V O L U M E 39, N U M B E R 24 PHYSICAL REVIEW LETTERS 12 DECEMBER 1977

Univers i ty , Cambr idge , M a s s . ^See, for example , A. De Rujula et aL, Phys . ReVo

D 2 2 , 147 (1975); B. W, Lee et cd., P h y s . Rev„ D L5, 157 (1977)o

^ J . -E . A u g u s t i n e (^., P h y s . Rev. Let t . 34, 233 (1975); F . Vannucci^oZo, Phys , Rev. D j ^ , 1814 (1977).

^A. B a r b a r o - G a l t i e r i et al., P h y s . Rev. Let t . _39, 1058 (1977).

lo P e r u z z i et ah, to be publ ished. ^G. Goldhaber ^ a Z . , Phys . Rev. Let t , ^ , 419 (1975);

L P e r u z z i ^ ^ . , P h y s . Rev. Let t . _37, 569 (1976); G. J .

F e l d m a n ^ a Z o , P h y s . Rev. Let t . 3 ^ , 1313 (1977); G. Goldhaber ^ a / . , P h y s . Let t . 69B, 503 (1977).

^Jo Siegr is t ^ a / o , Phys . Rev. Let t . _36, 526 (1976); R. F . Schwi t te rs , in Proceedings of the International Symposium on Lepton and Photon Interactions at High Energies, Stanford, California^ 1975^ edited by W. T, Kirk (Stanford L inear Acce le ra to r Cente r , Stanford, Calif., 1975); P . A. Rapidis et aL, P h y s . Rev. Let t . 39 , 526, 974(E) (1977).

•^B. Knapp ^ a Z . , P h y s . Rev. Let t . _37, 882 (1976), ^E. G. Cazzoli e aZ., P h y s . Rev. Let t . 34, 1125 (1975)„

Possibility of Charmed Hypernuclei

C. B . D o v e r a n d S. H. K a h a n a Brookhaven National Laboratory, Upton, New York 11973

(Received 10 August 1977)

We suggest that both two-body and many-body bound s ta tes of a cha rmed baryon and nucleons should exis t . Es t ima te s indicate binding in the ^SQ s ta te of CiN (I = f) and SN (I = 1). We fur ther e s t ima te the binding energy of CQ,CI in var ious finite nuclei .

Recent experiments^ have established the exis­tence of a new class of mesons and baryons pos­sessing net charm. In particular, there is firm evidence for the charmed baryons B^= Cg, C (A^ and Z^) and the charmed mesons, D^D^^. It is hence of fundamental interest to establish the na­ture of the interactions of these charmed part i­cles with more familiar hadrons.

In this Letter we ask whether the charmed bary­ons will bind to a single nucleon or to finite nu­clei, producing charmed analogs of the deuteron or of hypernuclei. The estimated short lifetime of even the lowest-mass charmed baryon, r{C^) -10"^^-10"^^ sec, may make it difficult to estab­lish the existence of such analog bound states. These questions are discussed here in a nonrela-tivistic framework in which the JB^-nucleon inter­action is represented by a sum of single-boson-exchange potentials. The 5^-nucleus potential is obtained by averaging the B^-N interaction over the nuclear density. The large mass of the charmed baryon alters the dynamics appreciably, resulting in a reduction in the B^ kinetic energy for both two- and many-body systems. Thus, in the ^^-nucleus case, where we find potentials comparable to the nucleon-nucleus potential in depth, a rich spectroscopy emerges encompass­ing many bound levels.

The 5^-iV potential, V{r), is given by a sum of meson exchange potentials for r ^ r^ but taken phenomenologically as an infinite hard core for r ^r^. Such models have frequently been applied

to nucleon-nucleon {NN) scattering in the low-en­ergy region.^ Recently, a one-boson-exchange model has been constructed^ which successfully reproduces both low-energy NN and hyperon-nu-cleon [AN and SiV) data. This model includes the coupling of SU(3) nonets of pseudoscalar, scalar, and vector mesons to the 1* baryon octet of SU(3). We Use here a natural generalization of this mod­el which considers the coupling of mesons ob­tained by mixing the {is} and {l} representations of SU(4) with members of the {20} representation of i"*" baryons. The details are given by Dover, Kahana, and Trueman."*

We have restricted out attention to the four lightest singly charmed baryons CQ, C^, A, and S with isospin T=0, 1, i, and | , respectively. For the CQ (also called A^), we take the presumed experimental mass of M^ =2.26 GeV; for C (al­so called S^), we take M^ =2.42 GeV. For A and 5, we use theoretical estimates^ of M^=2.47 GeV and ^5=^2.57 GeV. Since only the C^ baryon is stable under strong interactions it is likely that the C^'N and Co-nucleus systems are of greatest interest, although the C nuclei are most strong­ly bound.

Using these potentials we have searched for B^A^ bound states by solving the nonrelativistic Schr6*dinger equation. The meson-nucleon and meson-J5^ coupling constants we use are listed in Table II of Ref. 4 and Table I of Ref. 3. We took r^ = 0.522 fm for ^S^ states.

We start by considering the ^S^ state of B^N

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VOLUME 39, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 12 DECEMBER 1977

r(fm)

FIG. 1. Two-body charmed-baryon—nucleon poten­tials in the SQ state. The model of Nagels, Rijken, and Swart (Ref. 3) was used, extended to SU(4) sym­metry as in Ref. 4.

s y s t e m s . In analogy to the si tuation for the deu-te ron , if a ^S^ B^N s ta te is bound, we might a lso expect the cor responding ^S^ s ta te to be bound, a s a r e su l t of t enso r coupling of ^S^ and ^D^, However, the rr-meson component of the la t te r , propor t ional to {m^/M^ f, is weakened and th i s quest ion must be answered in a s epa ra t e coupled-channel calculat ion. The diagonal potent ials V{r) for B^N-^B^N a r e shown in Fig . 1 for the ^SQ con­figurat ion of iViV, C^N, C^N, and 5iV. The i sovec ­tor p - and TT-exchange potent ia ls a r e always s e p ­a ra t e ly attractive for the maximum / a n d repul­sive for minimum /. The isospin dependence of the B^N potential i s appreciable ; thus binding i s mos t likely in s t a t e s of maximum /.

In fact we find that 'S^ C^N {/= | ) and SN {1=1) s t a t e s a r e bound by 1.76 and 0.08 MeV, r e s p e c ­t ively. The somewhat s t ronge r potential obtained for the unbound ^S^ NN sys t em i s h e r e compensa t ­ed for by the lowered kinet ic energy of B^N. The l a r g e r reduced m a s s of B^N i s a lso respons ib le , however, for a lowering of the recoi l -dependent TT-meson exchange potential . Thus we find that nei ther the ^S^ (/= | ) C,N nor 'S^ C^N s t a t e s a r e bound in approximate calculat ions including t e n ­so r fo rces in the f o r m e r case and off-diagonal / = 1 exchange {C^N-C^N) fo rces in the la t te r . A

-20

-30

1 1

h

1

C o -

1

^N

1

1 1

^ c ,

1 1

1

"

i

J

r{fm)

FIG, 2„ The Hartree potential Vi^ir) for N, Co, and Ci in ^^O, calculated according to Eq, (1), using the spin-isospin averaged interaction vair) of Eq, (2).

slight i n c r e a s e in the s c a l a r meson coupling from ^g/"/47r= 5.03 to 5.23 does produce binding in ^S^ CQN when coupling to C\iV i s approximate ly i n ­cluded. Our conclusions about the two-body s y s ­tem a r e model dependent, but it i s c l ea r that no such s t a t e s will ever be strongly bound.

In cont ras t , the l a rge m a s s of the cha rmed b a -ryon should lead to many bound s t a t e s in the many-body sys tem, some of which will be deeply bound. In pa r t i cu l a r the spin and isospin averag ing which occu r s in a finite nucleus leads to s t rong C^-, C^-nucleus in te rac t ions comparab le to iV-nucleus. Thus charmed nuclear s t a t e s s table against s t rong decay will exis t . F o r sp in- , i so sp in - sa tu ra t ed nuclei ( J = 0 , 7=0), the H a r t r e e potential is^

V^{r)=Jp{P)v,{r^P)d'r', (1)

neglecting sp in-orb i t and Coulomb contr ibut ions . Only Wigner fo rces a r e genera ted, and these a r i s e from exchange of CJ, cp, e mesons . The ha rd core i s removed by use of the Moszkowski-Scott separa t ion procedure."^ This method cons i s t s of finding a cutoff r ad ius r^ for which the ^S^ phase shift vanishes near z e r o kinet ic energy. We may

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VOLUME 39, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 12 DECEMBER 1977

0

10

> 2 0

(S> u 30 2

O H Q 2 40 OD

50

60

- —1

- — 0 N=0

" 4He

-

-

— 2

— 1

— 0

N=0

K-I2C

N =

-1

-0

1

— 3 — 1

- ^ - 0 N = l

— 0

N=0

1*—'6o—H

— 6

— 5

— 4

— 3

— 2

— 0 N=0

— 4

— 3

— 2

— 1

— 0 N = l

-58Ni__

N

-2

-1

- 0

2

— i

— 0

N=3

H

— 12

— II

—10

— 9

— 8

— 7

— 6

— 5

_ 4

— 3

— 2 — 1

N 8 He

— 9

— 8

— 7

— 6

— 5

— 4

— 3

— 2

— 1

— 0

N = l

— 7

— 6

— 5

— 4

— 3

— 2

1

— 0

N=2

_208pb

— 5

— 4

— 3

— 2

— 1

— 0 N=3

— 3

~ 2

—-1

— 0 N = 4

— 1

— 0 N = 5

1

1

--

-

-

FIG. 3. The charmed hypernuelear spectrum for a Ci bound in various nuclei. The states are labeled by L value; each will be further split into two components by the L • S potential. Note that some states of high N and L are bound. The spectrum for Co is very similar in spacing, but fewer states are bound (Z. ^ 9 for N = 0 in ^^^Pb, for example).

then take

f(r) = 0, r<rQ,

v{r) = VQ{r), T^TQ,

with

(2)

(3)

In Eq; (3) jv^= fx^r where ii^ is a meson mass. Also ^^y47r= 15.29, 10.31, 11.74 while ^^V47r = 1.81, 1.56, 1.36 for the iViV, C^N, C iV systems, respectively. The scalar meson e is universally coupled to the {20} multiplet of SU(4) with ^e V47r = 25.32. The large width of the e is taken into ac­count by using a distributed mass spectrum.^ For each B^N channel, one must in general determine TQ independently. As a first approximation, we have used a single vsilne of r^, determined from the SQ channel of maximum /. For the iViV, C^N, and C^N systems, we obtained r^^ 1.025, 1.18, and 0.93 fm, respectively.

The^^-nucleus potentials were calculated for ' He, ^^C, ^^O, '«Ni, and ^^^Pb. Some typical r e ­sults are shown in Fig. 2 for ^^O. The B^-nucleus potentials are seen to be comparable in depth to the nucleon-nucleus potential. In Fig. 3 we dis­play the calculated spectrum of Cj-nucleus shell-model states. The high level density and deep binding resulting from the high charmed-baryon mass is evident.

Several points are worth mentioning: (1) The spacings between both radial and orbital excita­tions vary inversely as the reduced mass, M^

'^M^^ for heavy systems. Since M^ /M^ ^ 2.4, levels in a charmed nucleus will be much more closely spaced than in an ordinary nucleus. (2) Al­so for a heavy nucleus, the lowest state (iV = I, = 0) occurs close to the bottom of the well. (3) The close packing of B^ levels implies that states of rather high L may be bound in heavy charmed nu­clei. For example a C in a j = ^ state coupled to ^ ^3/2*^ neutron hole can produce states with spin up to 19 in 2^^Pbc .

We have already noted that C^N or C^-nucleus states are unstable with respect to strong inter­actions, in fact decaying to Co-nuclear states by single TT emission. One expects widths of order 1 MeV for such states, appreciably wider than the lowest Co-nuclear states which decay only weakly. The observation of any such states will be simi­lar, however, since C tracks before decay will be very short. A most likely mode of detection of the bound states will probably parallel closely the observation of the free charmed baryons (C^, C ) themselves. In an emulsion, for example, an event which can be sufficiently detailed to r e ­veal a charmed baryon mass lower than that a free C + nuclear system would signal binding. Presumably a good sign of charmed nuclear states would be the observation of a strange hy-pernucleus as an intermediate step after the pro­duction of a charmed baryon, even though this is an event that may occur with low probability. Clearly the most favorable systems to study are B^ nucleus for which binding is appreciable. Thus emissions placed in beams where slow charmed

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V O L U M E 39, N U M B E R 24 PHYSICAL R E V I E W LETTERS 12 DECEMBER 1977

baryons or mesons are produced are likely to ex­hibit bound B^ nuclei.

Finally one notes the strangeness analog states discussed by Kerman and Lipkin^ will have their counterparts in charm analog states, the latter resulting from the extension of the Sakata model to SU(4).

The authors are grateful to T. L. Trueman for many useful conversations. This work was sup­ported by the U. S. Energy Research and Develop­ment Administration under Contract No. EY-76-C-03-0016.

^E. G. Cazzoli et al., P h y s . Rev. Let t . ^ , 1125 (1975)5 66,

B . Knapp et al., P h y s . Rev. Le t t . £ 7 , 882 (1976); G. Gold-hsibeYetal., P h y s . Rev, L e t t . ^ , 255 (1976); L P e r u z -zi etalo, Physo Rev. Lett . 37 , 569 (1976).

^For a rev iew of one-boson-exchange potent ial mod­e l s , see K. E rke l enz , Phys . Rep, 13C, 191 (1974).

^M. M. Nage ls , T . A. Rijken, and J . J . de Swar t , P h y s . Rev. BU, lU (1975)..

^C. B. Dover , S. H. Kahana, and T. L. T r u e m a n , P h y s , ReVo D ^ , 799 (1977).

^A, De Rujula, H. Georgi , and S. L. Glashow, P h y s , Rev. D j ^ , 147 (1975).

^J. P . Vary and C. B. Dover , Phys , Rev. Let t . Sj., 1510 (1973),

• S. A, Moszkowski and B, L. Scott , Ann. P h y s . (N.Y.) j U , 65 (1960).

^A. K. Ke rman and H. J . Lipkin, Ann, P h y s . (N.Y.) 66, 738 (1971).

Hydrogen Surface Contamination and the Storage of Ultracold Neutrons

W. A. Lanford

A. W. Wright Nuclear Structure Laboratory, Yale University, New Haven, Connecticut 06520

and

R. Golub School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, Sussex, United Kingdom

(Received 2 August 1977)

The r e s u l t s of hydrogen profile n i ea su remen t s on samples p r e p a r e d by the s ame me th ­ods used for these m a t e r i a l s when making u l t r aco ld -neu t ron (UCN) s to rage bot t les show that the sur face hydrogen i s sufficient to account for the bulk of the anomalous shor tening of UCN s to rage t i m e s .

Ultracold neutrons (UCN) are neutrons with such low energy (~ 10"" eV) that they undergo to­tal reflection for all angles of incidence from many materials. As such, these neutrons can be stored in "bottles'' for substantial periods of time and may provide a source of neutrons upon which a variety of important fundamental meas­urements can be made, e.g., the accurate deter­mination of the neutron lifetime^ and a more sen­sitive search for its electric dipole moment.^

The first direct measurements of ultracold-neutron storage times^ revealed serious discrep­ancies between the observed storage times, r^, and those expected for an ideally flat, pure sur-face^ While neutrons below a critical energy undergo total reflection, there are mechanisms which result in the "loss' ' of these neutrons. Dur­ing a reflection, the neutron can be lost via nucle­ar capture or by being inelastically scattered to an energy above the critical energy.

Further experiments'^'^ have shown similar

anomalies and much effort has been devoted to understanding of these resultSo^*^*^"^^ Several authors^'^'"^'^^ have concluded that since the ob­served storage times appear to be independent of temperature, surface impurities cannot play an important role in the UCN losseSo The only pub­lished data on this point, Figo 1 (Refs„ 7a and 7c), is indeed consistent with UCN losses being inde­pendent of temperature, but it does not rule out variations in r up to a factor of 2 over the tem­perature range covered (dotted line. Fig. 1)

It is difficult to predict a unique temperature dependence of r if impurities are responsible for the anomalieSo If the impurity concentration was temperature independent and if the losses mainly result from inelastic scattering from a coherent scatterer or harmonically bound nucleus, the loss rate will be proportional to the Planck oscillator occupation number, i.e., ~ T for large T and '^exp[- ^Q/T] for small T (with respect to the oscillator frequency O^Q). ^ For incoherent

1509