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10 October 1996

Physics Letters B 387 (1996) 195-198

PHYSICS LETTERS B

On the determination of Am and Al? in tagged Do/Do decays

W. Grimusa, L. Lavoura b a Institutfiir Theoretische Physik, Universitiit Wien Boltunanngasse 5, A-1090 Wien, Austria

b Universidade T&nica de Lisboa CFIF Institato Superior T&nico, Edifcio Cit?ncia (fisica) P-1096 Lisboa Codex, Portugal

Received 19 April 1996; revised manuscript received 20 May 1996

Editor: R. Gatto

Abstract

We consider the time dependence of the decays of tagged Do and I)’ to arbitrary decay modes. We expand each decay width as exp(-It) times a series in It, where I is the average decay width of the mass eigenstates DH and DL, and examine the first three terms of the series. We show that experimental information on the coefficients of these terms allows in principle to compute the mixing parameters Am, AI and q = /q/p/, i.e., the mass and decay-width differences of the mass eigenstates, and CP violation in mixing. In particular, we discuss the possibility of extracting these parameters if one disposes of the measurement of the time-dependent decay widths of Do(t) /D,“( t) to a single final state f and make a comparison with the situation when information about two different final states f and f’ is available.

In this paper we consider the system of the charmed neutral mesons Do and Do. The mass eigenstates of

this system are given by

I&) = PIDO) + 41~“>Y

I&) = PIDO) - 41~“). (1)

where the index H refers to heavy and L to light. We denote by l? = ( IH + I,) /2 the average decay width of DH and DL. The standard model (SM) predicts very small parameters [ 1 ]

mH-mL X=

rH-FL

r and y= ~ 2r .

The parameter x is positive by definition. Experimen- tally, it is already known that (x2 + y*) 5 lo-* [ 21. Therefore, even if physics beyond the SM is very im- portant in the Do-Do system, at most the onset of oscillations can be discovered [ 3,4]. This fact is ex- ploited in discussions of tagged Do/Do decays with

decay-time information, where in the time-dependent decay widths one performs an expansion [ 3-61 with respect to the quantity (X - iy)It, which is small as long as It is of order one. It is reasonable to truncate the expansion at order (It) *. There is some hope that in future experiments the coefficients of such expan- sions will be measured. In this paper we propose to use these coefficients to get information on x and y and on CP violation. We show that if one measures them up to order (I?)* in the widths l?(DO(t)/Do(t) --f f)

to a given final state f, one can unambiguously extract X, y, and information on CP violation in mixing. We also discuss the additional information to be gained if we would be able to measure these coefficients for two different final states f and f’. Finally, we con- sider how our methods would be affected if CP were conserved in the decays and in Do--Do mixing.

To study phenomenologically the time dependence of the widths of tagged Do and Do decays into an arbitrary final state f two amplitudes are relevant:

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196 W. Grimus, L. Lavoura/Physics Letters B 387 (1996) 195-198

d(D" + f) E A, d(D” --) f) = B. (3)

As mentioned above the starting point of our analysis is an expansion of the time-dependent Do/Do decay widths with respect to (X - iy ) I?. The first three terms in this expansion are given by

(-) T(D” (t) -+ f) = ewrr[‘a) (f)+ (0’ (f) rt

+ (c) (f) (rt>* + . . .]

with the coefficients

(4)

a(f) = IAl*, b(f) = Im[ (x - iy) 4A*B], P

zt(f) = IB12> 6(f) =I~[(x--iy)gAB*], 9

c(f) = :[r1*1Bl*(~~ + r2> - IA12b2 - r*)l,

C(f) = $$AI*(x~ + y*> - IB12k2 - y’>l, (5)

where 7 = lq/pI parametrizes T and CP violation in Do-Do mixing (r] = 1 if CP is conserved). We assume that I and the coefficients a(f), b(f), c(f), Z(f), 6(f) and C(f) can be obtained from a fit of Eq. (4) to the Do/Do decays with decay-time information. In the following we shah leave out the label f in the

coefficients. The expansion in Eq. (4) supplies us with six exper-

imental quantities. This number has to be confronted

with the number of independent physical quantities constructed from A, B and q/p, which is four [7]: q, IAl, IBJ, and the phase of (q/p)A*B. Therefore, at least in principle, one can obtain x and y from such measurements. Indeed, in a first step we find

X2=2172F-c $F+C

a-rf*ii’ y2=2

a+q*Cz’

Clearly, v cannot be obtained from the coefficients ‘a’

and (cl alone. On the other hand, from b and 6 we get

$(b-~?g)*+$(b+~*b)3=4a~_r12. (7)

This is the equation of an ellipse in the variables l/x and l/y. Taking Eqs. (6) and (7) together we can compute

174 = t* (ac - a?) + 2bbac - 4ak2

b*( aZ - ac) + 2bh - 4aS ’ (8)

and then from Eqs. (6) the values of x and y can be obtained.

We stress that the determination of 7 must precede the one of x and y. The mixing parameters can only be extracted from the expansion coefficients via Eqs. (6) when we already know the mixing parameter 77, for instance from Eq. (8).

It is important to note that the determination of q depends not only on the observation of the shape of the decay curves of Do(t) and B’(t) , but also on the relative normalization of those decay curves. This means that it is not just the ratios c(f) : b(f) : a(f) and F(f) : 6(f) : ?I( f) which are important; for the computation of q the relative normalization a(f) : i?(f) is fundamental. This is a general point concern- ing the observation of T violation in the mixing of any neutral-meson system [S]. On the other hand, it is easy to check by inserting Eq. (8) into Eqs. (6) that in principle the knowledge of this relative normaliza- tion is not necessary for the determination of x and y.

We next want to consider what happens when we are able to measure the time dependence of the widths of tagged Do and Iso decays into two different decay modes f and f’. A typical example would be f’ = f. In this case, besides the amplitudes in Eq. (3)) we

have

A( Do --f f’) = A’, d(D" --+ f') E B’. (9)

The total number of parameters to be determined is now nine: besides /Ai, IBl, arg[(q/p)A*B], x, y and 77, we also have IA’I, IB’I and arg[ (q/p)A’*B’]. On the other hand, we dispose of twelve experimentally measured quantities. Therefore, we expect three con- straint equations; indeed, these come from the fact that the values of 77, x and y extracted from the decay widths to f must equal the ones extracted from the de- cay widths to f’. In particular, analogous to Eqs. (6) and (7) we have

x2 = 2 r12~(f> -c(f) = 2 dew’) - c(f’)

a(f) - r1*w> 4.79 - $a(.?> ’

Y2 = 2 11*ct.f> + c(f) = 2 qwf’) + c(f’)

a(f) + q2Z(f> 4.f’) + v%(f’) ’ (10)

W. Grimus, L. Lauoura/Physics Letters B 387 (1996) 195-198 197

and

$ [b(f) - 11zg(f)]2 + $ [b(f) +772i;(f)12

f [Mf’) - r12m’)]2 + Jy [b(f’) + 7ja(f’)]2

= 4a(f’)ii(f’)r12, (11)

respectively. Eqs. ( 10) lead to the consistency condi- tion

c(f)n(f’) -tE(f)a(f’) = C(f’)a(f) +c(f%(f).

(12)

Moreover, q can be extracted by

df’>4f> - c(fMf’> v4 = C(f)cl(f’) - C(f’)ii(f) .

(13)

This way of extracting v looks simpler than the one provided by Eq. (8). Once again, we note that the de- termination of 7 depends on the relative normalization of the decay curves of Do(t) and Do(t) .

From the system of Eqs. ( 11) it is also possible to extract x and 1 y I. This method for the determination of the mass and decay-width differences only works, however, if 71 is already known. This could be from Eq. ( 13) or Eq. (8)) for instance. Also, if we combine Eqs. (11) with Eqs. (10) we get two further consis- tency conditions analogous to Eq. ( 12).

Let us now consider the impact of CP conservation on our methods for the determination of x and y. If one has at one’s disposal a single final state f which is not a CP eigenstate, or two final states f and f’ which are not CP-conjugate to each other, then our previous analyses go through without change. Only the parameter 7 takes the special value 7 = 1. Now, from Eq. (8) this means that in this case there is a

constraint among the coefficients (a’ (f), (b) (f)

and (cl (f) for any decay mode f. This constraint reads

4U@&C2) = @2+&2)(a?-ac) +2b6(Z-uc).

(14)

This means that we can test CP conservation in mixing using the time dependence of the decays to any single

decay mode f, even if this decay mode is not a CP eigenstate.

On the other hand, if the time dependence of the decays of Do/Do to a CP eigenstate f is measured, CP conservation implies not only

a = a, b=6, c=E, (15)

but also

lb] = lylu and c= ;y2u,

and therefore

(16)

2ac = b2. (17)

For convenience we have omitted the label f. In this

case the expressions for x and 7 in Eqs. (6) and (8) are of the type O/O and thus undefined. This means that, if CP violation in mixing and amplitudes is small, it will be difficult to constrain x and 7 from the de- cays of tagged Do/Do into CP eigenstates. However, Eq. (16) shows that CP eigenstates are well suited to get information on y [ 41.

Finally we consider the case in which the time de- pendence of the decays of Do/Do to two CP-conjugate states f and f is measured, while CP is conserved. We make the identification f’ = f. The conditions of

CP invariance are

a(f) = a(f’)* ii(f) = a(f’),

b(f) = b(f), b(f) = b(f’),

c(f) = E(f’), E(f) = c(f’). (18)

Then the two ellipses in Eqs. ( 11) coincide and x and y cannot be disentangled in those equations. There- fore, Eqs. ( 11) constitute another version of the ob- servation that large CP violation can be quite helpful in getting a hold on mixing in the Do-Do system [ 61. Nevertheless, even in the case of CP conservation one could have an interesting restriction in the x-y plane, and consequently lower bounds on x and y.

On the other hand, the method of determining x and

y by using (cl (f) and (c’ (f) (see Eqs. ( 10) and ( 13)) does not suffer from the above problem, i.e., x and y can be determined even when CP is conserved. In this case, the two ways (see Eqs. ( 10) ) of obtaining x and y via the final states f and f’ G f coincide (7 must be set equal to 1) . Using CP non-eigenstates one

198 W. Grimus, L. Lauoura/Physics Letters B 387 (1996) 195-198

does not face the problem of undefined expressions in the case of CP conservation. The denominators in Eqs. (10) and (13) are different from zero.

If CP is conserved the constraint of Eq. (12) is automatically satisfied because of the equalities in

Es. (18). In conclusion, we have studied the conditions under

which we can extract the mixing parameters x, y and 7 = /q/p 1 of the Do-Do system from the measurement of the first three terms in an expansion in (x - iy) Tt of the decay widths of tagged Do(t) and D’(t) . We have shown that in general this task can be performed if one considers the decays into an arbitrary final state f, except if CP is conserved and f is a CP eigenstate. If one uses the coefficients of (Tt)’ and ( Tt)2 one can compute x and y, provided the parameter q mea- suring CP violation in Do-Do mixing is determined from elsewhere. If information on these coefficients is available for two decay modes, then this is sufficient to determine 17 as well and, in addition, a consistency condition among the coefficients has to be fulfilled. Experimental knowledge of q and of the coefficients of (lYt)O and (I?) 1 for two final states f and f’ al- lows the extraction of x and y; however, if f’ = f, x can be obtained with this method only if CP is vio- lated. In any case, in the framework discussed here, the determination of ‘17 seems to be the most difficult experimental problem. It also has priority, because x and y can only be extracted once q is known. Need- less to say, all the different methods we put forward to get information on Do-Do mixing can be combined in various ways. The experimental situation will pin down the most useful strategy.

One of us (L.L.) wants to thank the University of Granada for its warm hospitality and the opportunity to give there a seminar, which lead to the development of some ideas contained in this letter.

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