113

ASTROPHYSICAL AND COSMOLOGICAL CONSTRAINTS ON SUPERSYMMETRIC THEORIES Savas Dimopoulos Department of Physics Harvard University 13 Cambridge, MA 02 8 and Michael S. Turner Astronomy and Astrophysics Center The University of Chicago Chicago, IL 60637

ABSTRACT We summarize the astrophysical and cosmological constraints on theories in which a global superysmmetry remains unbroken down to an energy scale of the order of TeV. The emission of Goldstinos (and possibley photinos) from red giant stars constrain A (the scale of supersymmetry breaking) to be greater than 0(50 GeV) . If A is 0(1 TeV) , then it follows from big bang nucleo­ synthesis that at least one of the following particles must be more massive than 0(10 MeV : the photino, or the Goldstino . If is greater than ) VT , A 0(10 TeV) , then the big bang nucleosynthesis constraints are much less restric­ tive. The mass density contributed by relic photinos and the contribution of photons from their decays to various diffuse photon backgrounds restrict the photino mass to be either greater than 0(10 MeV) or less than 0(200 eV) . I. INTRODUCTION AND SUMMARY

One of the outstanding problems in particle physics is the gauge hier­ archy problem1) , i.e., the difficulties associated with having two (or more) 15 widely separated scales of symmetry breaking (1000 GeV, 10 GeV, ?). In the last year attempts have been made to solve this problem by imposing a global 2) supersymmetry which remains unbroken down to an energy scale of O(TeV) . In this paper we shall discuss the constraints on such theories that arise from astrophysical and cosmological considerations . It is only fair to point out that at present there does not exist a compelling supersymmetric model of the world . Thus it may appear premature to put constraints on a theory that does not exist. However, we shall attempt to focus on the most generic and model-independent consequences of super­ symmetric theories, so that our results will have general applicability . At the very least, our work will EErve as a 'case study' which illustrates the power and diversity of the astrophysical and cosmological restrictions on particle physics theories. We shall focus on the two new light particles which are predicted to exist in theories with a low-energy global supe1·symmetry: the photino (�) and the Goldstino (�) . If A (the scale of supersymmetry breaking) z O(TeV) , then the photino and the Goldstino couple to ordinary matter with roughly the same strength as a neutrino species and each have two degrees of freedom 3 4 (like a neutrino species) . The observed abundances of D, He , and He require 0(10 that the number of light � MeV) ), two-component neutrino species be 4 .0 (ref. 3) . Since v and v are known to be light , at most two of the _2 e i� rv rv 0(10 three species : v, , y, G, can he light. However, if /\ :\, TeV) , then the photino and Goldstino 'decouple' early enough so that during primordial nucleosynthesis their temperature is much less than that of the neutrinos and photons, and thereby contribute � 1/8 (each) , to the 'neutrino count ' which must be < 4.0. We will discuss these constraints in § III.

As a star evolves the timescale for its evolution is detennined by the rate at which it can radiate away the nuclear energy it produces (in the usual case by the emission of photons or neutrinos) . The Goldstino and photino 4 (1f less massive than a few keV) will be radiated by stars at a rate � /',- . 0 (50 If A z GeV) , then He-burning stars would lose energy primarily by Goldstino (or photino) emission and would evolve so rapidly that such stars would not exist for a long enough time to be seen - contrary to observation.

The stellar evolution constraint� ori supe:rsymmetric theories are derived iri ll s

IV. The photino can decay into a photon and Goldstino . Since the coupling strength of a photino to ordinary matter is about the same as that of a neu­ trino (for A � 0(1 TeV) ), the abundance of relic photinos should be about

the same as a neutrino species of the same mass. Therefore, wit1: slight modi­ fication the astrophysical and cosmological constraints on unstable, massive neutrinos are applicable to photinos . We examine these constraints in detail in § V, and find that if A� O(TeV) , then the photino must be either less

ma ssive than !"\.· 200 eV or more massive than about '\_, 10 MeV . In the next sec­ tion ( § II) we briefly summarize the generic aspects of a low-energy super­ 4) symmetric theory. Elsewhere , we will discuss big bang baryogenesis and supersymmetric GUT s.

II. LOW ENERGY SUPERSYMMETRY

In a supersymrnetric (SS) world fermions and bosons come in degenerate (in mass) pairs . Thus, in a SS theory there exist superpartners for quarks , leptons and gauge bosons : scalar quarks , scalar leptons, and spin-� gauge fermions . [There also exist spin-Yz partners for the Higgs bosons, however , they will not be of interest to us here .] In Table I we exhibit the particle content of a SS model of the low-energy world, and in Figures 1 and 2 we show the couplings of ordinary particles to their superpartners for the processes of interest to us in this paper.

Table I. - Particle content of the simplest SS extension of the standard low energy theory, SU(3) x SU (2) x U(l) . The arrows on the symbols indicate chir­ ality. Ordinary Particle SuEeri�artner Symbol

scalar quark ------quark q ¢q ;> lepton 1 scalar lepton ¢1 ----;> -- - photino photon y � ....., ...... , .,...,...... d � gluon g gluino g u oa �uu �w w Wino � z Zino z � Higgs Higgino H -> � Goldstino = G ······> ···· ··· It is clear that SS cannot be an exact symmetry of the world. At low energies it must be broken. We shall assume that it is broken at an energy

scale A � 0 (100 GeV - 100 TeV) . If SS is broken spontaneously, then there must be an exactly massless spin-\;: fermion. called the Goldstino and denoted .5) by (; . The couplings of the Goldstino which are of interest to u.s are shown 116

' e \ \

� \ e

e e

\ Q 'Pq \ � \ � \ eq

Q Q Figure The couplings of quarks, scalar quarks , leptons , and scalar leptons 1 - to gauge bosons and gauge fermions. Here eq is the charge of the quark.

(A) (8)

""-' .· G

.· . .

. . . 2 My/2A

(A) Coupling of the Goldstino to a chiral supermultiplet scalar, Figure 2 - (¢= o/ = fermion) ; (B) Coupling of the Goldstino to the photon and photino. 117

in Figure 2 . The quantity A which is associated with the breaking of SS is defined by the relation: (1)

where S µ is the supercurrent, a is a spinor index, y is a Dirac matrix, and is theµa Goldstino wave-function. �� III. BIG BANG NUCLEOSYNTHESIS CONSTRAINTS

The primordial mass fraction of 4He in the Universe is � 0.23 - 0.25, -5 and of Dis� 3 x 10 . The only viable astrophysical or cosmological site for their production is the big bang. Detailed calculations show that precise­ ly these amounts should have been synthesized in a period of primordial nucleo­ 6) synthesis which occurred from t � 0.01 s - 200 s when T � 10 MeV - 0.1 MeV . Because of the concordance of the calculations with observation, big bang nucleosynthesis has been used as a probe of conditions in the Universe when To, O(Mev) 3!6 ' l) In particular, the expansion �ate which is determined by the number of light ( 10 MeV) species affects the production of 4He. If � 4He is not to be overproduced, then the number of light, two-component neu­ 3) trino species (Nv ) must be -< 4 . We emphasize that this is a very firm con-

straint - N = 4 .1 cannot be tolerated. If v 0(1 TeV) , then both the photino and Goldstino couple to matter with A"' about the same strength as a neutrino, have two degrees of freedom, and thus for purposes of primordial nucleosynthesis 'count ' just like additional neutrino species. In particular, and v are known from laboratory experiments to be v e lighter than 1 MeV, and the Goldstinoµ must be massless5 ), so only one of the following particles can be less massive than "' 10 MeV : '\, v v . '\,G . y . T' ? i (i , ., N; N .::. 8). The particles denoted by '\,G are the Goldstinos = 2 i associated with the other broken supe.symmetric generators in larger super- symmetric theories, which of course must also be massless. Thus, if the scale of supersymmetry breaking is 0(1 TeV) , we are restricted to SS theories with N < 2 . If the scale of supersymmetry breaking is 0(10 TeV) then the photino � and Goldstino couple significantly more weakly than neutrinos and will contri­ 8) bute less than 1 to Nv . Recall that neutrinos are kept in thermal contact with the rest of the Universe by the neutral current and charged (for veve) current weak interactions, e+ e- The rate for these interactions + t vi + \J1 . is I ') 0 ! c .8 T-' for v ' G e F 2 S (2) v 0. 2 G T for v v l F µ' T' 2 R/R When f v < H 5.4T /m (H = = the expansion rate of the Universe) , neu - pl - trinos are no longer in thermal contact with the rest of the Universe, i.e., they are 'decoupled '. This occurs for T T ( � 2 MeV for and � 4 MeV � d Ve for V , VT, ...), where f (T ) � H(T ). After they decouple, neutrinos )J v d d evolve as a non-interacting Fermi-Dirac gas with RTV = constant (as long as they are relativistic) . In particular, they do not share in the 'heating ' (entropy release) when e + - pairs annihilate (T � 0.5 MeV) , and for that reason should have a lower temperature today than the photons: 1/3 (4/11) T � T . v y The Goldstino and photino stay in thermal contact with the Universe by reactions which involve the exchange of the scalar partners of e and V: e-j or \ + \J :_ (; + e+ e- -:-_ The rate for these reactions is : e+ + \ i 'C;; + Y+ Y. s -4 T � » 3 r -,· \ i ' ( ) 2 where = m / m. ). We take m to be the (assumed) :\y q c../2'P.) , and :\2 = A I f./2 p "' cp common mass of cp and cp , and assume that G couples with roughly the same e v strength to e and vi . Comparing (2) with (3) we find that the photino and Goldstino should decouple at temperdtures (4)

If T 100 MeV, then decouplin + 150MeV , d .\, � occurs before 1.T annihilations , if Td f(, then decoupling occurs before Tf- n° annihilation , and so on. Table II summa­ ( 1 rizes A, Td , and '6N\1 ', the effective contribution of a light � 0 MeV) photino or Goldstino to the 'neutrino count ' N, [ '.6.Nv ' vs . T was taken J d from ref . 8] .

�ble LI. - h.Nv , the effective contribution of the photino or Goldstino to Nv .

* I A T (dec0upling temp .) 'D,N d 'J

> 210 GeV > 2 MeV < 1 "' ,, o, > TeV '"> 100 MeV <' 0.69 c 6.6 " 0 3 > 9.0 TeV > 150 MeV '" .5 " ·1, < 11 > TeV "'> A � 200 MeV < 6.12 •1, QCD c ' .-.,> 38 TeV "'> l GeV "' 0.08 "* o.os "> 350 TeV m < z t 'v A= kFor the photino m /< e) and for the Goldstino � ., -J2' ,\ .\:';0,/z-. **For definiteness we take m t 20 GeV.

For �j z AQCD (the quark/hadron transition is assumed to take place for T A , many new degrees of freedom become available for the Universe � QCD (uu, dd , ss, gluons, etc.), and because of this 'AN ' falls to about 1/8. In v A order to have Td z AQCD "' 200 MeV, must be Z 11 TeV ; in the simplest models \ should be roughly of order A. For \ z 0(10 TeV) it is possible to have 3 light neutrinos, a light photino , and at least 7 massless Goldstinos. However ,

even with A as large as � 300 TeV it is not possible to have a supersyrnme tric 4 4 . theory with light neutrinos since NV must be strictly less than l IV. CONSTRAINTS FROM STELLAR EVOLUTION O)

As a star evolves it releases its nuclear free energy by successive stages of nuclear burning. Each stage proceeds at a higher temperature: hydrogen 7 8 K; 8 K, burning, T rv 10 K; helium burning, T "' 10 carbon burning , T v I x 10 etc. 11) The rate of its evolution depends on how rapidly it can radiate the free energy which is being released by the nuclear reactions . The dominant means of energy loss are: (i) photons, which of course must diffuse out (time­ 8 scale of order 10 - 109 yrs) ; (ii) neutrinos, which freestream out . When photon radiation is the dominant loss mechanism, the timescale for evolution is 8 9 long (10 - 10 yrs) ; when neutrinos are the dominant loss mechanism the -3 timescale becomes very short (as short as � 10 sec for the final stages of stellar evolution) . Owing to their different temperature dependences, there is a crossover from photon- to neutrino- dominated energy loss near temperatures corresponding to carbon burning. This aspect of stellar evolution has been 'experimentally verified ' in the sense that helium burning stars are observed , while carbon burning are not . It has been argued that if CF were larger (smaller) by a factor of 3, this crossover would occur during helium (after carbon) burning - in conflict with observationl�)

Light ( ,;:;10 keV) , weakly-interacting particles are potential energy-loss mechanisms for stars, and thus their properties can be constrained by stellar evolution. Such arguments have been vigorously applied to axions, and have 1 ) resulted in stringent constraints on their properties � Supersymme tric theories have two candidates for stellar energy loss: the photino (if !lf� ,;:; 10 keV) and the Goldstino . Both couple to ordinary matter with roughly

the same strength as a neutrino H \ � 200 GeV . For T 109 K the dominant process for neutrino production is: :�' e + y ? -3 8 v The energy loss rate for this process cr Cf m T , e + + ') Qv e wore 120

precisely14) s -1 0) Goldstino and photino pair production proceed via similar processes (e- + y + e 2� or e 2y) with a virtual ¢ being exchanged rather than a W boson. + + e � � The rates for G and y production are (1 GeV/A 4 (6) � 85 y) Q v 4 � (131 GeV/A�) Q (7) 2 v where as before A m !!:./i'e) and A A /t:./Z'm ). Of course, equation (7) y ¢ � = ¢ is only applicable if my�lO keV, otherwise its production is suppressed by a Boltzmann factor since for He-burning stars T � 108 K � 10 keV . Based on the argument that GF could not differ by more than a factor of 3 from its measured value since we see He-burning stars and do not see C-burn­ ing stars, we shall demand that 10 - otherwise helium burning stars �· � � Q (red giants) would be dominated by �- or y- lossv and evolve so rapidly that they would not be observed. A more conservative argument is to demand that l l �· � not exceed 100 erg g- s- - the nuclear energy output of a T 108 K - � and 104 g cm 3 0.5 M He star (this was the criterion adopted for axion p � 0 emission in ref. 13). For photinos (if my �10 keV ) this results in the constraint m ( 0 - 8 ) GeV, (8) ¢ � 5 5 where the range reflects the different criteria used : / 10 or 100. � Q � For Goldstinos the contraint is v A2 m (60 - 110) GeV (9) � ¢ For SS models where m eY, A, this implies that A � 0(35 - 64 GeV) . ¢ � V. ASTROPHYSICAL AND COSMOLOGICAL CONS INTS ON THE PROPERTIES OF THE PHOTIN01 �� As we have discussed earlier, the photino is a light, unstable gauge ferm­ ion which couples to ordinary matter with roughly the same strength as a neu­ trino. Therefore, most of the astrophysical and cosmological constraints on the mass and lifetime of an unstable neutrino species can be directly applied to the photino [for a review of these constraints see ref. 16]. If A� < lO TeV � photinos will decouple after + annihilations and will have the same yrelic µ- abundance as a neutrino species (see § III) . Photino decay proceeds by: with a lifetime given bylS) Y + Y + � 121

5 1! A4 5 1.5 (A/100 GeV)4 ( /l MeV) - sec. (10) T"­ 8 � - "' � y The astrophysical and cosmological constraints on photino lifetimes and masses are shown in Figure 3 ; in addition, the mass-lifetime relation given by 12 (10) is plotted for A= 0.1, 1.0, 10., 100. TeV. For 104 s �Ty � 10 s, photons from photino decay can potentially lead to a distorted 3K microwave 1 background; for TY� 10 2 s, photons from photino decay can 'overcontribute ' to various photon backgrounds (IR, visible, UV, X and Y ray) ; for Ty� T 1 Universe 3 x 10 7 s, relic photinos may contribute too much mass density today.

I TeV I TeV

I GeV

UJ m 'VieV I MeV

1 meV 1 meV

4 8 4 10 10 1012 1016 1020 102 PHOTINO LIFETIME (SEC )

Figure 3 - Summary of the astrophysical and cosmological constraints on un­ stable photinos, as a function of mass and lifetime. The cross-hatched re­ gions are forbidden. The four lines indicate the mass - lifetime relation­ ship predicted by (10) for A = 100 GeV, 1 TeV, 10 TeV, and 100 TeV.

During supernova collapse neutrinos of all flavors are copiously produced by reactions like: e+ e- and carry away the bulk of the gravita- + t V.1 + v. , 5 1 tional binding energy "' 10 3 ergs) released in the formation of a compact object (neutron star or black hole) . The average energy of a supernova-pro­ duced neutrino is "' 10 MeV. For A"- "'A< 0 (1 TeV) , photinos will also be pro-

- - 53 duced copiously by: e+ e � '\, :t (� exchange), and will share in the lO + + y + y e ergs of energy released. In this case the supernova constraints17) on neu­ trino properties are also applicable. From Figure 3 it is clear that the pho­ tino mass must be greater than a mass of order 1 - 10 MeV, determined by the supernova constraint, or less than a mass of order 200 eV, determined by either 122

the present mass density constraint, or the diffuse photon background con­ straint.

Table III.- Astrophysic�l and cosmological bounds on photino masses; the "safe masses" are those less than m1 and greater than m2 . We have used A y = mr'p;JZe "' A in order to compute the decoupling temperature , Td . A

100 GeV 10 eV 3 MeV (0.1 MeV) * 1 TeV 200 eV 10 MeV (0.5 MeV) * 10 TeV 350 eV MeV 100 TeV 1. 3 keV 30 MeV

* ForA �0(1 TeV) photinos should be copiously produced by supernovae , and thus the value of m2 is determined by the supernova constraint . The value of m2 listed in parenthesis is derived from the 3K background constraint, i.e., is independent of the supernova constraint.

For Ay �A� 0(1 TeV) , photinos will not be produced as copiously in supernovae as neutrinos, and the supernova constraints derived for neutrinos are not applicable. The upper boundary of the disallowed range is determined by the constraint that photino decays not distort the 3K background , and is my � 0(10 MeV) - depending upon A. In addition, for "� � 0(10 TeV) , photinos decouple at a temperature Td � 100 MeV , so that they should be less abundant than a neutrino species (see § III) . In this case the mass density constraint on light photinos is less restrictive since they are less abundant . For ex­

ample, with A "' 10 TeV, photinos are a factor of 0.6 less abundant than neu­ trinos, and the corresponding constraint from the mass density of the Universe is : my � 350 eV. The constraints on photino masses we have discussed are summarized in Table III . Roughly speaking, to be 'astrophysically and cos­ mologically safe' a photino must be less massive than 0(200 eV) or more massive than 0(10 MeV) .

This work was supported in part by the DOE through contract AC02-80ER- 10773 (at Chicago).

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