V O L U M E 12, N U M B E R 7 PHYSICAL REVIEW LETTERS 17 FEBRUARY 1964

is weak, however. Lately, there seems to be some evidence17 for a scalar particle at 380 MeV with 0 , which was first suggested by Brown and Singer.18 This particle can contribute a positive self-mass to K^, thus reinforcing our conclusion regarding the sign of the mass difference. An assumption regarding the transformation property of weak vertices similar to ours has recently been applied to the problem of nonleptonic hyperon de­cays in pole approximation by Sugawara19 who finds it possible to correlate presently known ex­perimental results on the basis of such a model. We are aware of the limitations of a restrictive model like the pole approximation for the (K^0 -K2°) mass difference, whose understanding might even­tually turn out to require a consideration of more complicated intermediate states than those con­sidered here. This possibility is underlined by our inability to convincingly calculate the magni­tude of the mass difference. We feel, however, that since the present model, apart from being simple and giving finite results, leads to definite predictions which relate the sign of (K^ - K2°) mass difference to other effects, it might still be of some interest to check its conclusions with ex­perimental results .

We are thankful to Professor G. Feinberg and Professor B. M. Udgaonkar for several discus­sions.

1 M. Gell-Mann, California Inst i tute of Technology R e ­por t No. CTSL-20, 1961 (unpublished); Phys . Rev. L25, 1067 (1962).

2Y. Ne 'eman, Nucl. Phys . 26, 222 (1961). 3 S. K. Bose , Phys . L e t t e r s 2, 92 (1962). 4 S. Oneda, S. Hori , M. Nakagawa, and A. Toyoda,

P h y s . Le t t e r s 2, 243(1962) .

5 S. Okubo, P r o g r . Theore t . P h y s . (Kyoto) 27, 949 (1962).

6 This may be der ived from the condition (ir0\Hoo\Ki0) = O

which is a consequence of CP i nvar iance . The condition Gj°|jffcJjK'1

0):=O leads to the s a m e conclusion. 7Equation (4) has been obtained independently by S. Oku­

bo (pr ivate communicat ion) . 8M. Bake r and S. Glashow, Nuovo Cimento 25, 857

(1962). 9G. Alexander , S. Almeida, and F . Crawford, J r . ,

P h y s . Rev. Le t t e r s 9, 69 (1962). 1 0C. Bouchiat , J . Nuyts, and J . P ren tk i , Phys . L e t ­

t e r s 3 , 156 (1962). " S . Oneda and S. Hori , Phys . Rev. 132, 1800(1963) . 12M. Chre t ien , F . Bulos , H. Crouch, J r . , R. Lanou,

J r . , and J . Mass imo , Phys . Rev. L e t t e r s 9>, 127(1962) . 1 3C. A l f f e t a l . , P h y s . Rev. Le t t e r s £, 325(1962) . 1 4E. Fowler , F . Crawford, J r . , L. Lloyd, R. G r o s s ­

man, and L. P r i c e , P h y s . Rev. Le t t e r s 10, 111 (1963). 1 5This may be eas i ly seen by wri t ing the phenomeno-

logical two-par t i c l e in teract ion as (dJC)Bn* K a n d B ^ denoting the field ampli tudes for the K meson and vec tor boson, respec t ive ly , and r e m e m b e r i n g the par i ty p r o p ­er ty ofB^y i . e . , PBip^^-Bi, i = l , 2 , 3 , a n d P B J P ~ i = B i . We would like to acknowledge the benefit of an i l lumina t ­ing conversa t ion rega rd ing th is point with P r o f e s s o r G. Fe inberg .

16We have not included the recen t ly repor ted [M. Abo-l ins , R. Lander , W. Mehlhop, N. Xuong, and P . Yager , P h y s . Rev. L e t t e r s 11 , 381 (1963)] B pa r t i c l e into con­s idera t ion as i t s quantum numbers a r e yet to be d e t e r ­mined. In any c a s e , s ince the B pa r t i c l e has no na tura l p lace in the SU3 s cheme , the possibi l i ty of i t s belonging to some octet o ther than the vec tor octet does not s e e m to be ru led out. In pa r t i cu l a r , it could be a Regge r e ­c u r r e n c e of one of the known vec tor m e s o n s . The spin of B will then be 3 .

1 7 F . Crawford, J r . , R. Gros sman , L. Lloyd, L. P r i c e , and E . Fowler , Phys . Rev. L e t t e r s 11 , 564(1963) .

1 8 L. Brown and P . Singer , Phys . Rev. Le t t e r s 8, 460 (1962).

19H. Sugawara (to be published).

ARE THE f° AND B MESONS TWO DECAY MODES OF THE SAME PARTICLE?*

William R. Frazer , ' Sharashchandra H. Patil, and Nguyen-huu Xuong Universi ty of Cal i fornia , San Diego, L a Jo l l a , California

(Received 19 December 1963)

The striking fact that the recently discovered B meson has the same mass and width (within statistics) as t h e / 0 suggests the hypothesis that these two resonances are , in fact, different de­cay modes of the same particle. x~4 In this Letter we examine the consequences of this hypothesis and find that it is not incompatible with the cur­rently available experimental data. In fact, it seems to be easier to reconcile the data to this

viewpoint than to the usual one in which the / ° is assigned 1 = 0.

A particle which decays strongly, as does the B, into a n and an a>, must have 1=1. If it also decays strongly into two pions, as does the/ 0 , it must have odd J and negative parity. We shall confine our attention largely to the 1" assignment, and refer to this hypothetical particle as the p ' meson. We shall use the name B to designate the

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VOLUME 12, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 17 FEBRUARY 1964

770) decay mode of the p ' meson, a n d / to designate the 777T decay mode, just as 6 and T designated different decay modes of the K. We now examine the implications of the p ' hypothesis.

(I) Angular distribution i n / 0 decay . -The / 0 is observed in the reaction

77 + />-7T +7T + W. (1)

Dist r ibut ions have been measu red in c o s f l ^ , the angle between incoming and outgoing n~ in the barycentric system of the final pions. It seems quite plausible to assume that for low momentum transfers to the nucleon the above reaction pro­ceeds primarily through one-pion exchange. Then the angular distribution should be [Pjfcosfl^)]2, where J is the spin of the / 0 . The data on cosfl^ of Guiragossian4 and of Bondar eit a l . 3 are re ­produced in Figs. 1(a) and 1(b).5 In Fig. 1(c) are plotted the combined data for both experiments. We also plot the normalized curves predicted by J = 1 (solid line) and J = 2 (dashed line). It is ap­parent that the best fit is obtained with J = 1. A X2 test of the J = 1 curve fitted to the distribution in Fig. 1(c) has an expected x2 of 4. 0; we find 4 . 1 , which corresponds to a probability of 40%. The J =2 curve gives a x2 of 105; i . e . , a very small probability.6 The curve predicted by J = 3 also does not fit the data; we get x2 = 40.

Thus the angular distribution i n / 0 decay strong­ly favors the p ' hypothesis. The scarcity of events at cos0 7777 = 0 is a serious obstacle to the assignment of 1 = 0, and therefore J even, to the

A (II) Branching ra t ios . -The most obvious objec­

tion to the p ' hypothesis is the fact that the de­c a y / - 77 +77° has not been observed. In the re­action

7T~ +p-~TI~ +TI°+p, (2)

no peak has been seen in the invariant mass spec­trum of the final pions in the neighborhood of 1250 MeV. On this basis Selove et a l . 2 made the as ­signment 1=0 for the / 0 . Moreover, no such peak was seen by Alff <et a l . 7 and by Carmony et a l . 8

in the reaction

77++£-774" + 77°+/>. (3)

The p ' hypothesis, however, assigns /= 1 to the f° and therefore implies a triangle inequality

Va+ + /a_M2a0)1 / 2 , (4)

where <j± are the total cross sections for 7r± +/? and <70 is the total cross section for 77~ - /*+/> ,

+p-»f+n

J = 2 J = l

O 70| oc UJ 2° 60 r> z

50

40

30

20

10

'I -"'

il 1 \

k

\ -

\ -1 \ ' . \

\

/ ^ \ v \ R \ / \ \ il

i V K 1 / L \ i / l _ .

! t r

P. 1

l / \\ f r i 1 /

/ / ' '

/ / / /

\ 1 \L -1 -0.6 -0.2 0+0.2 40.6

COS GTTTT

+1

This apparent contradict ion of the p '

FIG. 1. Cosfl^ distribution of peripheral events from the reaction 7r"+/>-*« + 7r+ + 7r" with M^ in t h e / 0 peak: (a) from Guiragossian (reference 4), (b) from Bondar ^t al. (reference 3), (e) from both experiments. Also are plotted the normalized curves predicted by J = 2 and J = 1.

hypothesis cannot be explained by a rguments in­voking violation of i sospin conservat ion, because the l a rge width of the p ' i n s u r e s that i t s decay is via s t rong in te rac t ions . We propose that the de ­cays f± - 7r± +7T° do indeed exis t , but that the co r ­responding peaks in React ions (2) and (3) a r e much h a r d e r to r e so lve from the background than i s t h e / 0 peak in Reaction (1). Due to the exis tence of the p peak, detect ion of t h e / peak depends c r i t ica l ly on the depth of the valley between the two peaks . The valley between the p± a n d / * peaks could be much shal lower than the one be -

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VOLUME 12, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 17 FEBRUARY 1964

tween the p° and/ 0 . We shall attempt to make this plausible by considering a simple model of p ' production which exhibits this feature.

(Ill) Qne-pion and one-omega exchange model. In an earlier Letter by Watson and two of the present authors,9 the reaction in which the B was observed,

TI* +p-~ir+ +oo +/>, (5)

was analyzed by the one-pion-exchange model, Fig. 2(a). This model, by assuming that the B is produced from a rnr state, implies that its quan­tum numbers are 1" (or 3", etc.) and that it has a strong im decay mode; therefore, it is consist­ent with the p ' hypothesis. The left-hand vertex in Fig. 2(a) involves the reaction

77 + 7T —71 + 0>, (6)

whose amplitude in the J= 1 state we call Af 12. In reference 9 this amplitude, along with the ampli­tudes M n and M22 for the processes

and

77+71-71+77

77 +U>-77 +(x)

(7)

(8)

in the 1" state, was parametrized by a multichan­nel effective-range theory based on the ND~X

method. The parameters were adjusted to fit the positions and widths of the B and the p . A rough and very conservative prediction of the width of the a) was made, r ^ ^ 1 MeV,10 which was subse-

T

\ \

it) N TT 7T

/ \ / / \ /

FIG. 2. Diagrams included in the 0(P + 0)E model (see text).

quently confirmed by a direct measurement of the (A) width. n Despite this confirmation, it is ap­parent that some extension of the one-pion-ex-change (OPE) model is necessary. Although the Treiman-Yang test was reasonably well satisfied in Reaction (5), other tests for one-pion exchange in this reaction did not seem to be satisfied.10

The most natural extension is to include one-omega exchange (OOE), Fig. 2(c). The left-hand vertex involves M22, which is determined by our effective-range model. For the NNoo vertex we take the form and strength found by Scotti and Wong to give a good fit to the nucleon-nucleon scattering data.12 In addition, form factors, which are slowly varying functions of the momentum transfer to the nucleon, are included for both OPE and OOE. The Ferrari-Selleri1 3 result is used for the OPE form factor, and the extra parameter introduced in the OOE form factor is used to fit the distribution in costf^ observed at the p peak in Reaction (2). Further discussion of the model is relegated to the Appendix, and to a more de­tailed subsequent publication, since the essential arguments of this Letter depend only on the quali­tative features of the model.

Having used all the parameters of the model to fit the p and the B decay mode of the p ' , we are able to extract from the model predictions about the/-decay mode. These are calculated from the diagrams of Figs. 2(b) and 2(d). We empha­size that no experimental information about the / has been used as input. The resulting distribu­tions in M ^ are shown in Fig. 3 for an incident pion momentum of 4 BeV/c, as in the experiment of Bondar et a l . 3 [dashed line for Reaction (1) and

1.2

0.8h - Q

E

0.4

0.2

k

f-

r~

h

i

11 INCIDENT MOMENTUM:4 Bev/c

1 TT"+p—*»TT~+TT++n

1 TT"+ p — ^ T T " + T T 0 + P

' !

— 1 ^ ^ - J — 1 1 I 200 400 600 800 1000 1200 1400 1600 1800

M TTTT (Mev)

FIG. 3. Effective-mass distribution of the pion pair from the reaction irN-^Nini as predicted by the 0(P + 0)E model for an incident momentum of 4 BeV/c.

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VOLUME 12, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 17 FEBRUARY 1964

solid curve Reactions (2) and (3)]. The predic­tions for Reaction (1) are in good agreement with the experimental data; those for Reaction (2) do not seem to be inconsistent.

We call the reader 's attention to the striking difference between/0 a n d / * production. The ac­tual values of the cross sections at the / peak are, of course, consistent with the triangle in­equality, Eq. (4). In / 0 production, however, there is a much more pronounced valley between t h e / and the p ' than there is i n / * production. This effect, which makes t h e / * peak much more difficult to resolve from the background, provides a possible explanation of why t h e / * has escaped detection. We estimate from our model that the ratio of the apparent (above the valley) cross sec­tions f o r / * production in Reaction (2) or (3) and f° production in Reaction (1) is 0.20 at 3 BeV/c and 0. 25 at 4 BeV/c. The experimental ratio is <0. 3 at 3 BeV/c and £0.25 at 4 BeV/c.

The difference between the predictions of the model for / 0 a n d / * production ar ises from the fact that i n / * production via Reaction (2) or (3) both OPE and OOE contribute, whereas for / 0

production via Reaction (1) only OPE is possible (the exchanged particle must be charged). Thus the solid curve in Fig. 3 is a sum of the two con­tributions (the interference term vanishes): The OOE term, and the OPE term which for Reaction (2) is | times the dashed curve. The OOE term differs in two important respects from the OPE term: (a) Because of the spin of the omega it has a kinematical factor in the numerator which sup­presses high values of M ^ . Its contribution is large around the p peak, but decreases rapidly to become almost negligible at the / peak, (b) It does not vanish between the two peaks, as does the OPE term.1 4

Our model gives a branching ratio (p' - it +co)/ (p' - 7r +7r) = | , consistent with the experimental upper limit of 1.0 found by Carmony et a l . 8

(IV) Some experimental tests . - The p ' hypothe­sis can be tested in many ways. We conclude by listing some which have occurred to us or have been suggested to us.

(A) Determination of the spin and parity of the Bi Our hypothesis, of course, requires 1" for the B. Some tests for its spin and parity have been proposed by Zemach.15 Using these tests, Carmony et al. found evidence for J ^ 1 but could not distinguish between odd and even parity.8

(B) Search for/°~7r° + 7r0: This is allowed if the / ° has 7 = 0, forbidden if it has 7=1. This search can be done by comparing the missing-mass dis­

tribution Mx of the reaction

7T + +D- (P)+P +X

with the 7T+7r~ effective-mass distribution of the reaction

7 T + + D - (p)+P +7T++7T".

Because p° does not decay neutrally, a peak in the Mx distribution at the f° mass would be easily observable. This peak must be | of the corre­sponding 7r+7r" peak to satisfy the 7=0 hypothesis. With the p ' hypothesis one would expect a ratio £j^ (due to the B° decay mode with the a; decaying neutrally).

(C) Comparison of n +p -~D +/° and p+p~D +/**: If t h e / is the p ' , with 7= 1, these reactions occur in the ratio 1 to 2.

^ Search f o r / * : One can, of course, keep looking for a small bump in the cross sections of Reactions (2) or (3). Alternatively, one could investigate the distribution in costf^ as a func­tion of M^TJ. A plot of the forward-backward asymmetry in cosfl^ has been given by Bonder et a l . 3 The plot for Reaction (2), in which t h e / " is expected, is remarkably similar to (the nega­tive of) the plot for Reaction (1), in which the / 0

is observed. We have no theory of this asym­metry, since it involves interference with back­ground, but the similarity lends some support to our hypothesis.

Appendix.-For the amplitudes M^As), we use the "effective-range" parametrization M^ND'1*

where ij o if v c r

D.. = 6..~ SJI..-I —T i j ij 0 ij TiJs s'(s' +

i

p.(s')ds'

s0Hs>-s)>

where px = 2q13/s1/2, p 2 = 2q2

3s1/2, and qlf q2 a r e c m . momenta in the irir and TTCO states. Observ­ing that the integral in D2l and D22 diverges, we introduce a cutoff in the form

p 2 -p 2 [A/(s+A)] 2 .

We now can calculate My. as a function of five parameters: nn, nx s0, and A (MX2=M2l

implies n12=n21). We eliminate two of these by requiring the zero of the determinant of D to represent a p meson with the correct position and width. Another parameter is determined to give the experimental width of the a> meson as described in reference 9. The remaining two parameters are determined so as to fit the posi­tion and width of the B in the process (5) using

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VOLUME 12, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 17 FEBRUARY 1964

the 77, QJ exchange model. Expressions for differential cross sections in­

cluding diagrams in Fig. 2 can be calculated in terms of M^ elements and NNTI,NNOO couplings (taken to be of the form g^y^Nu^)- They give the following expressions for the production cross sections of process (5) and the two-pion produc­tion:

d2o 77 77

dsdA2 8v2m2q 2 m w ( A 2 - w 2)2

77 48

I A n l 2

\**)

2* (2p 2 s in 2 0' A2),.y (A2

•m 2 ) 2 24 ^ l / , (13)

d2o 7T0D

dsdA2 8 T T W ^1L-["U) A2 a - V

L L i i i» (A2-™ 2 ) 2 24 I A 12'

Uw ( 2 p 2

2 s i n 2 e ' - A 2 ) 9 . Y

( A 2 - w 2)2

GO 12 W 2 2 , f ,(14)

where i 4 n = 4877Mn, A1 2 = 24V27rM12, ,<422 = 247rM22, m =nucleon mass ; q^ and #/ are the magnitudes of the incoming and outgoing pion momenta, and 6' i s the angle between qx and p2, all of these be­ing evaluated in the c m . system of outgoing 7777 or 77 0,. All kinematical quantities are evaluated with the exchanged particles off the mass she l l . 1 3

Formulas (13) and (14) are extended to off-shell values of A2 by introducing form factors fn , / w

for the 77 and a) exchange amplitudes, respective­ly. We take

L ( m ^ - m J V K 2 - A 2 ) ,

/ = (m2-m 2)/(m2

GO Z aJ Z A 2 ) .

The analysis of Ferrari and Selleri gives m2~ 15, while a fit to the c o s f l ^ distribution in the p pro­duction gives ra2

2~46. In the curves shown in Fig. 3, the values of the

parameters are s0= 1 .92x10 6 , A = 120, ^ 2 / 4 T 7 2 .7 , nlx = 2.94x10"

= 5 . 3 7 x l 0 " 3 . 4 . 4 4 x l 0 ~ 3 , and nl

*Work done under the auspices of the U . S . Atomic

Energy Commission. f Alfred P . Sloan Foundation Fellow. 1M. Abolins, R. L. Lander, W. Mehlhop, Nguyen-

huu Xuong, and P . Yager, Phys. Rev. Letters 11, 381 (1963); J . Kirz et a l . , Proceedings of the Siena Confer­ence on Elementary Particles, Siena, Italy, 1963 (un­published); G. Goldhaber, S. Goldhaber, J . Brown, J . Kadyk, and G. Trilling (to be published).

2W. Selove, V. Hagopian, H. Brody, A. Baker, and E. Leboy, Phys. Rev. Letters 9, 272 (1962); J . Veillet et a l . , Phys. Rev. Letters TO, 29 (1963); V. Hagopian andW. Selove, Phys. Rev. Letters lj), 533 (1963); W. Selove (private communication).

3L. Bondar e t a l . , Aachen-Birmingham-Bonn-Ham-burg-London-Munchen Collaboration, Phys. Letters j>, 153 (1963); I. Derado (private communication). There is an error of scale in Fig. Kb). The scale of the

number of 7r~7r° combinations" must be doubled. 4 Z. G. T. Guiragossian, Phys. Rev. Letters U,, 85

(1963). 5We do not include the data of Hagopian and Selove

(reference 2) at 3 BeV/c because the marked asymmetry of their cosf?^ distributions, in contrast with the sym­metric distributions found at higher energies, seems to indicate a larger contamination from processes other than OPE. If, however, we divide the Hagopian and Selove data in t h e / 0 region into five intervals in cosfl^ and fit it with J = 1 and J = 2, we find x2 of 66 and 95, respectively (x2 =4 expected).

6A note of caution is in order here. We cannot rule out the possibility that the / ° has spin 2, and that the observed distribution in cosfl^ results from interfer­ence with a complicated background of some sort . The hypothesis of a p' meson produced by one-pion exchange does, however, provide a much simpler way of account­ing for the distribution.

7C. Alff et a l . , Phys. Rev. Letters 9, 322 (1962). 8D. Carmony, R. Lander, C. Rindfleisch, Nguyen-

huu Xuong, and P . Yager (to be published). *W. Frazer , S. Patil, andH. L. Watson, Phys. Rev.

Letters 1 1 , 231 (1963). 1 °Nguyen-huu Xuong, R. Lander, W. Mehlhop, and

P . Yager, Phys. Rev. Letters 1 1 , 227 (1963). UN. Gelfand et a l . , Phys. Rev. Letters IT, 4 3 6 (1963);

R. Armenteros et a l . , CERN and College de France Collaboration, Proceedings of the Siena Conference on Elementary Particles, Siena, Italy, 1963 (unpublished).

12A. Scotti and D. Y. Wong, Phys. Rev. Letters 1_0, 142 (1963).

13See, for example, F . Selleri, Phys. Rev. L e t t e r s ^ , 76 (1962).

14Between the two peaks \Mu\ very nearly vanishes, because the phase shift in this channel goes through TT in rising from 7r/2 at the p to around 37r/2 at the p ' .

15A. C . Zemach (to be published).

182