 + e ° B~/r41 = 2n{ CMc + 3 [ ~'(Mc) - 0 ' ( 0 ) ] ) ,

g(~) = g ( 0 ) _+x/2 BoG,

(19)

(20)

where g(0) is the asymptotic value of the coupling constant, and the energy of the configuration is

E=12x/2rCB°g(lc)

g(0)l I .

(21)

Setting Bo=n2rc/g(M~), with M~= 1 we find 12nn

E_oe( Mc) + n/x/~

(22)

so that E = 12z~V/2 for small o~(Mc). We have not investigated non-spherical solutions to (15 ) and ( 16 ), since we expect the most symmetric solution to be the stable one. It is important to realize that in the presence of

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monopoles, which indeed are there [ 14 ], g = const. is not a solution o f the field equations. However, in the asymptotic region r ~ we have a n o r m a l gauge theory with the ultraviolet properties essentially unchanged. However, processes involving long correlation lengths, such as instantons, m a y pick up m o n o p o l e - i n d u c e d modifications. F o r instance, if g(Mc) < g ( 0 ) , as might be expected for q u a n t u m corrected couplings constants, instanton density for large instanton size m a y be reduced because o f the exp[ - 1/g2(r)] factor. In realistic cases, however, the n u m b e r density o f m o n o p o l e s is constrained to be so small that this is unlikely to resolve the problem o f the large size instanton contributions [17] in E6 models, which could invalidate axionic solutions o f the strong CP problem. U p to now we have neglected the potential (4). It is obvious that for r sufficiently large it becomes important, a n d will change the asymptotic b e h a v i o r o f g ( r ) . However, if the cosmological constant vanishes when all the other fields are at their m i n i m a , U = 0 and we can forget the potential. W h e t h e r U = 0 is a solution o f the field equations is a question we do not a t t e m p t to answer here. It is also o f interest to consider the case SR = const. = 1/g2 a n d let S~ vary. Then the equations o f m o t i o n in the b a c k g r o u n d o f m o n o p o l e s read

OuOuSi = 0 ,

(23)

( OuSi) 2 = (1/g6)B 2 ,

(24)

which are solved in the spherically symmetric case by

SI =0 +_Bo/gSr ,

(25)

where 0 is a constant, which experimentally [ 17 ] we know to be 0 < 10- 9. We have no explanation for its smallness. The solution (25) allows for the interesting possibility o f having strong CP violation near the monopole, which m a y have i m p o r t a n t consequences for baryogenesis in the early universe. The energy is again finite, and with M c = l a n d Bo=2nn/g is E=32rcSn2/g 6. Note that as long as V= V(S+S*), eq. (23) does not pick any corrections from the potential even if U S 0. The pure gauge sector o f compactified superstrings, as given by eq. ( 2 ) , thus possesses a rich variety o f finite energy solutions in the background

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o f a m o n o p o l e field. These solutions m a y be o f great i m p o r t a n c e for understanding the true d y n a m i c a l b e h a v i o u r o f the physical gauge coupling. However, in view o f the absence o f instantons as solutions to the classical field equations the role o f non-perturbative contributions to V(S, S*) needs to be clarified before definite conclusions about supersymmetry breaking in the observable sector can be reached.

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[9] E. Cremmer, S. Ferrara, L. Girardello and A. van Proeyen, Phys. Lett. B 116 (1982) 231; Nucl. Phys. B 212 (1983) 413. [ 10] H.P. Nilles and L.E. Ib~i~ez, Phys. Lett. B 169 (1986) 354. [ 11 ] J. Ellis, C. Gomez and D.V. Nanopoulos, Phys. Lett. B 168 (1986) 215. [ 12] J. Ellis and L. Mizrachi, CERN preprint, in preparation. [ 13] M. Dine, R. Rohm, N. Seiberg and E. Witten, Phys. Lett. B 156 (1985) 55; J.P. Derendinger, L. Ib~i~ez and H.P. Nilles, Phys. Lett. B 155 (1985) 65;

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E. Cohen, J. Ellis, C. Gomez and D.V. Nanopoulos, Phys. Lett. B 160 (1985) 62. [ 14] K. Enqvist, L. Papantonopoulos and K. Tamvakis, Phys. Lett. B 165 (1985) 299; K. Enqvist, D.V. Nanopoulos, L. Papantonopoulos and K. Tamvakis, Phys. Lett. B 166 (1986) 41. [ 15] X.G. Wen and E. Witten, Nucl. Phys. B 261 (1985) 651. [ 16] G. 't Hooft, Nucl. Phys. B 79 (1974) 267; A.M. Polyakov, JETP Lett. 20 (1974) 194. [ 17 ] M. Dine and N. Seiberg, CCNY preprint CCNY-HEP-86/2.