HUTP-97/A021

Inclusive Decay Distributions of Coherent Two-body States

Hitoshi Yamamoto

Dept. of Physics, Harvard University, 42 Oxford St., Cambridge, MA 02138, U.S.A. (June 11, 1997)

Abstract

When a vector meson such as ¦Õ, J/¦·(3S ) or ¦´(4S ) decays to a particleantiparticle pair of neutral mesons, the time distribution of inclusive decay to a given ?nal state is naively expected to be the incoherent sum of those of the two mesons with opposite ?avors. In this paper, we show that this is in general not the case for arbitrary coherent two-body states of the mesons, and obtain conditions under which such a naive incoherent sum gives the correct distributions. The analysis is based on the Weisskopf-Wigner formalism, and applicable to the cases where there are more than two orthogonal states that can mix to form a set of eigenstates of mass and decay rate. 03.65.-w, 13.20.Gd, 13.25.He

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I. INTRODUCTION

? 0 [1], In the studies of CP violation in neutral meson systems such as ¦Õ ¡ú K 0 K ? 0 , or ¦´(4S ) ¡ú B 0 B ? 0 [2], one often deals with an inclusive decay disJ/¦·(3S ) ¡ú D0 D tribution where one of the mesons decay to a given ?nal state f at a given time t and the other meson can decay to any ?nal state at any time. It has been shown in a recent study that in the decay of ¦´(4S ), the inclusive decay time distribution of ¦´(4S ) ¡ú f is the incoherent sum of the decay time distribution of a pure B 0 at t = 0 decaying to f at time t (denoted ¦£B0 ¡úf (t)) and that of B (denoted ¦£B ? 0 ¡úf (t)) [3]. Such relation is critical in analyses of inclusive lepton asymmetries [3], or in relating observed inclusive yield asymmetries ? to the asymmetry of decay amplitudes Amp(B ? 0 ¡ú f ) and of ¦´(4S ) ¡ú f and ¦´(4S ) ¡ú f ?) [4]. When a ¦´(4S ) decays to a pair of neutral B mesons, it is generated in Amp(B 0 ¡ú f a coherent L = 1 state, which is antisymmetric under the exchange of the two mesons: 1 0 0 ¡Ì |B (k ) |B 0 (?k ) ? |B 0 (k ) |B (?k ) 2 , (1.1)

0

where the mesons are labeled by their momentum ¡Àk which we will drop hereafter and implicitly assume that left (right) side of the meson pair is in +k (?k ) direction. A natural question is then whether such an incoherent sum gives the correct inclusive distribution for a general two-body state given by ¦· = aB 0 B 0 + bB 0 B + cB B 0 + dB B , where a, b, c, d are arbitrary complex coe?cients with |a|2 + |b|2 + |c|2 + |d|2 = 1 . When the pair is generated with a de?nite orbital angular momentum, further symmetry relations must be satis?ed; in this study, however, we will keep the general form as above. As we will show below, the necessary and su?cient condition for the naive incoherent sum (2|a|2 + |b|2 + |c|2 ) ¦£B0 ¡úf (t) + (2|d|2 + |b|2 + |c|2 ) ¦£B ? 0 ¡úf (t) 2 (1.3)

0 0 0 0

(1.2)

to give the correct distribution for any ?nal state f (and independent of the details of the mixing) is G ¡Ô a? (b + c) + d(b + c)? = 0 .

def

(1.4)

II. GENERAL COHERENT TWO-BODY STATES

In the following, we study a system of n orthogonal states Bi (i = 1, . . . , n) mixing to form n eigenstates of mass and decay rate (physical states) B¦Á (¦Á = a, b, c . . .). The eigenstates B¦Á are not necessarily orthogonal when CP is violated. We use the WeisskopfWigner formalism [5], but no assumptions are made on CP or CP T symmetries unless otherwise stated. The essential approximation used in the formalism is that the oscillations caused by mass di?erences and the decay rates are su?ciently slower than the time scale of decay transitions, which is a very good assumption for the cases under study [6]. The eigenstates B¦Á then evolve as B¦Á ¡ú e¦Á (t)B¦Á , e¦Á (t) ¡Ô e?(¦Ã¦Á /2+im¦Á )t ,

def

(2.1)

where ¦Ã¦Á and m¦Á are the decay rate and mass of the physical state B¦Á . The time t is the proper time of the particle under consideration. The Weisskopf-Wigner formalism can be relativistically extended to moving particles; it can be shown, however, that it is equivalent to the evolution in the rest frame formulated as above [7]. We will hereafter consistently use the indices i, j for the orthogonal states B1 , B2 . . ., and Greek indexes ¦Á, ¦Â for the physical states Ba , Bb . . . : Orthogoanl states: i, j, i , j = 1, 2 . . . n, Physical states: ¦Á, ¦Â, ¦Á , ¦Â = a, b . . . (n total) . The eigenstates Bi can be written as linear combination of B¦Á ¡¯s: Bi =

¦Á

ri¦Á B¦Á . 3

(2.2)

? 0 , we have n = 2: For the system composed of B 0 and B B1 = B 0 , and the physical states are usually written as

? ? ? ? Ba ? ? ?

?0 , B2 = B

= pB 0 + qB

0

0 0

Bb = p B ? q B

,

(2.3)

? 0, or solving for B 0 and B

? ? ? ? ? ? ?

B 0 = c (q Ba + qBb ) B = c (p Ba ? pBb )

0

,

with c ¡Ô namely, r1a = cq , r1b = cq , r2a = cp , r2b = ?cp ,

def

1 ; p q + pq

(2.4)

Returning to the general case of n orthogonal states, the orthonormality of Bi ¡¯s can be expressed in terms of the physical eigenstates B¦Á as ¦Äi j = Bi |Bj =

¦Á¦Â ? ri¦Á rj¦Â B¦Á |B¦Â ,

(2.5)

where we have used (2.2). The decay amplitude of a pure Bi state at t = 0 decaying to a ?nal state f at time t is, from (2.2) and (2.1), ABi ¡úf (t) =

¦Á

ri¦Á a¦Áf e¦Á (t) ,

where a¦Áf is the amplitude of B¦Á decaying to f : a¦Áf ¡Ô Amp(B¦Á ¡ú f ) . The normalization is such that |a¦Áf |2 is the partial decay rate of B¦Á to f : 4

def

|a¦Áf |2 = ¦Ã¦Á .

f

(2.6)

Namely, the density of the ?nal states, more precisely the square root of it, is absorbed into the de?nition of the amplitude. The time dependent decay amplitudes ABi ¡úf (t) satisfy the following orthonormality relation [10], where the ¡®inner product¡¯ of ABi ¡úf and ABj ¡úf is de?ned by integration of A? Bi ¡úf ABj ¡úf over time followed by summation over all possible ?nal states:

¡Þ f 0

dt A? Bi ¡úf (t)ABj ¡úf (t) =

¦Á¦Â ? ri¦Á rj¦Â f ? ri¦Á rj¦Â ¦Á¦Â f ¦Ã¦Á +¦Ã¦Â 2

a? ¦Áf a¦Âf

¡Þ 0

dt e? ¦Á (t)e¦Â (t) (2.7)

=

a? ¦Áf a¦Âf

? i(m¦Á ? m¦Â )

= ¦Äi j . In deriving the above, where we have used the generalized Bell-Steinberger relation [8] given by

f ¦Ã¦Á +¦Ã¦Â 2

a? ¦Áf a¦Âf

? i(m¦Á ? m¦Â )

= B¦Á |B¦Â ,

(2.8)

together with the orthonormality of Bi ¡¯s (2.5). Note that the relation (2.8) reduces to the amplitude normalization condition (2.6) for ¦Á = ¦Â . While the Bell-Steinberger relation can be derived by requiring that unitarity is satis?ed [8], it can also be derived from the old-fashioned perturbation theory to the lowest non-trivial order [9]. The probability that a pure Bi at t = 0 decays to a ?nal state f at time t is simply the square of the time-dependent amplitude: ¦£Bi ¡úf (t) = |ABi ¡úf (t)|2 . The relation (2.7) with i = j shows that this decay distribution conserves probability:

¡Þ f 0

dt ¦£Bi ¡úf (t) = 1 .

Now, take a general coherent two-body state at t = 0 given by 5

¦·(t = 0) =

ij

ci j Bi Bj ,

where ci j are arbitrary complex coe?cients with |ci j |2 = 1 .

ij

For simplicity, we will hereafter label the left and right sides of the particle pair as north (N) and south (S). The speci?c names to distinguish the two sides are irrelevant; we just need some labels for the two orthogonal spaces. The probability that north side decays to a ?nal state fN at time tN and the south side to a ?nal state fS at time tS is then ¦£¦·¡úfN fS (tN , tS ) =

ij

ci j ABi ¡úfN (tN )ABj ¡úfS (tS ) .

2

(2.9)

From the orthonormality of the decay amplitude (2.7), one sees that this double-time decay distribution also conserves probability; namely, when integrated over the two decay times and summed over all possible ?nal states, it becomes unity:

¡Þ fN fS 0

dtN

¡Þ 0

dtS ¦£¦·¡úfN fS (tN , tS ) = 1 .

(2.10)

We now de?ne the inclusive decay distribution of ¦· to a ?nal state f , where f can come from either side of the decay: ¦£¦·¡úf (t) ¡Ô

def fN 0 ¡Þ 0 ¡Þ

dtN ¦£¦·¡úfN f (tN , t) (2.11) dtS ¦£¦·¡úf fS (t, tS ) ,

+

fS

which, due to (2.10), satis?es

¡Þ f 0

dt ¦£¦·¡úf (t) = 2 .

The number 2 comes from the fact that the ?nal state f can come from either side of the decay. The question is under what condition this is equal to the naive incoherent sum ¦£naive ¦·¡úf (t) ¡Ô =

ij def ij

|ci j |2 ¦£Bi ¡úf (t) + ¦£Bj ¡úf (t) |ci j |2 + |cj i |2 ¦£Bi ¡úf (t) , 6

which is the generalization of (1.3). Using the expression of the double-time distribution (2.9) and the orthonormality of the decay amplitude (2.7), the inclusive decay distribution (2.11) becomes ¦£¦·¡úf (t) =

ii j ? ? (c? i j ci j + cji cji )ABi ¡úf (t)ABi ¡úf (t)

= +

¦£naive ¦·¡úf (t)

? ? (c? i j ci j + cji cji )ABi ¡úf (t)ABi ¡úf (t) . i=i j

The necessary and su?cient condition for this to be equal to ¦£naive ¦·¡úf (t) is then Gii A? Bi ¡úf (t)ABi ¡úf (t) = 0 .

i=i

(2.12)

with Gii ¡Ô

def j ? (c? i j ci j + cji cji ) .

(2.13)

A su?cient condition for (2.12) to be satis?ed is clearly Gii = 0 (for all i = i ). The matrix Gii is ¡®hermitian¡¯ in the sense that Gii = G? ii , which guarantees that ¦£¦·¡úf (t) is a real quantity. Note also that the norm of Gii is re-phase invariant; namely, when the phase of Bi ¡¯s are re-de?ned, Gii simply changes its phase: Bi ¡ú Bi ei¦Õi ?¡ú Gii ¡ú Gii ei(¦Õi ?¦Õi ) . (2.14)

Thus, the condition (2.14) is re-phase invariant. In the case of n = 2, the condition (2.14) becomes Eq. (1.4) with G = G12 . In this case, it is straightforward to show that the condition G = 0 is the necessary as well as su?cient condition as long as ¦Ãa = ¦Ãb , ma = mb , and coe?cients p, q, p , q are all non-zero. We still require that ¦£naive ¦·¡úf (t) is correct independent of decay amplitudes a¦Áf . The proof for general 7

case is given in the appendix. The derivation is simple if CP T is conserved in the mixing; namely, p = p and q = q . Then the condition (2.12) becomes (Gpq ? ) (e?¦Ãa t |aaf |2 ? e?¦Ãb t |abf |2 ) +2 (Gpq ?) with ¦Ã+ ¡Ô ¦Ãa + ¦Ãb , 2 ¦Äm ¡Ô ma ? mb . e?(¦Ã+ ?i¦Äm)t a? af abf = 0 ,

(2.15)

The three terms in (2.15) have di?erent time dependences, and thus each term should be separately zero. When ¦Ãa = ¦Ãb , this condition is equivalent to simply Gpq ? = 0 (CP T ) . (Gpq ? ) = (Gpq ? ) = 0, or

Thus, if both p and q are non-zero (i.e. there is a mixing), G must be zero in order for the naive incoherent sum to be correct. Let¡¯s brie?y appreciate the meaning of the condition G = 0. This is satis?ed by any ? 0 and B ? 0 B 0 since then we have a = d = 0. It includes the ¦´(4S ) case coherent state of B 0 B ? 0 state with an even orbital angular momentum. Also, the condition given by (1.1), or a B 0 B is satis?ed if b + c = 0 regardless of the values of a and d. However, the condition is not satis?ed, for example, by the symmetric state B 0B 0 + B 0B + B B 0 + B B . Such a state cannot be readily produced in practice, but in principle it is possible if there exists an interaction with ?B = 2, such as the hypothetical superweak interaction. To summarize, we have studied inclusive decay time distributions of coherent two-body states. We ?nd that there is a set of orthonormality relations among decay time distributions of states that are pure and orthogonal to each other at t = 0. Using this, we have shown that the naive incoherent sum of single particle decay time distributions does not always give the correct inclusive distribution, and extracted conditions for it to be the case. Such 8

0 0 0 0

? 0 state regardless of the orbital angular incoherent sum was found to be correct for any B 0 B momentum.

ACKNOWLEDGMENTS

I would like to thank Y. Azimov, R. Briere, I. Dunietz, S. Glashow, S. Pakvasa, for useful discussions, and R. Madrak for reading the manuscript. This work was supported by the Department of Energy Grant DE-FG02-91ER40654.

APPENDIX

For n = 2, the condition (2.12) reduces to (Gq ? p ) e?¦Ãa t |aaf |2 ? (Gq ? p) e?¦Ãb t |abf |2 + (G p q ? Gq

? ? ?

p) e?(¦Ã+ ?i¦Äm)t a? af abf

(A1)

=0.

Since the three terms have di?erent time dependences, each term should separately be zero. The decay amplitudes are in general non-zero (we are requiring that the naive inclusive distribution be correct independent of decay amplitudes); thus, the above condition leads to

? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

(Gq ? p ) = 0 (Gq ? p) = 0 , G? p ? q ? Gq ? p = 0 (A2)

where the last condition is due to the fact that the term ei¦Ämt samples all possible phases. Now we de?ne G ¡Ô |G|eig ,

def

s ¡Ô qeig ,

def

s ¡Ô q eig .

def

(A3)

Substituting this in (A4), and dividing each equation by |s|2 , one obtains

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

|G|

|G|

p s p |G| s ? p p ? s s 9

=0 =0 ; =0 (A4)

namely,

? ? ? ? ?

|G| = 0,

or

p p , : pure imaginary, and s s? p ? p ? ? = ? s s

Thus, if |G| = 0,this leads to p p = , ?q q (A5)

which means that Ba and Bb are same physical states and it contradicts our assumption that they have di?erent decay rates and masses. Thus, G = 0 results from (A1). On the other hand, the condition G = 0 trivially leads to (A1); thus, G = 0 is the necessary and su?cient condition for the naive inclusive distribution to be correct assuming that ¦Ãa = ¦Ãb , ? 0 mix). ma = mb and p, q, p , q are non-zero (namely, B 0 and B

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REFERENCES

[1] See for example, C. D. Buchanan, R. Cousins, C. Dib, R.D. Peccei, and J. Quackenbush, Phys. Rev. D45, 4088 (1992); M. Hayakawa and A.I. Sanda, Phys. Rev. D48, 1150 (1993). [2] Y. Nir and H.R. Quinn, in ¡®B Decays¡¯, 2nd ed., Ed. S. Stone, World Scienti?c, 1994, and references therein. [3] H. Yamamoto, hep-ph/9703336, to be published in Phys. Lett. B. ? 0 ¡ú K + ¦Ð ? ) = 0, [4] For example, if one assumes that Amp(B 0 ¡ú K ? ¦Ð + ) = Amp(B ? 0 mixing or CP violation, one can show that the inclusive yield then in spite of B 0 -B ratio measured on ¦´(4S ) is related to the amplitude ratio by N (K ? ¦Ð + )/N (K +¦Ð ? ) = ? 0 ¡ú K ? ¦Ð + )/Amp(B 0 ¡ú K + ¦Ð ? )|2 . |Amp(B [5] V.F. Weisskopf and E.P. Wigner, Z. Phys. 63, 54 (1930); ibid., 65, 18 (1930). [6] Typical deviations in decay distributions from the Weisskopf-Wigner approximation is of order ¦Äm/m 1. For more discussions, see for example, P.K. Kabir and A. Pilafsis,

Phys. Rev. A53, 66 (1996), and references therein. [7] J.S. Bell, Theory of Weak Interactions in Les Houches Summer School on Theoretical Physics, Les Houches, France, 1965, ed. C. de Witt and M. Jacob. [8] The relation can readily be obtained from

f

|Amp(¦Õ ¡ú f )|2 = ?d/dt ¦Õ|¦Õ , where ¦Õ

is an arbitrary linear combination of Bi . J.S. Bell and J. Steinberger, in Proceedings of the Oxford International Conference on Elementary Particles, 1965. [9] T.D. Lee, Particle Physics and Introduction to Field Theory, Harwood Academic Publishers, 1981 [10] I would like to thank Y. Azimov for pointing out this interpretation.

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