large radiative hyperon decays

Volume 160B, number 1,2,3 PHYSICS LETTERS 3 October 1985

LARGE RADIATIVE HYPERON DECAYS

Andrew C O H E N Lyman Laborato~ of Physics, Harvard Umverslty, Cambridge, MA 02138, USA

Received 12 June 1985

An effectwe lagrangmn for baryons, pmns, and photons is used to &scuss the weak radmt~ve decays of the hyperons Predlctxons for the s-wave parts of the decay rates are obtained, m reasonable agreement with expenment The p-waves are shown to be of the same order of magmtude as the s-waves, but no numerical predlchons are obtained

Although the strangeness changing radiative decays of the hyperons have received considerable attention [1], no satisfactory explanation of their decay amplitudes has been achieved. This is especially surprising since only one such decay mode has been measured with any substantial accuracy [2].

Such decays have been suggested as an im- por tant probe of the details of low energy strong interactions. The problem has been one of obtaining s- and p-wave amplitudes which are comparable, and a total rate which is the same order of magnitude as the observed 5~ +---, p- /decay rate. Using an effective field theory of baryons, pions, and photons, we obtain decay amplitudes in good agreement with current experiments.

We begin by writing down the effective lagrangian for low energy baryons, pions and photons *1. We use the following representations of the approximate global SUL(3 ) X SUR(3 ) flavor symmetry: Pions:

= exp (iqr/f,,), *r = Y',~r~T ~, a

T r ( T ~ T b) = l ~ a b

1~ ~ L~U t = U~R t, 4 2 ~ ~,

,1 For a review see ref [3]

Baryons:

B = ¢ ~ E B ~ T a , a

Photons:

B --* U*BU.

D r = 0 r + i e T . [ Q , ].

Ignoring weak interactions and symmetry breaking, the lowest order effective lagrangian is

*~ao= ¼f~ Tr ( D s ~ D r ~ t ) + Tr [B(iD + iV)B]

+ m B T r (BB) + D T r (B75 ( .4 ,B})

+ F Tr (BTs[~ ,B] ) .

We must also include symmetry breaking terms. In particular we have symmetry breaking mass terms proport ional to

M _ -

m,., 0 0 ]

0 m d 0 ,

0 0 m s

and anomalous magnetic moments for the baryons proport ional to

Q=

2 0 1 0 -~

0 0

0

0 , 1 - -7

0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (Nor th-Hol land Physics Publishing Division)

177


.£ax = ¼f2# Tr ( M E ) + h.c. + L 1 T rBM B

+ L 2 T r B B M

+ (e/2mp)amTr-Bo~Fgr(Q,B)

+ (e/2mp)cmTr-Bo~r~'[Q,B]

a~) Tr~o~r,~(Q, M,B) J e

+ 2rap j= l AxSB

+ (ebl/2mp)Tr (Be- FB) TrQM

+ (eb2/2m p) Tr (MB) o. F Tr (QB) + h.c.,

where (Q, M, B) s runs over the four distinct orderings of the three octets Q, M and B.

We must now include operators responsible for strangeness changing processes. These are conve- niently described in terms of the matrix

h = 0 • 0

The lowest order terms are

.~aS=l=oTrB({~th~,B}do + [~*h~,B] f0)

+h .c .

These operators are known to give an excellent description of the s-wave non-leptonic hyperon decays [4]. In addition, there are higher dimension operators responsible for the p-wave non-leptonic hyperon decays, but these operators all contain at least one pion and hence are irrelevant for the radiative decays in tree approximation.

So far the values of the coefficients appearing in these expressions are all known from experi- ment, and are in good agreement with naive estimates for their values from the full SUc(3 ) × SUL(2 ) × Ur(1 ) theory.

Now we must include operators that give rise to strangeness changing radiative decays. The short distance contribution (fig. 1) comes from operators of the form

.LP~ s=l - Or t r B (o • F)Ql~thg;B + ""

"}- O r trBv5 (o" F ) Q~th ~B.

However, estimates of Or and o r lead to decay rates that are several orders of magnitude too small to account for the experimental E + ~ p'/

d w s

Fag 1 Short thstance contribution to the operator (e/2mp)TrB(o F)~*h~BQ

B1 81 B 2 B 1 B 1 B2

Fig 2 Anomalous magnetic moment contribution to AS = i

hyperon radiative decay The cross represents a vertex from .wAS=I

decay rate [5]. By assuming some enhancement of Pr we could obtain the right Y.+~ p'y rate but would then be in conflict with the upper bounds on several of the o~ther decay modes .2. If this higher dimension operator is not responsible for the radiative decays, we must ask how else the decays might arise. In fact, we note that the operator which gives rise to the s-wave non-leptonic hyperon decays contains a pion-independent piece by chiral symmetry (setting ~ = ~+= 1)

p t rB{ h, B} d, + # T r B [ h , B ] f 0 + h.c.

Thus a AS = 1 transition from these operators followed by the radiation of a photon will give rise to a AS = I radiative decay (fig. 2). At first sight this contribution appears to vanish. If the pho ton-ba ryon interaction takes the pointlike form

t rBDB - T rB~B + i e T r B ~ [ Q , B ] ,

we can perform a V-spin rotation to remove the off-diagonal AS = 1 baryon-baryon vertex without affecting the baryon-photon interaction (since Q commutes with V-spin).

Of course, this is not the only photon-baryon interaction - the baryons do not look much like point particles. They have large anomalous magnetic moments, which violate V-spin invariance and hence contribute to the decay. The s-wave

:~2 These calculations have been done in ref [5] using an SU(6) non-relatiwstic quark model. The results there are easdy adapted to the effective field theory language used here

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Table 1

Decay F s (GeV)

~2+~ pT 3 8 5 X 1 0 18 A° --+ N7 609 X 10 18 Eo __+ N'y 1 44 X 10 -17 .-o _.., y.o y 6 57 X 10 -20 .-o __, AOy 2 00 X 10-17 E- - - , E - y 1.89 x 10 -18

decay rate for B 1 ~ B23 , from this s o u r c e is .3

F s ( B 1 ~ B2Y)=aem(p2/mp)[(M1 + M2)/Mx] 2

2 2 X ( ~ l - - i f 2 ) C12[q[/mp,

where C1=, a function of f0 and d o, is the SU(3) Clebsch-Gordan coefficient for the B1-B = transition and [q[ is the three-momentum of the photon. As promised this depends only on the anomalous parts of the magnetic moments - the non-anomalous parts cancel in the difference ~1-~2- Using p and C~2 from the non-leptonic decays, we get the s-wave decay rates of table 1. These are seen to be of the right order of magnitude to explain the experimental data.

What about the p-waves? At first sight we might just try the same th ing - the o term in L#as= t has a pion-independent piece of the form B1YsB= and this will give a contribution to the decay with radiation of a photon off the initial or final baryon. Since we could remove this term by a chiral V-spin rotation, this contribution to the p-wave amplitude will be proportional to the difference of the baryon magnetic moments, as was the s-wave contribution. However, these contributions are suppressed due to the ")'5, making this contribution to the p-wave decay rate too small by a factor

[( M 1 - M=)/( M~ + M2)] = - 10 -=.

Thus although this source was the most important

,3 The rate for decay modes revolving a y:o or A ° must be appropriately modified to mclude the effect of the A°-F- ° transmon magnetic moment The sxgn of the A°-Z ° transmon moment has been fixed relatwe to the &agonal moments by using the signs of the coetfiments of the operators m .LP 1 known from the measured magnetac moments

contribution to the s-waves, it is negligible for the p-waves.

As already indicated, the short distance contribution to the higher dimension operatorB(o • F ) × 3'shB had a coefficient estimated to be too small to account for the decay rate. However, these estimates must be modified to take into account medium and long-distance contributions m our effective field theory.

How large do we expect these effects to be? In a consistent effective field theory we expect higher dimension operators to be suppressed by powers of some mass scale which is large compared to the energies of the processes we are considering. There is good evidence that the relevant scale here is [6]

A x S B = 41rf,, - 1 GeV.

Hence we would expect the dimension-five operator B(o - F)3"shB to make contributions to the p-wave amplitude suppressed by a factor

Iql /AxsB = ( M 1 - M2)/AxsB,

e.g.

r. . . . . . - rs . . . . . [(M1-M2)/AxsB] 2.

This is too small by several orders of magnitude. However, this estimate is not right. The s-wave

amplitudes constructed from the tree diagrams of fig. 2 are actually smaller than we might have thought since they are only proportional to the anomalous parts of the baryon magnetic moments, /£1 - - ~2" But there is absolutely no reason to beheve that a similar suppression would hold for the higher dimension operators. In partmular the one-loop diagrams of fig. 3 contain a divergent piece (with a cutoff of Axs a = 4rrf,,)

( M 1 + M2)(/z 1 +/- t2)(p/A2sB)(e/mp)4~Ts,

which requires a counterterm of the form

tr [ B [ ( o . F ) / 2 m p ] 3',Q~th~B]

×[(M, + M2)/AxsB]

X (ep/AxsB)(l~ , + #2).

It would be inconsistent to have the coefficient of this &mension-five operator much smaller than the counterterm necessary to renormalize the effective theory. But the coefficient above now yields a

179


1 / \ 2 1 / \ 2

Fig 3 Contnbutaon to p-wave hyperon radmtlve decay The dashed line represents a meson The uncrossed baryon- baryon-meson vertex comes from the D and F terms an "~o

contribution to the decay rate of order

rp-wave - L-wave [(~1 + ~ 2 ) / ( ~ 1 - ~ 2 ) ] 2

x[(M1-M:) /A , , s . ] 2

Since (/z 1 +/~2)/(#1 -/~2), (3/1 - M:)/A×sB is about one, this yields a contribution to the p-wave decay rate of the same order of magnitude as the s-wave rates we calculated previously.

Although the loop diagrams indicate the size of coefficients of higher dimension operators, there are similar contributions from the matching condition, and we should not take the loop graphs as anything more than an indication. We have four octets, B, B, ~th~, and Q, hence we can make six operators. However, since h and Q commute, only four are necessary. Since there are six decay modes this yields two predictions for the p-waves.

A(SO__, A0~) +A(ZO_~ ~:0~) = 2A(r0_~ N~),

A(A °-, Nv) +A(ro- Nv) =A(Z Unfortunately, both these predictions involve decay modes with neutrons in the final state. Since these decay modes are difficult to see, these predictions are not particularly useful, and we get no predictions for the remaining four decay modes.

Conclusions. Using an effective field theory of baryons, pions, and photons, we have been able to predict the six AS = 1 weak radiatwe s-wave decay modes of the hyperons consistent with current experimental data. The experimental result for Fs(E ÷---) p~/), Fs(X +---) py)expt = (3.36 + 1.7)× 10 -18 GeV is in good agreement with our predmtion of 3.85 × 10-18 GeV. In addition, our effective field theory predicts p-waves of the same order of magnitude as the s-waves, although we are unable to make quantitative predictions for the specific p-wave decay rates.

It should be noted that since the amplitude for .-0 ~ E0), involves the y0 magnetic moment, a measurement of the s-wave decay rate will allow an experimental determination of the y0 magnetic moment.

Why did the higher dimension operators not get induced for the s-waves? Since the pion-baryon coupling involves a ~/5, the loop graphs which give rise to s-wave contributions are suppressed. Hence the higher dimension operators are only a small correction to the s-wave decay rates.

How accurate do we expect our results to be? We have been ignoring contributions suppressed by powers of ( M 1 - M 2 ) / A × s B, so our results should be accurate to about 20%. If the experimental values of the baryon magnetic moments change, then the rates here would also be altered significantly. Note that the decay rates depend on the difference of moments, not just the moments themselves.

If we had a detailed model of the matching condition, we could say more about the coefficients of the four dimension-five operators responsible for the p-wave decays. For the moment we must content ourselves with the less ambitious result that these coefficients are of the right order of magnitude.

This research is supported in part by the National Science Foundation under Grant Num- ber PHY-82-15249. I would like to thank H. Georgi for useful discussions and E. Wang for Chicken Curry.

References

[1] See, e g F E Close and H Rubenstem, Nucl Phys B173 (1980) 477; M B Gavela et al , Phys. Lett 161B (1981) 417, and references thereto; see also M B Wise, Ph D Thesxs, Stanford Umvers]ty (1980)

[2] A Manz, Phys. Lett. 96B (1980) 217; T. Yeh, Phys Rev D10 (1974) 3545, E.C Booth et al, Brookhaven National Laboratory proposal (1985).

[3] H GeorgL Weak mteracUons and modern particle theory (Benjarmn/Cummmgs, Menlo Park, 1984)

[4] A Manohar and H Georgi, Nucl Phys B234 (1984) 189 [5] M B Wise, Ph D Thesis, Stanford Umverslty (1980) [6] A. Manohar and H Georgl, Nucl Phys B234 (1984) 189;

J Donoghue et al , Phys Rev D30 (1984) 587

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large radiative hyperon decays

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