XJ0400115

MY76

E2-2004-4

M. I. Shirokov*

DECAY LAW OF MOVING UNSTABLE PARTICLE

Submitted to International Journal of Theoretical Physics>>

*E-mail: shirokovthsunljinr.ru

1. INTRODUCTION

Experimenters showed that the lifetime of unstable particles moving with velocity v is equal to 7-0-y, where To is the lifetime of the particle at rest and - = 1 _ V2/C2)-1/2. Usual explanation of the fact is based on the special theory of relativity. For example, Moller 1972) sets forth it as follows:

"In view of the fact that an arbitrary physical system can be used as a clock, we see that any physical system which is moving relative to a system of inertia must have a slower course of development than the same system at rest. Consider for instance a radioactive process. The mean life of the radioactive substance, when moving with a velocity v, will thus be larger than the mean life o when the substance is at rest. From 2.36) we obtain immediately = - v 2 IC2) - 12, ro".

This argumentation may be complemented by the following possible defin- ition of the unit of time which radioactive substance provides: this is the time interval during which the amount of the substance decreases twice, e.g. However, the standard clocks of the relativity theory are used when obtaining Eq. 2.36) At = t2 - t = 7(t2 - t' V2/C2)-1/2 Ar which Moller mentions. He begins the derivation of this equation with the phrase:

"Consider a standard clock C' which is placed at rest in S' at a point on the x-axis with the coordinate x = x"'.I However, such a quantum clock as an unstable particle cannot be at rest (i.e., cannot have zero velocity or zero momentum) and simultaneously be at a definite point (due to the quantum uncertainty relation). So, the standard derivation of the moving clock dilation is inapplicable for the quantum clock. The related quantum-mechanical derivation must contain some reservations and corrections. Here another way of deriving the time dilation is used: to find relativistic quantum decay law of a moving unstable particle (momentum p --4 0) and to compare it with the decay law of the particle at rest. Both the laws refer to one Lorentz frame (e.g., laboratory frame). Lorentz transformations of the space-time coordinates are not needed as well as the space coordinates themselves.

I This approach to the derivation of the time dilation was used by Exner ( 983) and Stefanovich 1996). A simple derivation of the probability amplitude Ap(t) of the decay (more exactly nondecay) of the particle with momentum p is suggested in Sect. 2. Distinctions from Exner and Stefanovich derivations are discussed. Let us define the terminology used below. The equation p = T-Y is equiva- lent to the the equation jAp(t) 12 = Ao(t/-y) 12 if JA, 12 and Ao 12 depend upon t as

IA,(t)l' - exp(-t/-rp), JAo(t) 12 _ XP(_t/70).

Eq. (1) or the substitution t - t/-y used in Eq. (1) will be called Einstein dilation ED (in this respect see also Eq. 2.37) in [Moller, 1972]). Calculations of quantum probabilities JA, (t)12 and jAo (t)12 show that ED does not hold exactly. Numerical calculations were performed by Stefanovich (1996). Here analytical evaluation of Ap(t) is carried out, see Sect. 3 and Appen- dix. It allows us to determine the region of times t, where ED is approximately valid, and its accuracy. It also shows that ED fails under some conditions. In particular, I shall consider in Sect. 4 a variant of the quantum clock for which appreciable deviations from ED take place at all times. For conclusion see Sect.'5. The analytical evaluation of Ap(t) is presented in Appendix.

2. NONDECAY LAW OF MOVING UNSTABLE PARTICLE

Let us consider a relativistic theory which describes unstable particles, prod- ucts of their decay, and the corresponding interactions. A field theory may be an example. Such a theory must contain operators of total energy and momentum 1' 0 (the generators of time and space translations), total angular momentum, and generators of Lorentz boosts. Suppose that at the initial moment t = there is one unstable system (particle) having definite momentum p'. As there are no other particles p is an eigenvalue of the total momentum P. The state vector of this system is the related eigenvector which may be represented in the momentum representation as

(14Y = 6#"700- (2)

Here V50 describes the unstable system in its rest frame. For example, *0 may describe an excited state of the hydrogen atom in the atom rest frame. In Eq. 2), 6#,fi, is the Kronecker symbol: I suppose that P has a discrete spectrum (the

2 system is in a large space volume and usual periodicity conditions are imposed or the volume opposite boundaries are identified). Thus, Tp, has the unit norm. Consider a common eigenvector p,,, of and 1. Similarly to Eq. 2) one has (PI(DPI') (3) Here 0, is the common I and P eigenvector corresponding to the zero eigen- value of P. The related ft eigenvalue may be called the mass p. Let us expand IPP, in vectors fi,, TP = ACUL),Dfi", (4) Here S, denotes sum and/or integral over (it will be refined below). I suppose that c(p) in Eq. 4) does not depend on p'. It follows from Eqs. 2)-(4) that

0 = S'-cG')00"'- (5) One may consider Eq. (5) as a possible concrete definition of the vector . Let us determine the eigenvalue Ep, of ft which corresponds to the common eigenvector Dfi,, of ft and P, p 54 0. (Ep, assumes the value if p 0) In relativistic theory E'P - P" is the Lorentz invariant. From 2 /2 2 2 EP - P - E = 11 (6) the known relativistic formula Ep, follows. So we have

f14,P11 /p2 p2.p PtA (7) (the prime over p is omitted). It follows from Eqs. 4) and 7) that

lpp(t = e- p = c(p)1Dp,. exp( -itV'p2 +,u2). (8) So the probability amplitude of the nondecay (survival amplitude) is

Ap(t) (,Pp, e-' HtXPP = SAIC(tt)12 exp(-itN/p-2+p (9) The state qPP is called unstable if Ap(t - as t - oo. This property holds only if S, in Eq. 9) is integral over continual i values. Further, the spectrum of tile Hamiltonian H must be bounded from below. Let us choose the energy origin so that the lower bound in S, would be zero. Thus S, in Eq. 9) must denote the integral over from zero to, e.g., infinity. When p = Eq. (8) turns into the known equation for the probability amplitude of nondecay of the unstable system at rest, e.g., see [Fonda et al., 1978]. Eq. (8) was obtained by Stefanovich 1996) in a different way. He defined TP as Upoo where Up represents Lorentz transformation from the rest frame of unstable particle to the frame where its momentum is p. Here XPp is defined by Eq. 2) and only one relativistic formula, Eq. 6), is used.

3 3. DILATION OF MOVING UNSTABLE PARTICLE DECAY Let us calculate and compare the nondecay law Ap(t) of moving unstable particle and nonclecay law Ao(t) of the particle at rest. For this purpose jc(A)j is needed, see Eq. 9). It is possible to calculate Ic(p)1' using solvable models, e.g., see [Levy, 19591, [Alzetta, d'Ambrogio, 1966], [Gi-Chol Cho, 1993], [Horwitz, 19951. Here the known simplified representation is used (cf. [Stefanovich, 1996]) r ICUL) I' = [( - m 4] (10) 27r

In general r' andm are functions of jL, see [Messiah, 1961], [Goldberger, Watson, 19641. Here F and ?7i are supposed to be constants such that F/m I (this is true for all observable unstable systems). I hope that using Eq. (10) allows us to obtain a faithful qualitative notion on the relation of A,(t) and Ao(t). 3.1. In the Appendix the main contributions to Ap(t) and Ao(t) are calcu- [ated using Eq. (10) for all times except extremely small ones t < 1/rn.

AO () = rf d1i [(tL M)2 + 2 /41 exp(-ittt) 27r0

- -itrn.- Ft/2 tM > 1 (I ) 27r m Tnt

[( _ M2 + 2 Ap (t) = 0c' dit /41 exp(-it V1p2 + p2) 27r 0 - exp[-i VpT-+m2(1 - ,)l exp[-I't( + a/2-y] (12)

I (27r) - 12 (Pt) - 1,2 exp(-ipt + i7r/4). 2 Tn Eq. 12) is true if t > 1/m and t 1. The latter condition does not make it possible to get Eq. (I 1) from Eq. 12) by putting p = ; VP-2+ M2/M = _ V2/Cl)-1/2,

1 1,2 P2 1 T,2 ly 2 - i (13) Cf = - - 8 2 + n2 P2 + M2 8 M2 ^/4

Eqs. (11) and 12) contain terms proportional to exp(-rt/2) and exp(-rt(I + a)/2-y). Let us call them exponential terms (ET). Besides ET the equations include terms containing small factors r,/,m and proportional to the inverse powers of t. They are small as compared to ET during several decades of lifetimes. However, they dominate at sufficiently large t. For their discussion see, e.g., [Fonda et al., 1978], [Norman et al., 19881 and references therein.

4 3.2. The exponential term in Eq. 12) is close to exp(-itm-y - t/2-y) because a < 1. Therefore, JAp(t) 12 almost coincides with A(t/-Y)12 at the times t when ETs dominate. So, ED approximately holds at these times. Let us estimate corrections to the dilation. One obtains using Eqs. (I 1) and 12):

f A, (t)12 _ IAO(tl-Y)12 I/jAo(t/-y) 12 rtapy. (14) Here the difference in the curly brackets decreases when t increases because both minuend and subtrahend decrease as exp(-ft/-y). Therefore, the deviation of IA, (t) 12 from Ao (t /-Y 12 should be characterized by the ratio of this difference to jAo(t1-y) 12 . The ratio is equal to 1'ta/- and grows as t increases. However, it is extremely small even if, e.g., - because of a << 1. One cannot consider still greater times because then nonexponential terms will dominate in Eqs. (I 1) and (I 2. It follows from Eqs. (I 4 and (I 3 that the maximal deviation occurs when -y2= 53. The estimation shows that measuring deviation from ED in the exponential time region needs much more accuracy than is achieved in existing experiments (which is about 0.1 - 02 %), see [Bailey et al., 1977], [Farley, 19921. The same conclusion was obtained by Stefanovich 1996) by means of nu- merical calculations of the difference Ap(t) 12 - Ao(tl-y )12 for /Tn = 2 I - 4 and several values of t and -y. Note that my analytical estimate gives much smaller values for the difference, as compared to those presented in Fig. I of Stefanovich's paper. 3.3. ED decisively fails for asymptotically large times when nonexponential terms in IA,(t)l' and jAo(t)j' dominate. Indeed, then IA,(t)l' - t-' while Ao(t)1' - t-', see Eqs. (II) and 12). So, the former is delayed as compared to the latter but this is not ED t --- t/-y. For the sarne reason ED fails also in the transient (preasymptotical) region of times when exponential terms are of the same order as the asymptotical ones. It is appropriate to note here that the asymptotic behaviour of the nondecay amplitude is sensitive to the F(M) behaviour at small (remind that r' is here supposed to be a constant). It is likely that A(t) and Ao(t) have different asymptotic inverse power behaviours at any rji). However, when asymptotic terms begin to dominate the decay is almost completed ad we have nothing to observe. So far experimenters fail to measure nonexporiential asymptotics of the decay law, e.g., see [Norman et al., 1988 ad references therein. In the next section, I consider another variant of the quantum clock in which ED fails for all times under some condition. 3.4. Exner 1983) clairried the validity of ED for he decay law of moving unstable particles (see his section V). Let us comment on his approach. Instead of the initial state with definite momentum Exner considered a packet with almost definite momentum. In this case, the definition 9 of the

5 survival amplitude is not suitable. Indeed, then the survival amplitude changes with time not only because of the decay but also because of the packet diffusion. Besides, the corresponding amplitude of moving particles changes due to the initial packet displacement at the vector W, v being the packet average velocity (under the displacement our %Ppdoes not change up to an unessential phase factor). So, in the case one must use a more general definition of the survival law, e.g., see Eq. (3a) in [Exner, 1983]. Nevertheless, Exner used definition 9 which in his approach may be considered as a simplified approximate definition of the nondecay law. He stipulated that his claim on the ED validity holds provided this approximation is admissible. In particular, he made the reservation that the decay law is considered in the time region where the decay law is exponential. Exner did not estimate the validity of his approximation. A refinement of Exner's approach may result in ED violation, although small. This would agree with the conclusion obtained in Subsect 32: ED holds only approximately when exponential terms dominate.

4. Ko-MESON-LIKE SYSTEMS AND EINSTEIN DILATION

Consider two unstable particles K and K with different masses and life- times, i.e., different distributions ICs [1 12 and IC, tt 12, see Eq- (10). I call the particles K., and Ki because the known mesons K., and K, may be familiar examples. However, here their masses m, and ml are not supposed to be close. As usual, let us suppose that ,/m, << I and rl/ml < 1. Let K,(t) and K(t) denote the corresponding survival amplitudes of the particles at rest, see Eq. (I ) where and Tit are replaced by 17', m and F, ml, respectively. Let us suppose that at t = one may prepare the state

[jK.) + IKI)l /v' _ KO). (15)

For example, in the case of real K meson the state IKo) is the product of the reaction T- + p - K AO. Consider the survival amplitude of the state

K (t) I([IK.,) + KI)] , exp(-iHt) [K.) + KI)J) 2 1 1 -(IK,,), exp(-0-1t) IK,,)) + - (IKj),exp(-iHt)IKj)) (16) 2 2 1 1 -K, (t) + - KI (t). 2 2 1 supposed above that the vector IK,) (which describes the particle K.) as well as all vectors of the K, decay products are orthogonal to IKI) and to its decay product vectors. This supposition is valid for the real mesons K, and

6 KI because K, and KI and their decay products have different CP-parities (CP conservation holds almost exactly). Neglecting nonexponential terms in Eq. (I ) for K,(t) and Kl(t) one obtains-for the K survival probability IK(t)l'

JK(t) 12 1 e-tr, + e-'r" 2 exp(-t(r, + 'l)/2) cos(m - mf] (17) 4 cf. the Eq. 7.83) in [Perkins, 19871. Eq. 17) may be represented in the form

JK(t) 12 = exp(-t r,+ ,)/2) (COS2 tM, - ml)/2 + sinh2 tj' - rl)/4] (8) which presents more visually oscillations which K(t) 12 has. For example, when t1', < I and tr'l < we have JK(t) 12 -= S2 tM, - ml)/2. Analogously, consider the moving Ko system having nonzero definite p. The related survival amplitude Kp(t) is a superposition [Kp(t) + Kip(t)) //2, where K,,,(t) and Kp(t) are defined by equations of the kind of Eq. 12) F and n being replaced by r,, m, and F m. As above, only exponential terms are retained. The small corrections ce, and al are also neglected. Thus, p2+ in2 2 + 1112 Kp(t) _- exp it 5 e:xp(-t1',/27,), -y, _ p (19)

Klp(t) -_ exp it p2 + M2I exp(-t1'j/2-yl), -yj V/p2 + in2/in,, (20)

1Kp (t) 12 IKP(t) + KIP(t)12 2 lexp + [COS2 ( -t ( P2+ M2 F2 + M2 2 -y, -yj 2

+sinh 2 t (r,_1 FN (21) 4 -/., -/I Note. The oscillations which are present in K(t) and Kp(t) allow us to use the unstable system described by the vector Ko = K.,) + KI)j /vF2 as a quantum clock in a different way as compared with the usual unstable system (see Introduction). Namely, the unit of time provided by IK(t)l' may be defined as the period n - ml)-' of IK(t)l' oscillations, in - ml being the oscillation frequency. Analogously, the moving K determines the unit of time P-', AP in, p + , being the oscillation frequency of K,(t)l 2 e Eq. 2 Vlp se We can see from Eqs. (18) and 21) that lKp(t)12 can be obtained from IK(t)12 by the replacements

F., - Ir"PY." Fl - Am AP, (22)

Am n, - ml, AP /p2+M2 - F- (23)

7 Multiplying and dividing AP by Vp2+m2 + VrpT+7nl one obtains for AP

AP = (m 2 _ m2) [Vp2 +,n,2 2

= [Vp2 + M2 + p2 + M2 ] /M +ml). (24)

So the last replacement in 22) may be represented as Am - Am/. The replacements 22) cannot be reduced to one replacement t --4 th if -Y i Y- So ED decisively fails in this case. The inequality -y, 54 -l holds if 7n - ml (provided p is not too small: if p < m, ml then -y,, --- -j -- -- 1). ED holds approximately if m -- ml (this is the case for real K, and K). Then -y, y and 22) may be reduced to t - th.

5. CONCLUSION

Evaluations of the decay law of moving unstable particles show that Einstein time dilation t t/-y (ED) is not the exact kinematical law of relativistic quantum mechanics. The deviation from ED is small for usual unstable systems when the time t of the decay does not exceed several decades of lifetimes. The estimation of the value of the deviation is obtained in Sect. 3 see Eqs. 14) and 13) It linearly grows as t increases.(in the above region) and is maximal at -Y2 53. It imuch less than the achieved experimental accuracy. It was shown that ED does not hold decisively for larger times when the decay is nonexponential ED fails for all times in the case of unstable system of the kind K = K, + KI)l V'2 with appreciably different masses m, and ml. More explicitly, in both two letter cases we have dilations, see Subsect 33 and Eq 22) in Sect. 4 bt the dilations are not Einsteinian. However, experimenters fail to observe nonexponential decay while K-meson-like system with n, ) 7n, are unknown. Thus, for te ustable decays available now ED turns out to hold approximately with great accuracy determined here by analytical calculations.

APPENDIX: EVALUATION OF SURVIVAL AMPLITUDE

The survival amplitude of the moving unstable particle follows from Eqs. 9) and (10)

Al W = or dxw (x) exp (-- i t -,Ip + X2), (A. )

r W(X) = - [(X _ M2 + r 2/4] (A. 2) 27r

8 (tL is replaced by x). To calcu- Y late (A.1) the integration contour +iP C is chosen, see Fig. 1. The inte- grand in (A.1) is the single-valued analytical function inside C (with M -ir/2 x the pole at the point m - iF/2). Its branch is chosen which is pos- _1P itive on the real positive sermaxis x: arg V/z2+ p2 = when z = x. Then on the quarter of the circle CR the argument of VZ2-+p2 co- Ci R incides with argz (the radius R of CR being large) and is zero at the point x = R, y = 0, and is equal to -7/2 at the point x = 0, y = -iR. When z moves along C from iR Fig. 1. Integration contour for A,(t). Dotted up to the branch point -ip the line shows the chosen branch cut for the function arg V/z2 + p2 does not change re- V2_+ p2 maining to be equal to -7r/2. This follows from the equation

arg VZ2 + p2 larg(z - ip) + arg(z + ip)]. (A.3) 2

So on the interval (-iR, -ip) we have fz2 + p = -ij V/p2j. Further as z passes the point (-ip) the argument of (z - ip) does not change as before while arg(z + ip) changes by +,7r in the vicinity of (-ip). Therefore, arg fZ-2-+p2 changes by 7/2, see Eq. (A.3), and Fz2 + p2 becomes positive real being equal to I p2-- y2 1. The above consideration of the behaviour of the chosen branch Of V/_Z2 + p2 gives

IC dzw(z) exp(-it Vz2 + p2 = fo"o dxw(x) exp(-it V1x2 + p2)

fR' )exp(-tlVy -p2j) (A.4)

0(-idy)w(-iy) exp(-itl Vp2 y21 = -27riRes.

Here Res is the residue of the integrand at the pole m - ir/2:

Res = (-27ri)-l exp[-itv/( i/2)2 + p2j. (A.5)

The integral over CR is not written in Eq. (A.4) because it vanishes as R - 00 due to the Jordan lemma.

9 To calculate Ap(t), see Eq. (A.1) and (A.4), we must now compute the integrals

-[(i + M)2 + 2 I".P = i dy /4] -1 exp(-tj V/y2 p2j), (A.6) 27r

-[(i + n)2 2 1P0 = if dy /4] -1 exp(- it V/p2 y2). (A.7) 27r

Let us estimate I,,p. The main contribution to I ...p is due to y values close to p, if t is not too small. One may suppose, e.g., that t >> 1/m or t > 0.1-r = 0.1/r. In order to obtain an approximate value for I,p, let us replace (iY + M) + r2 /4 in the integrand by (ip + M2 . Then

I00P -- ill i + n)-2jp, j.P 27r TO dy exp (- t V/y 2 p2). (A. )

After the change of variables u = Vy2-- p2 the integral Jp reduces to the table one 00 u tu J.P du '/-U2-+p2 C- (A.9) e.g., see 3366.3 in Gradshtein, Ryzbik, 19621. shall need the value of J,,, at pt > I (this means that p is supposed to be not too small). Using formulae 3.366.3 and 8554 in the cited tables (or immediately by integrating (A.9 by parts) we get

J.,,, AM)-', (A. I 0)

11-PI 1 F I I , mt > 1, pt >> 1. (A. I I) 2 7r Vlp2 + 2 X/ +M 2 t pt

Now let us estimate po, see Eq. (A.7). After the change of variables v Vp-2 y2 we get (neglecting r2 /4)

ji VIP _ 2 + M)-2 eXp(_itV). IV = ir f dv v (A. 2) 27r 0 Vp2 V2

The values of v close to p bring the main contribution due to the factor (P 2 V 2)-1/2. So,

1P0 E lfpdvv costv -- t' fp dvv sin t11- Vp2 V2 0 Vl V2

ir {IC isl (A. 3) 27rm2

I 0 For the integrals IC and Is see the formula 25.8.2 (with = 12 in [Prudnikov, 2003] (note that in the first edition of the book there is a misprint in this formula) 7r ir is = _Pj (Pt), IC -pH I Pt) + . 2 2 Here J is the Besse] function and H, is the Struve function. Using their asymp- totic expansions (see, e.g., 8451.1 and 8554, 8451.2 in [Gradshtein, Ryzhik, 19621) one gets at t >> 1

1S - _ 5 cos(pt + 7/4), - Ir sin(pt + 7/4), (A. 14) 7r2 VptP Ic - V(2 VIpt

1/2 P I ,Po (2 7r) - i7r/4), Pt . 2 M r,,M IM___t exp(-ipt+

We see that the integral I,,.p, Eq. (A. II) can be neglected as compared to ipo in the case t 1. Therefore, I,,,p is omitted in Eq. 12) for A,(t). One can see that Res, Eq. (A.5), dominates in Ap(t) during several decades of lifetimes while ,o gives Ap(t) asymptotics as t -+ oc. The amplitude Ao(t) is calculated in the same way, the result being given by Eq. (I 1). Our goal is the investigation of the dilation of A,(t) as compared to Ao(t). For this purpose, one has first of all to compare Res in A,(t), see Eq. (A.5), with Res in Ao(t), see the first term in Eq. (I 1). Let us represent Res, Eq. (A.5), in the form best suitable for this comparison (see Subsect 32) As F/m < one may expand

[(,rn - ir/2 )2 2 1/2 = /P2+M2[j _ i]prn + ,2 /4) (P2 + M 2) 111/2 in the series over powers of 1. Omitting the terms _ r4 and still lesser ones we obtain

Rm - ir/2 2 211/ 2 VP2 + m2(l - o,) - ir(1 + c,)/2-y + . . . (A. 5) for a and -y see Eq. (I 3. Eq. (I 2) follows from Eqs. (A. 1), (A.4), (A.5), (A. 14), (A. 15) Note. Instead of expansion (A. 5) Stefanovich 1996) used the expansion of the square root VP2-+x2 in the integrand of (A. ) in series over powers of the ratio (X _ M[P2 + M2]-1/2 . He made the implication that x values close to m predominate in the integrand because w(x) is peaked when -F < x - m < r, r'/,m < 1. Then only the first terms of the series dominate and Ap(t)1' turns out to be approximately equal to jAo(t1-y)j' notwithstanding t values, i.e., ED results for all t. However, my analytical calculation of A,(t) shows that ED decisively fails at large t when the decay is not exponential. So Stefanovich's implication actually is false for large t (e.g., because x values from the interval

I I (7 - F, in F) are suppressed by fast oscillations of exp(- it V/p2 + x2 at large t). Let us stress that expansion (A.15) (unlike Stefanovich's one) is done after analytical evaluation of integral (A. ). The validity of (A. 5) does not depend on Stefanovich's implication, only the smallness of F/m being essential. Note also that Stefanovich's expansion does not give an explicit estimation of the ED violation (which is characterized here by the quantity , see Eq. (15)). For this estimation he numerically calculated the involved integrals (for times not exceeding ten lifetimes, see his Fig. 1).

REFERENCES Alzetta, A. and d'Ambrogio, E. 1966). Evolution of a resonant state. Nuclear Physics 82, 683-689. Bailey J. et al. 1977). Measurement of relativistic time dilation for positive and negative muons in a circular orbit. Nature 268, 301-304. Exner, P. (I 983). Representations of the Poincat-6 group associated with unstable particles. Physical Review D 28, No. 10, 2621-2627. Farley, F. 1992). The CERN (g-2) measurements. Zeitschrift fijr Physik C 56, S88, Sect. 5. Fonda, L., Ghirardi, G., and Rimini, A. 1978). Decay theory of unstable quantum system. Rep. Progr. in Phys. 41, 587, Sect. 3. Gi-Chol Cho et al. 1993). The time evolution of unstable particles. Progr. Theor. Phys. 90, No. 4 803-816. Goldberger, M. and Watson, K. 1964). Collision Theory. John Wiley, New York, Ch. . Gradshtein, 1. and Ryzhik, 1. 1962). Tables of Integrals, Sums, Series and Products. GIFML, Moscow. Horwitz, L. 1995). The unstable system in relativistic quantum mechanics. Foundation of Physics 25, No. 1, 39-65. Levy, M. 1959). On the description of unstable particles in quantum field theory. Nuovo Cimento 13, No. 1 H . Messiah, A. 1961). Quantum Mechanics. North-Holland, Amsterdam, V. 2 Ch. 21.13. Moller, C.(1972). The Theory of Relativity. Clarendon Press, Oxford, Ch. 26. Norman, E. et al. 1988). Tests of the exponential decay law at short and long times. Phys. Rev. Lett. 60, No. 22, 2246. Perkins, D. 987). Introduction to High Energy Physics. Addison-Wesley Co., Inc. Pnidnikov, A. P. et al. 2003). Integrals and Series. V. 1, Second edition, Fizmatlit, Moscow. Stefanovich, E. 1996). Quantum effects in relativistic decays. Intern. Journ. of Theor. Phys. 35, No. 12, 2539-2554.

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Shirokov M. 1. E2-2004-4 Decay Law of Moving Unstable Particle

Quantum relativistic decay law of moving unstable particle is analytically cal- culated in the model case of the Breit-Wigner mass distribution. It turns out that Einstein time dilation of the moving particle decay holds approximately at times when the decay is exponential. The related correction is calculated analytically. Being very small at these times it is practically unobservable. It is shown that Ein- stein dilation fails for large times when decay is not exponential. An unstable sys- tem of the kind of K meson (which is the superposition of K, and K,) is also con- sidered. In this case, the violation of Einstein dilation is shown to be appreciable at all times under some condition.

The investigation has been performed at the Bogoliubov Laboratory of Theo- retical Physics, JINR.

Preprint of the Joint Institute for Nuclear Research. Dubna, 2004 KoppeKTop T. E. floneKO

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