# QFT Derivation of the Decay Law of an Unstable Particle with Nonzero Momentum

Authors: Francesco Giacosa
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

We present a quantum field theoretical derivation of the nondecay probability of an unstable particle with nonzero three-momentump. To this end, we use the (fully resummed) propagator of the unstable particle, denoted as S, to obtain the energy probability distribution, called dSp(E), as the imaginary part of the propagator. The nondecay probability amplitude of the particle S with momentum p turns out to be, as usual, its Fourier transform: aSp(t)=∫mth2+p2∞dEdSp(E)e-iEt (mth is the lowest energy threshold in the rest frame of S and corresponds to the sum of masses of the decay products). Upon a variable transformation, one can rewrite it as aSp(t)=∫mth∞dmdS0(m)e-imth2+p2t [here, dS0(m)≡dS(m) is the usual spectral function (or mass distribution) in the rest frame]. Hence, the latter expression, previously obtained by different approaches, is here confirmed in an independent and, most importantly, covariant QFT-based approach. Its consequences are not yet fully explored but appear to be quite surprising (such as the fact that the usual time-dilatation formula does not apply); thus its firm understanding and investigation can be a fruitful subject of future research.


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## Body

## 1. Introduction

The study of the decay law is a fundamental part of quantum mechanics (QM). It is now theoretically [1–9] and experimentally [10, 11] established that deviations from the exponential decay exist, but they are usually small. Such deviations are also present in quantum field theory (QFT) [9, 12].An interesting question addresses the decay of an unstable particle with nonzero momentump. In [13–17], it was shown—by using QM-based approaches enlarged to include special relativity—that the nondecay probability of an unstable particle S with momentum p is given by (in natural units)(1)PSpt=aSpt2withaSpt=∫mth∞dmdSme-im2+p2t,where dS(m) is the energy (or mass) distribution of S in its rest frame [dmdS(m) is the probability that the unstable state S has energy (or mass) between m and m+dm]. A review of the derivation is presented in Section 2. Quite remarkably, there are many interesting properties linked to this equation, which include deviations from the standard dilatation formula; see below.The purpose of this work is straightforward: we derive (1) in a QFT framework (see Section 3). We thus confirm its validity and, as a consequence, its peculiar features. For definiteness, an underlying Lagrangian involving scalar fields is introduced, but our discussion is valid for any spin of the unstable particle and of the decay products. A central quantity of our study is the relativistic propagator of the unstable particle S: the mass distribution dS(m) is then obtained by the imaginary part of the propagator.In this Introduction, we recall some basic and striking features connected to (1). The normalization(2)∫mth∞dmdSm=1is a crucial feature of the spectral function, implying that aSp(0)=1. It must be valid both in QM and in QFT. Here, without loss of generality, we set the lower limit of the integral to mth≥0. In fact, a minimal energy mth is present in each physical system; in particular, for a (relevant for us) relativistic system, it is given by the sum of the rest masses of the produced particles (mth=m1+m2+⋯≥0). Clearly, for p=0, (1) reduces to the usual expression(3)PSrestt=PS0t=aS0t2=∫mth∞dmdSme-imt2.A detailed study of (1) shows that the usual time dilatation does not hold:(4)PSpt≠PSresttMM2+p2,where M is the mass of the state S defined, for instance, as the position of the peak of the distribution dS(m); in general, however, other definitions are possible, such as the real part of the pole of the propagator; see Section 3. The point is that, no matter which definition one takes, expression (4) remains an inequality.It is always instructive to investigate the exponential limit, in which the spectral function of the stateS reads [18, 19](5)dSBWm=Γ2πm-M2+Γ24-1,where M is the “mass of the unstable state” corresponding to the peak. Even if the spectral function dSBW(m) is clearly unphysical because there is no minimal energy (mth→-∞), in many physical cases it is a good approximation for a quite broad energy range. Here, the decay amplitude and the decay law in the rest frame of the decaying particle notoriously read(6)aSBW,0t=e-iMt-Γt/2⟶PSBW,restt=e-Γt.When a nonzero momentum is considered, the nondecay probability is still an exponential given by(7)PSBW,pt=e-Γptwhere the width is [17](8)Γp=2M2-Γ24+p22+M2Γ21/2-M2-Γ24+p2.Clearly, Γp=0=Γ. One realizes, however, that Γpdiffers from the naively expected standard time-dilatation formula, according to which the decay width of an unstable state with momentum p should simply be(9)ΓMp2+M2=γΓ.Namely, the quantity γ=p2+M2/M=1/1-v2 is the usual dilatation factor for a state with (definite) energy M. Deviations between (8) and (9) are very small, as the numerical discussion in [17] shows. Although not measurable by current experiments [20], the very fact that deviations exist is very interesting and deserves further study.As a last point, it should be stressed that in this work we consider unstable states with a definite momentump. This is a subtle point: while for a state with definite energy, a boost and a momentum translation are equivalent, this is not so for an unstable state, since it is not an energy eigenstate. Even more surprisingly, a boost of an unstable state is a quantum state whose nondecay probability is actually zero: it is already decayed (on the contrary, its survival probability presents a peculiar time contraction [21]). In other words, a boosted muon consists of an electron and two neutrinos [15, 17]. In this sense, the boost mixes the Hilbert subspace of the undecayed states with the subspace of the decay products; see [17] for details. There, it is also discussed why the basis of unstable states contains states with definite three-momentum. Indeed, the investigation of this paper also confirms this aspect: unstable states with definite momentum naturally follow from the study of its propagator in QFT.The paper is organized in this way: in Section2 we recall the QM derivation of (1), while in Section 3—the key part of this paper—we present this derivation in a QFT context. In the end, in Section 4 we describe our conclusions.


## 2. Recall of the QM-Based Derivation of (1)

For completeness, we report here the “standard” derivation of (1). To this end, we use the arguments presented in [17], but similar ones can be found in [13–16].We consider a system described by the HamiltonianH, whose eigenstates are denoted as(10)m,p=Upm,0,where Up is the unitary operator associated with the translation in momentum space. Standard normalization expressions are assumed:(11)m1,p1∣m2,p2=δm1-m2δp1-p2.The state m,p has definite energy,(12)Hm,p=p2+m2m,p,definite momentum,(13)Pm,p=pm,p,and definite velocity p/p2+m2. Note, assuming that the energy of m,p isp2+m2, we have a relativistic spectrum.Formally, the Hamiltonian can be written as(14)H=∫d3p∫mth∞dmp2+m2m,pm,p=∫d3pHpwhere(15)Hp=∫mth∞dmp2+m2m,pm,pis the effective Hamiltonian in the subspace of states with definite momentum p.Let us now consider an unstable stateS in its rest frame. The corresponding quantum state at rest is assumed to be(16)S,0=∫mth∞dmαSmm,0,where αS(m) is the probability amplitude that the state S has energy m. Hence, it is natural that the quantity dS(m)=αS(m)2 is the mass distribution: dS(m)dm is the probability that the unstable particle S has a mass between m and m+dm. As a consequence, ∫0∞dmdS(m)=1, as already discussed in the Introduction.For the states of zero momentum, the HamiltonianHp=0 can be expressed in terms of the undecayed state S,0 and its decay products in the form of a Lee Hamiltonian [22] (similar effective Hamiltonians are used also in quantum mechanics [6, 19] and quantum field theory [9, 23]):(17)Hp=0=∫mth∞dmmm,0m,0=M0S,0S,0+∫d3kωkk,0k,0+∫d3k2π3/2gfkS,0k,0+k,0S,0,where k,0 represents a decay product with vanishing total momentum: in the two-body decay case, k,0 describes two particles, the first with momentum k and the second with momentum -k, hence(18)ωk=k2+m12+k2+m22.The last term in (17) represents the “mixing” between S,0 and k,0, which causes the decay of the former into the latter. Moreover, g is a coupling constant and f(k) encodes the dependence of the mixing on the momentum of the produced particles. The explicit expressions connecting the states k,0 to m,0 formally read(19)k,0=∫mth∞dmβkmm,0where βk(m) can be found by diagonalizing the Hamiltonian (17).Let us now consider an unstable state with definite momentump, which is denoted as S,p:(20)S,p=UpS,0=∫mth∞dmαSmm,p.The normalization(21)S,p1∣S,p2=δp1-p2follows. Note that (20) isnot a state with definite velocity. This is due to the fact that each state m,p in the superposition has a different velocity p/p2+m2. The subset of Hilbert space given by {S,p∀p⊂R2} represents the set of all undecayed quantum states of the system under study.The form of the HamiltonianHp in terms of the states S,p and Upk,0=k,p can be in principle derived by using the expressions above. Together with (20), one shall also take (19) and apply Up in order to get(22)Upk,0=k,p=∫mth∞dmβkmm,p.Then, one should invert (20) and (22) and insert it into Hp of (15). However, its explicit expression is definitely not trivial but, fortunately, also not needed in the present work. Hence, we do not attempt to write it down here.We now turn to the nondecay amplitude of the stateS. In its rest frame (p=0), upon starting from a properly normalized state with zero momentum, S,0/δ(0), one obtains the usual expression(23)aS0t=1δ0S,0e-iHtS,0=1δ0∫mth∞dm1dm2m1,0e-iHtm2,0=∫mth∞dmdSme-imt,in agreement with (3). The theory of decay is discussed in great detail for the case p=0 in [4, 6, 7, 9] and references therein. Note here the nondecay probability coincides with the survival probability (that is, the probability that the state did not change), but in general this is not the case [17].Next, we consider a normalized unstable stateS with nonzero momentum: S,p/δ(0). The resulting nondecay probability amplitude(24)aSpt=1δ0S,pe-iHtS,p=1δ0∫mth∞dm1dm2m1,pe-iHtm2,p=∫mth∞dmdSme-im2+p2tcoincides with (1), hence concluding our derivation.In principle, one could also start from the HamiltonianHp and obtain the energy distribution associated with this state, denoted as dSp(E). Then, aSp(t) should also emerge as the Fourier transform of the latter. This is hard to do here, since the explicit expression of Hp in terms of S,p and k,p was not written down (as mentioned previously, this is not an easy task). Quite interestingly, in the framework of QFT, the function dSp(E) can be easily determined; see Section 3.As a last comment of this section, we recall that the general nondecay probability of an arbitrary stateΨ reads(25)PΨt=∫d3pS,pe-iHtΨ2,whose interpretation is straightforward: we project Ψ onto the basis of undecayed states. In general, PΨ(0) is not unity. Notice also that PΨ(t) isnot the survival probability of the state Ψ (a state can change with time but still be undecayed if it is a different superposition of S,p).When a boostUv on the state with zero momentum (and hence with zero velocity) S,0 is considered, the resulting state reads [17](26)φv=UvS,0=∫mth∞dmαSmmγ3/2m,mγv,where γ=(1-v2)-1/2. In fact, each element of the superposition, m,mγv, has velocity v. Of course, φv is not an eigenstate of momentum, since each element in (26) has a different momentum p=mγv. In this respect, the state S,0 is special: it is the only state which has at the same time definite momentum and definite velocity (both of them vanishing). As mentioned in the Introduction, the nondecay probability associated with φv vanishes:(27)Pφvt=0∀v≠0.As soon as a nonzero velocity is considered, the state has decayed. This result is quite surprising but also rather “delicate”: when a wave packet is considered, Pndφv(t) is nonzero (even if it is not 1 for t=0) [17].


## 3. Covariant QFT Derivation of (1)

Let us consider an unstable particle described by the scalar fieldS(x)≡S(t,x). For an illustrative example, one can couple S with bare mass M0 to two scalar fields φ1 (with mass m1) and φ2 (with mass m2) via the interaction term gSφ1φ2, leading to the QFT Lagrangian(28)L=12∂μφ12-m12φ12+12∂μφ22-m22φ22+12∂μS2-M02S2+gSφ1φ2.This is the QFT counterpart of the QM system of the previous section. Note we use scalar fields for simplicity, but our discussion is in no way limited to it.The (full) propagator of the stateS (details in [24]) reads(29)ΔSp2=1p2-M02+Πp2+iεwithp2=E2-p2,where E=p0 is the energy and p the three-momentum. Because of covariance, ΔS(p2) depends only on p2. The quantity Π(p2) is the one-particle irreducible diagram. Its calculation is of course nontrivial (it requires a proper regularization), but it is not needed for our purposes. The imaginary part is(30)ImΠp2=p2Γp2=k8πp2g2fΛ2k+⋯,where dots refer to higher orders, which are however typically very small [25]. Once ImΠ(p2) is fixed, ReΠ(p2) can be determined by dispersion relations (for an example of this technique, see, e.g., [26]). The quantity Γtl=Γ(p2=M) is the usual tree-level decay width; hence in the exponential limit the decay law PS(t)=e-Γtlt must be reobtained. As mentioned in the Introduction, an unstable state has not a definite mass: this is why different definitions for M (which is not the bare mass M0 entering in (29)) are possible: ReΔS-1(p2=M2)=0 (zero of the real part of the denominator), or Respole, with ΔS-1(spole)=0 (real part of the pole), or the maximum of the spectral function defined below.We also recall that(31)k=p4+m12-m222-2p2m12+m224p2coincides, for the on-shell decay, with the modulus of the three-momentum of one of the outgoing particles. The vertex function fΛ(k) is a proper regularization which fulfills the condition fΛ(k→0)=1 and describes the high-energy behavior of the theory (its UV completion); hence the parameter Λ is some high-energy scale; fΛ(k) is formally not present in (28) since it appears in the regularization procedure, but it can be included directly in the Lagrangian by rendering it nonlocal [27] in a way that fulfills covariance [28]. In a renormalizable theory (such as the one of (28)), the dependence on Λ disappears in the low-energy limit.The properties outlined above, although in general very important in specific calculations, turn out to be actually secondary to the proof that we present below, where only the formal expression of the propagator of (29) is relevant. Moreover, even when the unstable particle is not a scalar, one can always define a scalar part of the propagator which looks just as in (29), then the outlined properties apply,mutatis mutandis, to each QFT Lagrangian.As a next step, upon introducing the Mandelstam variables=p2, the function F(s) defined as(32)Fs=1πImΔSp2=sfulfills the normalization condition:(33)∫sth∞dsFs=1,where sth=mth2 is the minimal squared energy. For the case of (28), one has obviously sth=m1+m22. The normalization (33) is a consequence of the Källén–Lehmann representation [29](34)ΔSp2=∫sth∞dsFsp2-s+iε,in which the propagator ΔS(p2) has been rewritten as the “sum” of free propagators p2-s+iε-1, each one of them weighted by F(s): dsF(s) is the probability that the squared mass lies between s and s+ds. Of course, the normalization (33) is a very important feature of our approach. For a detailed proof of its validity, we refer to [30]. Here we recall a simple version of it, which is obtained by assuming the rather strong requirement Π(p2)=0 for p2>Λ2, where Λ is a high-energy scale (no matter how large). Under this assumption(35)ΔSp2=1p2-M02+Πp2+iε=∫sthΛ2dsFsp2-s+iε.Then, upon taking a certain value p2≫Λ2, the previous equation reduces to(36)1p2=∫sthΛ2dsFsp2⟶∫sthΛ2dsFs=1The general case in which Π(p2→∞)=0 smoothly requires more steps, but the final result of (33) still holds [30].Let us now consider the rest frame of the decaying particle:p=0, s=p2=E2=m2. Here, upon a simple variable change (m=s), we obtain the mass distribution (or spectral function) dSp=0(m) through the equation(37)dmdSp=0m=dsFs,out of which(38)dSm=dSp=0m=2mFs=m2.As already mentioned, dmdS(m) is the probability that the particle S has a mass between m and m+dm [24, 31]. In this context, the normalization(39)∫mth∞dmdSm=1follows from (33). Once the function dS(m) is identified as the mass distribution of the undecayed quantum state, the nondecay probability’s amplitude aS0(t) can be obtained by repeating the steps of Section 2. The result coincides, as expected, with (3). Yet, it should be stressed that the unstable quantum state S,0 characterized by the distribution dS(m) is not simply given by a0†0PT, where 0PT is the perturbative vacuum and ap† the creator operator of the noninteracting field S. The case of neutrino oscillations shows a similar situation: the state corresponding to a certain flavour, such as the neutrino νe, must be constructed with due care by making use of Bogolyubov transformations [32]. Along this line, the exact and formal determination of the state S,0, corresponding to the mass distribution dS(m), in the context of QFT requires a generalization of Bogolyubov transformations and was, to our knowledge, not yet explicitly done (it is left for the future). Nevertheless, it is not needed for the purpose of this paper.Let us now consider the particleS moving with a certain momentum p. Upon using s=E2-p2, the energy distribution—as function of E—is obtained by(40)dEdSpE=dsFs,leading to(41)dSpE=2EFs=E2-p2=EE2-p2dSE2-p2.The quantity dEdSp(E) is the probability that the particle S with definite momentum p has an energy between E and E+dE (clearly, dSp=0(E)=dS(m=E)). Also in this case, the normalization(42)∫mth2+p2∞dEdSpE=1is a consequence of (33). When dS(m) has a maximum at M, then dSp(E) has a maximum at ~M2+p2. Note the very fact that the propagator depends on p2=E2-p2 allows to determine the spectral function dSp(E) for a definite momentum p, that corresponds to the state S,p of Section 2.The nondecay probability’s amplitude for a stateS moving with momentum p is then given by(43)aSpt=∫mth2+p2∞dEdSpEe-iEt,where we have taken into account that the minimal energy is now given by mth2+p2.This expression can be manipulated by using (41) and via a change of variable:(44)aSpt=∫mth2+p2∞dEdSpEe-iEt=∫mth2+p2∞dEEE2-p2dSE2-p2e-iEt=∫mth∞dmdSme-im2+p2t,which coincidesexactly with (1), as we wanted to demonstrate. Thus, we confirm the validity of (1) in a covariant QFT-based framework.


## 4. Conclusions

The decay law of a moving unstable particle is an interesting subject that connects special relativity to QM and QFT. An important aspect is the validity of (1), which expresses the nondecay probability of a state with nonzero momentum and whose standard derivation is reviewed in Section 2.The main contribution of this paper has been the derivation of a quantum field theoretical proof of (1). To this end, we started from the (scalar part of the) propagator of an unstable quantum field, denoted as S. Then, we have determined the energy distribution of the state S with definite momentum p, out of which the survival’s probability amplitude is calculated.As discussed in the Introduction, there are interesting and peculiar consequences of (1). Future studies are definitely needed to further understand the properties of a decay of a moving unstable particle and to look for feasible experimental tests.



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