Volume 131B, number 4,5,6 PHYSICS LETTERS 17 November 1983

TESTING LORENTZ INVARIANCE IN BARYON DECAY

R. HUERTA ’

Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA

and

J.L. LUCIO M. Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA

and Centro de Inuestigation y Estudios Avanzados I.P.N. Apdo. Postal 14-740, Mexico DF 07000, Mexico

Received 27 May 1983

We show that, in contrast with p decay, measurements of the baryon lifetime and semileptonic branching ratios for

AB = 0 and AS = 1 decays at different energies lead to a test to Lorentz invariance.

Recently Nielsen and Picek [l] proposed a way of introducing Lorentz noninvariant inter- actions (LNI) in gauge theories. Using the extended (to include LNI) standard model [2] the authors of ref. [l] showed that p decay is almost insensitive to the LNI t1 and therefore the existing tests of Lorentz invariance based on measurements of muon lifetime in flight are not very reliable. On the other hand meson decays are sensitive to LNI and have been considered by Nielsen and Picek as the most promising place to test Lorentz invariance.

In this letter we report on the influence of LNI in baryon decay. The relevance of such an alternative for typical Fermilab energies has been the motiwtion for this research.

Following Nielsen and Picek [l] we start with the effective hamiltonian

H,, = (G/d/z)&“” + X”“)[J;(x)J,(x) + h.c.] , (1)

where

Xfiu = 0, if p f u, ~00 = Ly, Xii = i(y (2)

and so cx characterizes the strength of the LNI.

’ Present address: Dept. de Fisica, CINVESTAV Apdo Post.

14-740 Mexico DF 07000, Mexico.

” Terms of order (m,lm,)2 as well as O(a*) have been neglected in ref. [l].

Let us now consider two different decay modes.

(a) Semileptonic baryon decay B -+ B’G,. In this case we write for the hadronic current

(B’IJ, IB> = u,,(gv + g,ys)r, Ug . (3)

We could consider a more general expression in eq. (3) however this will not modify our basic conclusions *2. Using eqs. (l)-(3) it is straight- forward to derive the modification of the partial decay rate due to the LNI t3

A wr,, = (G2/72~3)(~ly)(y2 - $(g$ - g?,)m&,

(44

10 = 5[(1 - p- $(l- 5’)3+ a(1 - 5”)(1 - .$2)2

+4t210g [I 3 (4b)

with (Y defined in eq. (2) y = EB/mB and 5 = mBt/ms. It is clear that AWrNi vanishes if lgvj = lgAj independently of mBs. On the other hand if lgv] f /g,/ then A W,,, only vanishes if mBt = 0. Therefore we conclude that the selection rule [l] stating that the decay rate for p type decays

” The contribution of the remaining form factors only modify the 0(S2) terms. See eq. (5).

u In this equation we have neglected the masses of the

leptons in the final state as well as terms of order a2(y2 -

a,.

0 031-9163/83/0000-0000/$03.00 @ 1983 North-Holland 471

Volume 131B, number 4,5,6 PHYSICS LETTERS 17 November 1983

is not sensitive to LNI, is only valid if lgvl f IgA( or if it is possible to neglect the masses of the particles in the final state.

An approximated expression for the modified standard partial decay rate WO [3], due to the LNI, is obtained by expanding in powers of S=A/CwhereA=ma-mB,andC=mB+mrr

W = (W”/Y)[I + C1cy(Y2 - $1 + O(S2) ,

WO = (G2/60,rr3) A5(Z/2mrr)3(gZ + 3g%), (5)

G = %g’v - gsJ/k:, + %A) . (6)

The importance of the LNI term depends both on the value of cx and the energies at which the experiment is done.

(b) Weak nonleptonic decays B + B’TT. The

first step in describing weak nonleptonic decays is to incorporate the strong interaction cor- rections into the weak hamiltonian eq. (1). Using the standard procedure [4] we get f4

JCff = - 2 $, (akg,, + bk&) G?? , (7)

where

0:” = f dr,(l - Ys)UUY,(I - 75)s

- d$Aay,(l- ys)uuthay,(l - 75)s)

o:, = 3 dr, (I - Y&m?%0 - Ys)S

+ dthay,(l- ys)uU tAarY(l - ys)s ,

ak = [%(~2)/~s(EI.2)]dk ,

bk = [%@f’>/%(P’>~“” ,

with d, = - ;, d2 = +, di = &, di = - &.

(8)

The second step is the evaluation of the matrix elements of this hamiltonian. In the case of S wave parity violating process there are two main contributions: (1) Factorization term which can be written

(B’+%@) = - 2 c (akgwv f b!&) k

x(B’IJ”IB)(dJ”~O) , (9)

(B, - Bo)/Bo = (c2 - c&v:Bo . (13)

The Fermilab charged hyperon beam will run at energies around 300 GeV. It is reasonable to expect measurements of the lifetime and semileptonic branching ratio to about 5% [7]. This will imply a limit on (Y - lo-’ to lo-’ t5.

In conclusion we have shown that the semileptonic baryon decay is sensitive to the

LNI. On the other hand we have argued that

M We are neglecting SU(4) symmetry breaking. A study of ” We are assuming Q - 1. This is not unreasonable. Fur-

such effects as well as the evaluation of the matrix ele- thermore as we increase cl we obtain a smaller value for

ments in the Ko-%I system is given in ref. [5]. a.

472

(2) soft pion contribution. In this case the matrix element (B’rr]H&lB) can be expressed in terms of the matrix elements of the effective hamiltonian between baryonic states

(B’?r]&(B) - (B’~O~,IB)(ak~P” + bkXPV) . w

Instead of considering a particular model to evaluate the right hand part of eq. (10) we parametrize the partial width in a similar fashion as the obtained in the semileptonic case

W(B + B’n) = W”(B + B’r)[ 1 + C*(Y (r2 - a)] .

(11)

It is clear that eq. (9) gives a nonvanishing contribution to c2 in eq. (11). So in writing eq. (11) we assume that the soft pion term gives a similar result. It is worth noticing that although the contribution proportional to g@” in eq. (lo), when evaluated in the bag model, is of opposite sign that the similar term in eq. (9) the latter

represents only 15 to 20% of the total amplitude [6] and therefore we expect c2 # 0.

Let.us now consider the possibility of measuring the LNI effects above discussed in the lifetime and semileptonic branching ratio. For the sake of definiteness we will consider Z- decay. The lifetime is dominated by the non- leptonic decay C- -+ nr-. Therefore comparing the results obtained at high energy with those at low energy it is possible to set a limit on CZ(Y

(r 0 0 ly )(yl/rl - yO/rO) = c2ffy: ,

on the other hand measurements of the semileptonic branching ratio B = W(B + B’&),)/ W(B + all) restrict

(12)

Volume 131B. number 4,5,6 PHYSICS LETTERS 17 November 1983

nonleptonic baryon decay will also be sensitive to the LNI but we are unable to make a definite prediction for c2 [see eq. (ll)]. The consequence of this is that either the semileptonic branching ratio and/or the baryon lifetime as measured at high energies could be different from the values obtained at low energies. Considering typical Fermilab energies and assuming that the measurements are done to about 5% we obtain the limits cy - 10m5, 1O-7 from the branching ratio and lifetime respectively. This is an im-

provement of two to three orders of magnitude on the existing limits [l,S] *6 on the LNI.

We wish to thank J.D. Bjorken for calling our attention to this problem; J. Lath for useful

conversations and the Theory Group at Fer- milab for its warm hospitality.

“‘There exist data on the K0-G that could be interpreted as

evidence of LNI [9]. Analysis based only on the energy

dependence of Amk, gives the limit cy = 3 x 10m6 [1,5].

References

[ll

PI

[31

[41

PI P51

[71 PI

[91

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S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: Elementary particle theory, ed. Svartholm

(Almqvist and Wiksell, Stockholm, 1968).

H. Pietschmann, Formulae and results in weak inter-

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M.K. Gaillard and B.W. Lee, Phys. Rev. Lett. 33 (1974)

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L. Marani and G. Altarelli, Phys. Lett. 59B (1975) 293;

M.A. Shifman, A.I. Vanishtein and V.I. Zakharov,

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J. Lath, private communication. R. Huerta and J.L. Lucia M., Fermilab-Pub-83/18-THY

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