Phenomenology of Strange Quark Matter

G. FIORENTINI

LN.F.N. Sezione di Pisa, 56100 Pisa, Italy and Physics Department, University of Cagliari, Italy

ABSTRACT

I review the arguments for the stability of strange quark matter. I discuss the origin of this hypothetical species of matter and its main properties. Finally I examine possible ways of detecting this kind of material, particularly searches using mass spectrometry techniques and ion activation experiments.

KEYWORDS

Quark matter;, dark matter;, mass sp~trometry; ion activation; cosmic rays.

INTRODUCTION

Strange Quark Matter (SQM) is a hypothetical species of matter consisting of about the same amount of three quark flavors - up, down and strange - , which could be more bound than ordinary nuclear matter. The basic idea is that at fixed baryon number A the Fermi energy of a quark gas is lower ff one can have three quarks instead of two in a single cell of phase space. Thus a 3-quark species matter can be more bound than quark matter consisting of up and down quarks only and even more bound than ordinary nuclear matter. If this is true, it means that we have misidentified so far the true zero pressure ground slate of strong interactions, which is then strange matter and not iron. As I discuss in the next section, there are several arguments in favour of this possibility, first enlarteined by E. Witten (1984), but no actual proof and I will consider it, in the rest of my laik, as an assumption.

SQM could have been produced abundantly during the Big Bang and could be produced even now in the interior of neutron stars. SQM can thus be around us either as a remnant of the Big Bang or as a product of catastrophic collisions of stellar objects. The former possibility was rather popular few years ago as an explanation of the dark-matter problem, presently, however, it is rather unlikely after the estimate of the evaporation rate in the hot unive~e presented by Alccck and Farhi (1985).

For an experimental search of SQM one has to know where to look at and what to look for. For this purpose I discuss some properties of aggregates of SQM and their interactions with ordinary matter. The phenomenology of SQM, which can exist in aggregates ranging in size from a few fermis up to a few kilometers, is an extremely wide subiect and is mostly unexplored. More extended studies than presently available should be performed.

Several methods for the detection of SQM have been proposed. The list includes, as two extreme examples, the search of anomalously heavy nuclei u-appeal in ordinary material by using mass spectrometry techniques and the observation of high energy cosmic rays emitted from compact stars composed of SQM (strange stars). So far there is no positive and unambiguous signal of SQM whatsoever. On the other hand, it has to be remarked that no experiment precisely aimed at the detection of SQM has been completed so far and the upper bounds on concentration

343

344 G. Fiorent ini

and/or flux of SQM are derived as by products of experiments with different goals, see De Rujula and Glashow (1984) and De Rujula (1985) for a clever analysis of present indirect information on the abundancy of SQM as well as for several suggestions about direct searches of this species of matter. I will discuss the methods of some devoted experiments which are now either in preparation or in a data analysis stage and which will give in the future more precise information.

THE STABILITY OF SQM: THREE IS BETTER THAN TWO

As a pedagogical starting point let us consider ordinary quark matter (consisting of up and down quarks only) in a naive bag model. The problem is to find the configuration which minimizes the total energy E of a given and large number of 3A quarks. The energy of the system is:

E = Elmg + Equarks , (1)

where Elm_ is the energy associated with the color field in the volume V where quarks are confined. In the bag model the colort~ld is clmractefized by an energy density B, with B 1/4 - 150 - 200 MeV, so that

Elmg = B V. (2)

The energy carried by up and down quarks, E~u. ks, is easily calculated by assuming that they form a degenerate • I " 1 . .

system of massless fermtons, and that any cell o~hase space (with volume 8~3), ts occupied by 6 quarks (2 spin states x 3 color states) of each quark flavor (i), up to the Fermi momentum K F

E quarks -- E(U)quarks + E(d)quarks (3)

E(i)quarks = ~ d3x dp p 6/87t 2 = 3/4x 2 V Ki4. (4)

The Fermi momentum K i is related to the density of quarks ni:

K i = ( n i re2) 1/3". (5)

F_.q. (1) can now be written as:

E = V [ B + 31t2/3/4 ( nu 4/3 + n d 4/3 ) ].

It is clear that at a given n -- n u + n d , the energy is minimal for n u = n u = n/2, so that :

E ( n ) = 3 A [ B / n + 3 C n 1/3 ] , C--~2/3 /27/3

(6)

This equation means that quarks like to stay at the lowest possible density, so that the Fermi energy is as small as possible; on the other hand,the field prefers small volume configurations, since its energy density is fixed. In conclusion, energy is minimized for

and an energy per unit volume:

n = (B/C) 3/4 . (8)

At this value corresponds an energy per unit baryon number:

e = 3*t 1/2 21/4 B 1/4 , (9)

O = 4 B. (10)

Note that in the last equation a factor 3B comes from the energy density of quarks.

As a numerical example, by taking B 1/4 = 150 MeV, one finds e -, 950 MeV. Note that the Fermi momentum

(7)

Strange Quark Mat ter 345

comes out to be K ,* 300 MeV, i.e. a value def'mitely larger than what is expected for the masses of up and down quarks, a few MeV, and this is consistent with the approximation of massless fermions used above.

Nuclear physics tells us that ordinary matter is in the form of nuclei built up of nucleons and not in the form of quark matter, so we deduce that the energy density of this form of quark matter at zero pressure is higher than that of nuclear matter.

Let us see what occurs when a third quark species, with non negligible quark mass m s enters into the play. The question is: is it convenient (and how much is it convenient )to add strange quarks instead of up and down? We have to compare the energy of a massless quark at the top of the Fermi sea,

Eu, d = K = (n n2/2) 1/3 (11)

with the ground state energy of the strange quark in the bag which we assume to be spherical, with radius R = (9A / 4rm) 1/3 :

Es = ( ms 2+ Ps 2 )1/2 = ( ms 2+ rc2/R 2 )1/2. (12)

We find E s -- Eu, d for:

A = 8/9 rc 2 ( 1- m s 2 /K 2)-3/2. (13)

This equation has a real solution for ms/K < 1 This is the ease for the strange quark, which has a mass m of about 250 MeV. Thus at sufficiently large baryon number (volume) addition of strange quarks is energetically favoured with respect to the addition of up and down quarks. Strange quark matter looks as more bound than ordinary quark matter.

Let us note, in passing that for heavier quarks, like c or b, one has mq/K > 1 and the analogue of eq. 14 has no real solutions. This is why one considers strange quark matter and not charmed quark matter.

For an attempt towards a realistic calculation of the binding of SQM, several improvements with respect to the naive model shown above are clearly necessary. Since an ab initio calculation in the framework of Quantum CromoDynamics (QCD) is impossible at present, one must resort to refinements of the bag model. One tries to incorporate the effect of short distance interactions by using perturbative QCD to lowest order in the coupling constant ct c and fixes the values of B, m s and ct c from fits to the hadronie spectrum performed in the framework of bag model calculation. The weak point in this approach is that a¢ comes out too large for a perturbative calculation to be reliable, see Table 1.

TABLE 1. Parameters obtained from ba~-model fits to light hadron soectra

Reference B 1/4 (MeV) m s (MeV) (z c

DeGrand (1975) 145 280 2.2 Bartelski (1984) 149 283 2.0 Chanovitz (1983) 120 340 2.8 Carlson (1983) 200-220 288 <1

In this framework, Farhi and Jaffe (1984) calculated the energy per unit baryon number, e = E/A, for several values of the three parameters, 0to, B and ms. They found (see Fig. 1) that for a wide interval of the parameters, consistent with the values which can be derived from bag model calculations of hadron properties, e in SQM is lower than in

PPNP--L

346 G. Fiorentini

nuclear matter, i.e. it is possible that SQM is more bound than ordinary nuclei. This energy difference, I, is of the order of tens of MeV. Clearly this cannot be considered as a proof of the stability of SQM, but just as an indication.

300

200

100

145

M S

(MEV)

300

200

i00

MS (M~V)

, 150 155 160 14 140 145 150 B / (rIEV)

879 99

130 135 140 145" 120

' (A) j j ' ' ' (S)

e'c=0 ~ . ac=0'3

155

' ' ( D )

a c -0,9

919 39

879889

125 130 135

B I/'~MEV)

Fig. 1 Contours of fixed E/A in the B 1/4- m s plane for different values of a c. The vertical line at the left of each fi re, 1/4 gu "s the minimum value of B for which non strange quark matter is unbound.

One may wonder as to why ordinary nuclei nuclei do not decay into strange matter, if the process is energetically allowed. For example, one might expect:

56Fe --~ (56 u + 56 d + 56 s) + energy. (14)

The answer is that such a process occurs in an extremely long time ( i.e. much longer than the Universe age), as it eorresl~nds to the 56th order in the weak interaction coupling constantl

THE MAIN PROPERTIES OF SQM

If SQM consists of precisely the same number of u,d and s quarks, it is electrically neutral, since qu=2/3 e and

Strange Quark Matter 347

qdfqs=-l/3e. Actually one expects the number of the heavier quarks to be somehow reduced and SQM to have a srnall~, positive charge. On the other hand, one gluon exchange is an attractive interaction for slow particles, whereas it is repulsive for relativistic quarks, thus favouring the presence of heavier quarks. For small and moderate values of % Farhi and Jaffe (1984) find:

z - ( 5 -15) A I/3 . (15)

One may note that at large % the effect of one giuon exchange is even dominating, thus favouring a net negative charge. This result, which is not physically significant as it is derived in a regime where perturbation theory is not applicable, shows that a calculation of the electric charge of SQM is actually beyond present theoretical methods.

Eq. (16), giving a Z/A ratio asymptotically vanishing as A -1/3, is to be contrasted with Z/A -, 1/2 of ordinary nuclei.

Stability uv to extremely large A values

As a consequence of the small value of the electric charge density, there is no analogue to the fission instability of ordinary nuclei. The contribution of the Coulomb energy per unit baryon number,

ECou/A - (few MeV) A -2/3, (16)

is negligible at large A and SQM can be stable as long as gravitational collapse does not start, i.e. when the gravitational energy is comparable to the mass. This corresponds to a baryon number A ,, 1057 1

In conclusion, aggregates of SQM span a mass range from heavy nuclei up to stellar objects. The interaction of these systems with ordinary matter is a rich field of phenomenology which so far is essentially unexplored and which has to be investigated in view of planning experimental searches of SQM.

SOM likes to eat ordinary nuclear matter

Consider a single neutron slowly entering a lump of SQM. Since E/A in SQM is smaller by O(10) MeV, it is convenient to dissolve the neutron and put its quarks on the top of the Fermi sea:

SQM(A) --~ SQM*(A+I) + energy. (17)

Later, with a weak interaction time scale, SQM* will deexcite to a new equilibrium configuration through the r tion

d + u ~ s+ u . (18)

See Berger and Jaffe (1986) for a discussion of radioactivity in strange quark matter.

Similar processes can occur (energetically) for a proton or a nucleus. Normally, however, they are inhibited by the Coulomb barrier which exists provided that SQM is positively charged. Note in passing that the stability of ordinary matter poses severe limits against the existence around us of aggregates of SQM with negative electric charge.

WHEN AND WHERE IS SQM PRODUCED

SOM in the early universe

Witten (1984) proposed a scenario such that large quantities of SQM are formed during cooling through the QCD phase transition at the time the Universe temperature was about 200 MeV. This SQM is non luminous - not like the stuff of ordinary stars - and does not take part in the primordial nucleosynthesis. As emphasized by De Rujula (1985), SQM could account for the so called dark matter ( Mdark- 10 Mlu m ) which is needed to explain the

348 G. Fiorentini

velocity distribution of outer stars in galaxies without spoiling the success of the theory of primordial nucleosynthesis, which requires Mbaryon - Mlu m for the prediction of the correct fraction of He and other light nuclei.

This scenario, although very appealing, is essentially ruled out by an argument of Alcock and Farhi (1985). On the grounds of thermodynamic cosiderations, they were able to calculate the evaporation rate of aggregates of SQM,

SQM(A) --~ SQM(A-1) + n , (19)

and they found that any SQM with A < 1052 evaporated before the universe temperature drops belowl MeV.

SOM in swan~e stars

According to Alcock, Farhi and Olinto (1986), it is conceivable that the interior of a neutron star actually consists of SQM. Conversion from neutron matter to SQM can occur in several ways:

a)Density fluctuations in the hot phase of a neutron star can provide a region for the spontaneous formation of a seed of SQM, which then aceretes through the absorption of surrounding neutrons.

b)Neutrino sparking: a high energy ( E > 1014 MeV) cosmic ray can deposit its energy in a small region inside the neutron star, where a local transition to a seed of SQM occurs and this then grows extending over almost all the star.

c)Seeding from outside: a lump of SQM can enter the neutron star and then gradually convert all existing neutron matter into SQM.

For a rather detailed study of the properties of strange stars the reader is referred to Alcock, Farhi and Olinto (1986), Olinto (1987), Baym and colleagues (1985).

It is worth remarking that collisions between stars are not an unfrequent event. Collisions involving strange stars could liberate SQM, which eventually could reach Earth.

SEARCHES FOR SQM

There are several experimental methods which are suitable, at least in principle, for the detection of SQM. The interested reader can find a wide discussion in a paper by Glashow and De Rujula (1984). Basically, one can take three different approaches:

a)Look for SQM falling presently onto Earth: one needs large area detectors, similar to some apparatuses built or planned for the detection of magnetic monopoles, see the paper by Giacomelli and colleagues (1986) for a discussion of this possibility.

b)Look for the traces of SQM which has fallen onto Earth in the past. In particular if the piece of SQM is not to heavy, it can be trapped by chemical forces inside ordinary matter.One has to perform refined analyses of the composition of terrestrial samples which have been exposed to cosmic rays for very long times, in search of impurities which can be ascribed to SQM.

c)Find clear signatures which distinguish between strange stars and ordinary neutron stars.

It is hard to compare the different methods. Concerning e), it comes out that astrophysical quantifies which are calculable and observable, like the mass to radius ratio, the moment of inertia and so on, come out to be essentially the same for strange stars as for ordinary neutrg_n stars, see the paper by Farhi (1986). Just as an example for a comparison of a) and b), a flux of SQM ~ ,- 10 "13 particle cm "z s" lwi l l give N- 3 signal in a detector with 100 o/o efficiency during an exposition time t = 1 year provided its surface S is :

S = N / ¢?t-. 104 m 2 . (20)

Strange Quark Matter 349

For the same flux, a sample with a surface S .. 1 cm 2 which has been exposed for a geological time, say t = 109 years, and which traps any arriving particle, has accumulated a total of about 30 SQM particles in its interior, which represent however a very small concentration, compared to the number of ordinary nuclei. Also one has to remark that in the latter approach it is crucial to know the history of the material and its preparation before the analysis.

In the following. I will concentrate on the third method, the search of SQM trapped in the interior of Earth material.

This search refers to the aggregates of SQM such that chemical f orLces OVchem - 0.1 eV/~ ) exceed the gravitational force (Fm.av ,- g m,, A) . This restricts to aggregates with A < 10 lb. According to eqs. 15 and 8, this correspond to electric ~ g e Q:~ 10°e and to a radius R ~ 1 ~. Essentially, these objects behave as (very) heavy nuclei and one has to perform a "heavy isotope" search.

The best way to look for these objects is to resort to mass spectrometry of the highest possible sensitivity, as can be done for example at a tandem accelerator. The principle of the technique is schematically shown in Fig. 2.

Ion Source Sample Tandem T.O.F. & Energy

Accel erator Measurement

1

l Stripper

Fig. 2 A plan view of a mass spectrometry apparatus.

Negative ions emitted from the sample through sputtering, are accelerated inside the tandem. They emerge as positively charged. One measures then the energy and the velocity of the ion .to_ determine its mass. For a given sample, one can reach a sensitivity to the level of a concentration of 10"15-10 - r / depending on the Z value of the isotope one is looking for. Also, the sample can be enriched in its content of heavy isotopes by a factor R varying between 102-109, depending on the Z value and on the technique. A summary of experimental data on upper bounds for the concentration of heavy isotopes of different elements is shown in Fig. 3, taken from a recent paper (Elmore 1986). The quantity on the vertical axis is the effective concentration, i.e feff = f R.

350

FEFF

i0-i0

10-15

10-20

i0-25

i0-30

~ ' H _

G. Fiorentini

I I

H 0

\ \

\

I NA --

. . . . B~-

t.L ~ C~

~ - ' ~ _ - - ~ - - - - _ - - F~B

0 m

i00 104 I I I

101 10 2 10 3

MASS (A,M,U)

Fig. 3 Concentration limits (90 o/o confidence level) for the existence of heavy isotopes in matter. The dashed lines correspond to preliminaly results of Elmore and colleagues (1986).

One has to remark that mass spectrometry techniques are not sensitive for masses M > 104 a.m.u. The reason is that atomic reactions, such as stripping in the tandem and ionization in. the detector, cannot be induced if the ion velocity is too low in comparison with the velocity of the atomic electrons, v ,~ ~c.

For higher A, a complementary investigation is based on the activation by heavy ions (Farhi and Jaffe, 1985). The basic idea is that, due to the small value of Z/A in SQM, the Coulomb barrier at the surface of a strangelet (fflag&regate of SQM) is smaller than at the surface of a ordinary nucleus. Heavy ions impinging onto a sample can react with SQM impurities and not with ordinary nuclei if their energy is conveniently chosen.

The interaction of a heavy ion with a strangelet has specific signatures. Nucleons in contact with SQM find more convenient to dissolve into SQM and an energy per nucleon of about 10 MeV per nucleon is released.This energy heats the strangelet to a temperature T s given by:

T s - ( 2 K A b I / A s n2) , (21)

where A b and A s are the baryon number of the impinging ion and of the strangelet respectively.

The strangelet then radiates the energy Abl - few Gev into photons with average energy E. . - T s. As an example, for A s ffi 10 6 one has about 10 4 photons with energy around 0.5 MeV, whic~ is a spectacular signal.

An experiment along this line is_in progress by the group of Stevens and colleagues at the LBL SupcrHILAC, see (Farhi 1986). The ion beam is t97Au, accelerated at 903 MeV with a current of 85 nA.

Strange Quark Matter 351

CONCLUSIONS

There is really no definite conclusion about SQM at this moment, but a few statements are possible:

1)The stability of SQM is a reasonable hypothesis. There is no proof, however, since the theoretical calculations use pertubation methods in an unsafe region.

2)SQM production during the Big Bang as an explanation of dark matter is substantially ruled out

3)Several experimental methods for the search of SQM have been suggested and some experiments are in progress. No affirmative result has been claimed so far. The phenomenology of the interaction of SQM with ordinary matter should be extensively investigated, as this can be very valuable to several future experiments.

REFERENCES

Alcock, C. and E. Farhi (1985). The evaporation of strange matter in the early universe. Phys. Rev., 32,1273. Alcock, C., E. Farhi and A. Olinto (1986). Strange stars. MIT preprint CTP # 1239, to appear in The Astrophysical

Journal. Bartelski, J., A. Szymacha, Z. Ryzak, L. Mankiewicz and S. Tatur (1984).The influence of c.m. motion and of chiral

symmetry on the massaes and the electro-weak parameters of hadrons in the bag model.Nucl. Phys., A424. 484-494.

Baym, G., E. W. Kolb, L. McLerran and T.P. Walker (1985). Is Cygnus X-3 strange? Phys. Lett., 106B, 181-187. Berger, M. S. and R. L. Jaffe (1986). Radioactivity in swange quark matter. MIT preprint CTP # 1360, submitted to

Physical Review C. Carlson, C. E., T. H. Hansson and C. Peterson (1983). Meson, baryon and glueball masses in the MIT bag model.

Phys. Rev., 27D, 1556-1564. Chanovitz, M. and S. Sharpe (1983). Hybrids, mixed states of quarks and gluons. Nucl. Phys., B222, 211-244. DeGrand, T.A., R. L. Jaffe, K. Johnson and J. Kiskis (1975). Masses and other parameters of the light hadrons.Phys.

Rev., 17D, 2060-2076. De Rujula, A. (1985). Aborigenes in the nuclear desert. Nucl. Phys., 434A. 605-626. De Rujula, A. and S. L. Glashow (1984). Nuclearites: a novel form of cosmic radiation. Nature, 312, 734. Elmore, D., D. Nitz, D. Ciampa, T. Hemmick, P. W. Kubik, S.L. Olsen, T. Gentile, H. Kagam, P. Haas and P. F.

Smith (1986). A search for anomalously heavy isotopes of low Z nuclei. University of Michigan preprint. Farhi, E. and R. L. Jaffe (1984). Strange matter. Phys. Rev. 30D. 2379-2390. Farhi, E. and R. L. Jaffe (1985). Searching for strange matter by heavy-ion activation. Phys. Rev., 32D, 2452-2455. Farhi, E. (1986). The physics and astrophysics of strange matter. MIT preprint CTP # 1343, to appear on

Comments Nucl. Particle Phys.. Giacomelli, G., G. Mandrioli, A. Margiotta, L. Patrizi, P. Serra and M. Spurio (1986). Detection of Strange quark

matter in MACRO. University of Bologna Internal Note MACRO-13/86. Olinto, A.V. (1987). On the conversion of neutron stars into strange stars. Phys. Lett., 192B, 71-75. Witten, E. (1985). Cosmic separation of phases. Phys. Rev., 30D. 272-283.