Saving the axions in superstring models

Volume 161B, number 4,5,6

PHYSICS LETTERS

31 October 1985

SAVING T H E A X I O N S IN S U P E R S T R I N G M O D E L S ~' Katsuji Y A M A M O T O Institute of Field Physics, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA Received 28 June 1985

In the effective four-dimensional supersymmetric theories derived from the superstring theories, we investigate a possible way to evade the cosmological bound fa -< 1012 GeV on the axion decay Constant. A significant amount of entropy could be produced by the out-of-equilibrium decay of the Higgs field which is responsible for the breaking of the extra gauge symmetry at an intermediate mass scale M t. The appearance of such a Higgs field is a general feature of the superstring models. We find that if the gravitino mass m3/2 -1-102 GeV and M t - 107-10 l° GeV, the axions are diluted efficiently by the entropy production, and fd - 0( 1016 GeV) will still be acceptable.

Superstring theories based on the 0(32) or E(8) × E(8) gauge group have been investigated extensively since the recent discovery of the cancellation of Y a n g Mills and gravitational anomalies [ 1 - 8 ] . A phenomenologically interesting prediction of the superstring models is the appearance of axions in the effective four-dimensional theories, as discussed by Witten [2]. The first type axion ¢n appears as a part of the antisymmetric tensor field BAB of gauge singlet such that aUen ~ egV°PHz,ap (A, B, ... = 1 - 1 0 and/a, v .... = 1-4) where HABC is the gauge invariant field strength for BAB defined in terms of differential forms by H = dB + I2 with the Chern-Simons form I2. This type of axion couples to the gauge fields through the Bianchi identity dH = - T r F 2 + Tr R 2. The second type axion q~a c o m e s from the gauge supermultiplet of G = 0(32) or E(8) X E(8). A local U(1) subgroup of G breaks spontaneously if its gauge field develops a topologically nontrivial configuration tangent to the extra six dimensions so as to realize the compactification. This local U(1) symmetry, however, may not break completely, but appear as a global U(1) symmetry in the ordinary four dimensions, since the U(1) gauge field is Research supported in part by the D.O.E. under Grant No. DE-AS05-79ER 10448. 1 Address after September 1, 1985: Department of Physics, The Johns Hopkins University, Baltimore, M D 21218, USA.

0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

neutral. If such a global U(1) symmetry has an SU(3)C anomaly, it can be regarded as the Peccei-Quinn symmetry [9,10]. Hence the identification of the second type axion depends on the embedding of the U(1) subgroup in G which is responsible for the compactification. The axion couplings to the gauge fields are given by ~,,v (1/32rrE)(C~n/Mn + ¢~a/VPO) p" i~,,"i = (1/321r2) f a 1 a Fff, where (~a) - VpQ and Mn is determined as MPlanek/ 96~r5/2 --~7 × 1015 GeV by reducing the ten-dimensional action to the effective four-dimensional one. Only one combination a - cos a ~a + sin a ~n (tan a = VpQ[Mn) couples to Fff, as pointed out by Choi and Kim [11 ]. This combination actually plays the role of "axion". It should be noticed that the scale of the axion decay constant fa depends on whether the second type axion exists or not: fa "" VpQ ('~ Mr0

with t~a,

" M n "" 7 × 1015 GeV

without Sa.

Hence the existence of the second type axion seems indispensable to satisfy the cosmological bound fa ~ 1012 GeV [12]. Unfortunately, it turned out [11] that Calabi-Yau manifolds with SU(3) holonomy for the 289

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spontaneous compactification cannot accommodate the second type axion since the nonvanishing gauge field configurations break completely the SU(3) subgroup of E(8) or 0(32). However, Calabi-Yau manifolds will be preferred for a consistent construction of string theories since they are Ricci fiat and may preserve the N = 1 supersymmetry in four dimensions [4]. Therefore, the axion apparently looks harmful in superstring theories. In this letter, we would like to investigate a possible way to evade this axion problem in superstring theories. It was suggested by Steinhardt and Turner [13] that entropy production at a temperature T"~ 1 M e V 1 GeV of the universe can dilute the axions significantly and relax the bound on the axion decay constant: fa ~ A9/ll X 1012 GeV,

(1)

where A represents the factor of entropy increase. If A ;~ 103,fa ~ 7 X 1015 GeV will still be acceptable. Entropy production could be due to out-of-equilibrium decays of a relic particle species. Such a particle species must decouple sufficiently from the light particles. We first discuss the fact that a light Higgs field with the desired decoupling property generally appears in the effective four-dimensional supersymmetric theories derived from superstring theories with the gauge group E(8) X E(8) ,1. Then, we calculate the entropy production by the I-Iiggs field. Due to the holonomy group SU(3) of Calabi-Yau manifolds, one of the E(8) gauge groups where the SU(3) is embedded breaks down to E(6). A multiply connected manifold N/F can be constructed by using a discrete symmetry F acting freely on the original Calabi-Yau manifold N. I f N / F is taken for the compactification, the E(6) further breaks at MG to a subgroup K with rank higher than four which commutes with F embedded in the E(6). The effective low-energy supersymmetric theory is described by the gauge group K × E(8), and some chiral supermultiplets of (Ri, 1) under the K X E(8) will survive as massless fields below M G where Ri is contained in 27 or 2-7 of E(6) [4,6,7]. Let us consider, for example, a case K = SU(3)C X

,1 We have encountered a similar kind of Higgsfield in a class of supersymmetric theories where the gauge symmetry breaking occurs in a continuum of supersymmetric minima due to the radiative corrections on the soft supersymmetry breaking terms [14,15]. 290

31 October 1985

SU(2)L X U(1)y X U(1)N X U(1)I. (The discrete subgroup P associated with the multiply connected mani. fold may be embedded as exp [2ni(pY + qN+ rl)/n] E Z n.) Then, we may have massless chiral superfields,

S(1, 1,0, 0, 1),

S(1, 1,0, 0 , - 1 ) ,

(2)

where the K quantum numbers are indicated in parentheses with a convenient normalization for the U(1)I charge so that QI(S) = 1. S (S-) is one of the two SU(3)C X SU(2)W X U(1)y sing,lets contained in a 27 (2-7) of E(6). These multiplets will break K to K0 SU(3)C X SU(2)W X U(1)y X U(1)N at an intermediate scale. (This intermediate scale will be also desired to give heavy masses to the extra quark (D and D e) and lepton (L and L c) supermultiplets transforming as 10 under SO(10) which is contained in 27 of E(6).) The relevant part of the superpotential for S and g will consist of only nonrenormalizable terms which might be regarded as the effects of superheavy fields and/or gravitational interactions [7] : W = {~MGI(gs) 2 + ....

(3)

Renormalizable terms such as S 3 are absent because of the U(1)I symmetry. The mass term SS has been excluded by the first assumption that S and S are massless below MG. Since the VEV's of K0 nonsinglets will vanish at this scale, the terms such as (K0 nonsinglets)2S are irrelevant at the tree level for the effective potential of S and S. (We should, however, remember that they may contribute to the radiative corrections to the effective potential, as discussed below.) Due to the specific form of the superpotential (3), there is a flat line of supersymmetric minima in the scalar potential corresponding to FS = b-~ = 0 and D(U(1)I) = 0 in the limit ofMp ~ ~ and the gravitino mass m3/2 "-,"0,2:

ISl = 1~1 = ¢ / 2 ,

(4)

where q~is undetermined in this approximation. (Hereafter we use the same notations for the chiral superfields and their scalar components.) It should be no¢2 Actually, since there am several fields with the quantum numbers of S and S, we have to look for a minimum in a hypersurface defined by D (U(1)I) = 0, Le. ~lSa 12ISil2 = 0. Then, we should regard that S and S axe suitable linear combinations of Sa and Si respectively so that (S) = (S) = (Y-ISa 12)1/2. This complexity does not change the essential results obtained in the text.

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ticed that the appearance of a flat direction in the scalar potential is a general feature of the superstrin~ models since the cubic couplings of the SU(3)W × SU(2)L × U(1)y singlets contained in 27 and ~ are absent in the couplings 273 and ~ 3 of the E(6) [6,7]. A minimum will be generated in the direction (4) by the effects of supersymmetry breakings. Soft supersymmetry breaking terms such as the scalar mass terms ~,im21¢i12may be induced by the supergravity couplings with the "hidden" E(8) sector where the supersymmetry could be spontaneously broken by the gaugino condensation [8]. The total scalar potential in the direction (4) becomes as follows, including such effective soft supersymmetry breakings: V(~b) = ½m2~b2 + a--~(C~+ C*~*) m3/2MGl(b4 + 3-~1~IZMGZq~6 + O(m3/2MG3~6),

(5)

where m 2 _=x~(m S2 + m~s) . The second term of(5) with ICl "" O(1) comes from the soft supersymmetry breaking term corresponding to the first term of (3). The nonrenormalizable terms in V(¢) become important beyond a scale (m3/2Ma)1/2 (109-1011) GeV, while the mass term dominate below that scale. The scalar (mass)2's of soft supersymmetry breakhags, which may take the same value dictated by super2 2 gravity, i.e. mi =m3/2 atMp, will be renormalized by the Yukawa couplings. For exam~ale, let us consider a coupling ADeDS contained in 2 7 a o f the E(6). (With the VEV of S, the quark supermultiples D and D e become heavy.) This coupling contributes to the renormalization of ms2. The relevant renormalization group equations are given by dA/dt = (5/16rr 2) ;k3 ,

(6)

dA/dt = (5/8 lr2) A X2 ,

(7)

dm2/dt = (1/8rt 2) ~,215m2 + (4 + 3.4 2) m3/2] 2

,

(8)

neglecting the contributions of other couplings for • .. 2 2 2 2 sLmpliclty (m D -- roDe = constant and m s - 5m D = constant), where t - ln(~/Mp) and A M)eDS is the soft breaking term. A solution for m 2 with m 2 = m]/2 at ~b=Mp (t = (3) is

31 October 1985

2 2 ms=m3/2[u+](u-1)+{A2(O)u(u-1)],

(9)

where u = (1 - [sxZ(0)/8~r2] t} -1 .

(10)

By taking X2(0)/4rr-,- 0.063 and A 2(0) "" 0.667 for example, ms2 turns negative at q~~ 5 × 109 GeV. A similar result will be obtained for rn~ by taking a term XI)CDg contained in (~-~)3. Therefore we may expect for the (mass) 2 term in V($) that m2($)=0

at

¢=Mo~Mpexp(-8n2/~2),

(11)

where ~, is some mean value of the relevant couplings. Since m 2 becomes negative at Mo by radiative corrections, a minimum of the potential can be generated at (q~)=MI ~ rnin(M0, (m3/2MG)l/2),

(12)

where min(A, B) indicates the smaller one of A and B. It should be noticed that the symmetry breaking scale MI can be much smaller than O(m3/2MG)1/2 even if the nonrenormalizable terms are considered. Furthermore, if the leading term in W is (SS)4 instead of (SS) 2 (i.e., ~ ~ 0), MI will become even larger than (m3/2MG)1/2. Hence we may generally regard that

m3/2 < M I


(13)

depending on the model parameters. The mass of the Higgs scalar, say "flaton", is given by m 2 = d2V/d~b2 I¢=MI ~

O(m]/2).

(14)

This shows a remarkable fact that the flaton is very light independently of its large VEVMI >> m3/2. Furthermore, since all the particles such as D and D e that couple directly with the flaton acquire heavy masses of O(MI), the flaton fairly decouples from the light fields including itself. These characteristic properties of the fiaton could bring interesting effects in the history of the early universe, as discussed below. We now investigate cosmological implications of the symmetry breaking induced by the flaton. First of all, since V(O) - V((q~)) ~ 0(m2/2 M2) ,<1~i due to the specific form of the fiaton potential (5), the critical temperature for the phase transition from the symmetric vacuum at $ = 0 to the asymmetric vacuum at = MI becomes much smaller than the symmetry breaking scale [16]: 29/

Volume 161B, number 4,5,6 T c "" ( m 3 / 2 M i ) 1/2 .

PHYSICS LETTERS (15)

Secondly, as seen in the above arguments, the flaton is light, while its couplings t o the light fields are fairly suppressed. The flaton may interact with the light fields, say f, including itself through heavy intermediate slates of O(MI). For example, a D-type interaction KMI2(StS fi'f)D will be generated by radiative corrections (supersymmetry is not broken by (4) in the limit of m3/2 ~ 0 andMp -* co) where r < O(1) is a relevant coupling constant depending on the Yukawa couplings and/or gauge couplings (typically r "" e2/167r2). This supersymmetric interaction includes the couplings such as MI'-2(S*S ~f~ua u ~f) in terms of the component fields. We also obtain an effective coupling KMI-I(sftf)D by substituting (S) = M I / 2 in the previous one. Therefore, the scattering cross section and the decay width of the flaton are given by o "~ g2(s/M~) ,

I'e~ ,,, r 2( m 33/ 2 / M ~2) ,

(16,17)

respectively (note m~ "~ m3/2), where s < M 2 is the square of center-of-mass energy. These results obviously show the sufficient de¢oupling of the flaton. For example, if we take m3/2 "" 100 GeV, MI ~" 1010 GeV and r "~ 10 -2, the lifetime of the flaton becomes of O(10 -6 s) corresponding to the temperature of the • universe T ~ 1 GeV, as desired for diluting the axions. By comparing the thermal-averaged interaction rate of the fiatons (nov) ~ K2(T5/M 4) with the Hubble constant H ~ T2[Mp for the radiation dominated universe, we find the decoupling temperature of the flatons: T d "" ( M ~ i / r 2 M p ) 1/3 .

will decay at a time t i ~ I ~ 1 producing a significant amount of entropy to reheat the universe, when the Hubble constant H ~ [P(Ti)[M2p] 1/2 becomes comparable with the flaton decay rate £~. The temperature at tijust before the reheating is T i " ~ 4 / 3 M 2 / 3 m l t ~ 6 M ~ 3 / 2 . The final temperature after the reheating is determined by T 4 "~ P(Ti): .,~,1/2~3/2 ~,t-1 T f " ~,,p ,,,3/2~,,i . (19) Consequently, the factor of entropy increase is given by A ~ (Tf/Ti) 3

.,-1 "~,t-1/2_-1 Ai3/2 • 'P "'3/2"'I

(20)

"

In order to save the axions, we need A ~ 103 as seen in (1). There are also some constraints on Tf and A from standard big bang cosmology [13]. Tf must be between the temperature of nucleosynthesis and the temperature when the axions begin to oscillate, i.e. 1MeV < Tf < 1 GeV. Since the entropy generation also dilutes the baryon-to-entropy ratio, A must be less than O(106). By taking account of these constraints, we show in fig. 1 the desired values of Mi and m3/2 (shaded region)to save the axions. We fred that if m3/2 ~ (1"102) GeV,

M I ~ (107-1010) GeV

for K "~ ( 1 - 1 0 - 2 ) , the axion decay constant fa of

id4 ,

,

lo,~t

.y/. Tf ~ I M e V

Tf ~ I G e V

(18)

I f M I ;~ 107 GeV with m3/2 "~ 102 GeV and K ~ 10 -2, the flatol~ ~.will already decouple for temperatures T

T~. After the phase transition is completed at T ~ Te, the flatons oscillate coherently around the true vacuum at @= MI like nonrelativistic matter due to the decoupling property. (We assume that tunnelling effects are significant just below the critical temperature so as to complete the phase transition quickly. The delay of the phase transition might produce much more entropy, as discussed later.) Since the universe is dominated by the flatons until they decay, the eJtergy density stored in the oscillation of the flaton field redshifts as p(T) p ( T e ) ( T [ T e ) 3 with P(Tc) "~ T 4, and the scale factor expands as R "" T - 1 "" ( Tc/M2)I/ 3 t 2/3. The flatons 292

31 October 1985

~W° ~ ' 77

."-

108

/

z~-lo3 ' ' ' ' ~ . - - - .- - . - -- "

....... _

T~>T~..---" (~--10 - 2 )

lo o"" 0

I

I

10 ~

10 2

f

~, 10 3

10 4

m3t2(GeV)

Fig. 1. The desired values of m3/2 and MI to save the axiom are shown in the shaded region, where the effoetive coupling K ~ 10- 2 is taken fox example.

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O(1016 GeV) will still be acceptable. This result may suggest that MI ~ O(m3/2MG) 1/2. If the tunnelling is not efficient at T ~ To, the phase transition might be delayed. Since the symmetric state = 0 becomes no longer a local minimum for T O(m3/2) due to the negative (mass) 2 term o f 0(m2/2) in the zero-temperature potential (m 2 < 0 for ~ < M 0 ) , the phase transition temperature Tt could be o f O(m3/2). In this case, the energy density stored in the false vacuum at ~ = 0 does not redshift until the temperature decreases to Tt. Hence the expansion rate of the universe after phase transition at Tt becomes larger by a factor o f O((Tc/Tt) 3/2) compared with that for the previous case where we assumed that the phase transition is completed at T ' ~ Tc. Correspondingly, T i is lowered by a factor o f O(Tt/Te), while T 4 ~ P(Ti) "" p(Tc)(Ti/Tt) 3 is almost unchanged. Therefore, ent r o p y production is increased by a large factor ~(Tc/Tt) 3 ~ (Mi/m3/2) 3/2, if T t ~ m3/2. I f Tf > 1 GeV, since the axions do not oscillate until the temperature decreases to about 1 GeV, the effective factor ~leff for diluting the axions is given b y ~eff A(1 GeV[Tf) 3 ~ (1 GeV/Ti) 3 [13]. Even excluding the baryogenesis problem, the reheating temperature must be marginal for nucleosynthesis. I f we take m3/2 100 GeV and K ~ 10 - 2 for example, we find that ~ f f ~ 103 f o r M i 2 7 × 107 GeV corresponding to Tf ~ 500 GeV and A 2 1011. Although we need to calculate the bounce action to determine precisely the phase transition temperature Tt (m3/2 ~ Tt ~ Tc), we can regard (20) at least as the lower bound on entropy production due to the flatons. I f Tt "¢ Tc, we might worry about baryogenesis at a low temperature ,,, O(10 2 G e V ) i n s t e a d o f the axion problem. In summary, we have investigated a possible way to evade the axion problem in superstring models. A significant amount o f entropy could be produced by the out-of-equilibrium decay o f the Higgs field (flaton) which is responsible for the breaking o f the extra gauge symmetry at an intermediate mass scale MI. The appearance o f such a Higgs field is a general feature o f the superstring models due to the fact that the absence of the cubic couplings o f the SU(3)C X SU(2)L × U ( 1 ) y singlets contained in 27 and 27 may give rise to a very flat hypersurface in the potential for such singlets. I f m3/2 ~ ( 1 - 1 0 2 ) GeV a n d M i ~ (107-1010) GeV, the axions are diluted efficiently by entropy production, and the axion decay constant fa O(1016 C-eV) will still be acceptable.

31 October 1985

I would like to thank P.H. Frampton, A.K. Kshirsagar, Y.J. Ng and P. Moxhay for discussion. [1] M.B. Green and J.H. Schwarz, Phys. Lett. 149B (1984) 117. [2] E. Witten, Phys. Lett. 149B (1984) 351. [3] P.H. Frampton, H. van Dam and K. Yamamoto, Phys. Rev. Lett. 54 (1985) 1114; K. Pileh and A.N. ScheUekens, Stony Brook preprint ITP-SB-85-14 (1985). [4] P. Candelas, G. Horowitz, A. Strominger and E. Witten, UCSB preprint (1984), NucL Phys. B., to be published. [5] D.J. Gross, J.A. Harvey, E. Martinee and R. Rohm, Phys. Rev. Lett. 54 (1985) 502; P.G.O. Freund, Phys. Lett. 151B (1985) 387. [6] E. Witten, Princeton preprint (1985); J.D. Breit, B.A. Ovmt and G.C. Segr~, University of Pennsylvania preprint, UPR-0279T 2/85 (1985); A. Sen, FERMILAB-PUB-85/41-T.(1985); S. Cecotti, J.-P. Derendinger, S. Ferrara, L. Girardello and M. Roncadelli, CERN-TH.4103/85 (1985). [7] M. Dine, V. Kaplunovsky, M. Mangano, C. Nappi and N. Seiberg, Institute for Advanced Study preprint (1985). [8] M. Dine, R. Rohm, N. Seiberg and E. Witten, Institute for Advanced Study preprint (1985); J.-P. Derendinger, L.E. Ib~ez and H.P. NiUes, Phys. Lett. 155B (1985) 65; R.I. Nepomeehie, Y.-S. Wu and A. Zee, University of Washington preprint 40048-02 P5 (1985). [9] R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38 (1977) 1440; Phys. Rev. D16 (1977) 1791; S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279. [10] J.E. Kim, Phys. Rev. Lett. 43 (1979) 103; M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B166 (1980) 493; M. Dine, W. Fischler and M. Srednicki, Phys. Lett. 104B (1981) 199; M.B. Wise, H. Georgi and S.L. Glashow, Phys. Rev. Lett. 47 (198I) 402. [11] K. Choi and J.E. Kim, Phys. Lett. 154B (1985) 393; S.M. Barr, University of Washington preprint 40048-07 P5 (1985). [12] J. Preskill, M.B. Wise and F. Wilczek, Phys. Lett. 120B (1983) 127; L.F. Abbott and P. Sikivie, Phys. Lett. 120B (1983) 133; M. Dine and W. Fischler, Phys. Lett. 120B (1983) 137. [13] P.J. Steinhardt and M.S. Turner, Phys. Lett. 129B (1983) 51. [14] S. Dimopoulos and H. C-corgi, Phys. LetL 117B (1982) 287; K. Tabata, I. Umemura and K. Yamamoto, Phys. Lett. 127B (1983) 90; Prog. Theor. Phys. 71 (1984) 615. [15] K. Yamamoto, Phys. Lett. 135B (1984) 63; P. Moxhay and K. Yamamoto, Phys. Lett. 148B (1984) 304; Phys. Lett. 151B (1985) 363; B. Gato, J. L6on and M. Quir6s, Phys. Lett. 136B (1984) 361. [16] K. Yamamoto, Phys. Lett. 133B (1983) 315. 293