Laser-intensity effect on beta decay of the free neutron

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IOP PUBLISHING JOURNAL OF PHYSICS G: NUCLEAR AND PARTICLE PHYSICS

J. Phys. G: Nucl. Part. Phys. 39 (2012) 065103 (10pp) doi:10.1088/0954-3899/39/6/065103

Laser-intensity effect on beta decay of the free neutron

Shahid Rashid

Center for Scientific Development, 10 Commercial Ave, 10M, New Brunswick, NJ 08901, USAandDepartment of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway,NJ 08854-8019, USA

E-mail: shahidr@rci.rutgers.edu

Received 11 January 2012Published 15 May 2012Online at stacks.iop.org/JPhysG/39/065103

AbstractUsing the exact solution of the minimally coupled Dirac equation as the statefor the lepton, the cross section for the weak interaction is calculated in thepresence of an intense laser field. A completely new term is discovered due tothe interaction of the spin of the electron with the intense laser field present.Moreover, the presence of the dressed lepton only in the initial or final stategives rise to new arguments for the Bessel functions which is different fromthose calculated by previous authors. We have computed the decay rate of afree neutron in the presence of an intense laser field. It is shown that with theincreasing intensity, the lifetime of the free neutron increases nonlinearly.

1. Introduction

It is a well-known fact that in 1930 Pauli was the first to suggest the existence of neutrino ν

to explain the continuous energy spectrum of β-decay from atomic nuclei [1]. Later in 1934,Fermi used Pauli’s conjecture to postulate the weak interaction as opposed to the nuclearinteraction [2] and proposed the interaction matrix as

Vfi = (ψnO ψp)(ψeOL ψν), (1)

where ψi is the wavefunction of the ith particle and the subscripts n, p, e and ν denote theneutron, proton, electron and neutrino, respectively.

O is the nucleon operator and OL is the lepton operator [3, 4]. In the absence of apropagator, equation (1) represents a point interaction as shown in figure 1.

On the evidence of weak interactions amongst the pairs (p, n), (e, νe) and (μ, νμ), whereμ denotes muon and νμ its associated neutrino, Feynman and Gell-Mann [5] postulated auniversal weak interaction given by the Hamiltonian

Hint =√

2gV J∗ · J. (2)

Here J is the weak interaction current and J∗ is its Hermitian conjugate. The coupling constantgV has the value [6]

gV = 1.41 × 10−49 erg . cm3 (3)

0954-3899/12/065103+10$33.00 © 2012 IOP Publishing Ltd Printed in the UK & the USA 1

J. Phys. G: Nucl. Part. Phys. 39 (2012) 065103 S Rashid

P

N

W

e

Figure 1. Feynman diagram of neutron decay in the presence of laser radiation (drawn with [8]).

The discovery by Lee and Yang [7] of the non-conservation of parity restricted the weakinteraction current J to having the vector minus axial vector or the (V − A) Lorentz structure,so that the β-decay interaction Hamiltonian Hβ is given by

Hβ =√

2gV [ψpγμ(1 − λγ5)ψn] · [ψeγμ(1 − γ5)ψν]. (4)

The constant factor λ = 1.26 is the ratio of the axial vector coupling constant gA to the vectorcoupling constant gV . The factor arises because of the strongly interacting character of thenucleons so that the hadronic part of the weak interaction is modified. γ s are the usual Diracmatrices [4].

The cross section calculated by using the Hamiltonian (equation(4)) gives rise to high-energy catastrophe because it is a point interaction involving four fermions. This problem hasbeen resolved by introducing the intermediate vector boson [9] in the context of gauge theoriesof the electro-weak interaction [10]. The present picture of the weak interaction is based onthe standard model [11], where for β-decay a down quark changes into an up quark via anintermediate vector boson W − which then breaks into the electron and the antineutrino.

Recently, the effect of the magnetic field on the lifetime of unstable particles has beenconsidered in some detail [12]. In this paper, we set out to analyze the β-decay of nucleonin the presence of intense laser light. Since the electron is light enough to be affected by theradiation field, this effect could be observed using the present day available laser systems. Theeffect on the electron by the laser field is characterized by the Kibble parameter, K, which isdefined as the average of the square root of the quiver energy of the electron divided by theelectron’s rest mass energy [13]. Since the intensity of the laser field required to observe thechange in the neutron’s lifetime is calculated for the conditions when the Kibble parameter Kis very small compared to 1, and also since the proton is very heavy compared to the electron,we ignore the coupling of the proton mass to the laser field. This is because the quiver energyof the proton in the laser field would be extremely small compared to the rest mass energy ofthe proton. Also since the neutron has no charge, we ignore the coupling of the neutron to thelaser field and treat it as a free particle. Also, as the nucleon is heavy compared to the lepton,it will not be affected by the laser field so long as the intensity of the radiation field is lowerthan the threshold, where the quiver energy of the electron is equal to the rest mass energy ofthe electron [14]. The neutrino being of zero charge would not interact with the radiation field.Although there is evidence that the neutrino has small mass, in this paper we will assume themass to be zero.

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J. Phys. G: Nucl. Part. Phys. 39 (2012) 065103 S Rashid

In section 2, we assume the electron to be in the Volkov state which is the exact solutionof the minimally coupled Dirac equation. This solution takes the interaction of the spin of theelectron with the radiation field fully into account and has been shown to be important by theauthor [14] in past papers. Because we are considering very high laser intensity and a softphoton (where wavelength λ0 = 0.1 μm), we treat the coherent laser field classically. Thequestion of asymptotic coupling and decoupling of the electron to the laser field is then treatedin the context of the time-averaged kinematical quantities using Volkov states.

In section 3, the transition rate is derived using the interaction Hamiltonian given byequation (4) above, but instead of the free-electron state we use the Volkov state for theelectron’s wavefunction. The differential scattering cross section is derived as a sum of termseach of which involves an infinite sum of products of the Bessel function whose arguments areproportional to the intensity of the laser field. We find that the spin of the electron contributesto the scattering process via the second term, but it is negligibly small unless the intensity ofthe laser field is very high.

In section 4, to obtain some tractable results, which can be verified by the experiment, weassume that the neutron is stationary and that the proton is non-relativistic. It is then shownthat the scattering cross section is the same as obtained by previous analysis [3]. To obtainthe average neutron decay rate, we integrate over the energy of the final state electron afterintroducing the phase-space contribution into the expression. For sufficiently low intensity ofthe laser field, we expand the relevant Bessel functions into a series and then plot the averageneutron lifetime with the corresponding laser intensity. The plot clearly shows the effect ofincreasing the lifetime of the neutron as the intensity of the laser field is increased.

The conclusion is given in section 5, where we analyze some of the results and outlinesome future investigations.

2. Volkov state for the electron

Since the laser field is designed to cover the whole of the interaction region, the electrons areinside the laser field after the neutron decays and hence we consider the electron to be in theVolkov state [15] which is the exact solution of the minimally coupled Dirac equation. Wetreat the electron relativistically to take into account the relativistic effect of the laser field.We use the Lorentz–Heaviside units [16] with � = c = 1 and the metric gμυ = (1,−1,−1,−1).The laser beam can be represented by a classical monochromatic field, the amplitude of whichis given by (bold print means 3-vector)

Aμ = (0, A) = εμA0 cos(k · x), (5)

where

kμ = (k0, k) =|k|(n0, n), (6)

and

εμ = (0, ε). (7)

The gauge is ε · k = 0. The exact solution for the electron in the intense laser field Aμ is givenby the Volkov state [15]

ψi =√

m

Ei

(1 + e

2n · piγ · nγ · A

)ui e−i(pi·x−Si ), (8)

where ui is a spinor satisfying the normalization condition

u+i · ui = |Ei|/m (9)

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J. Phys. G: Nucl. Part. Phys. 39 (2012) 065103 S Rashid

and

Si = 1

2n · pi

∫ +n·x

−∞{2e pi · A − (eA)2} dy. (10)

The subscript ‘i’ indicates the incident electron and the γ s are the usual Dirac matrices [4].Expression (8) is an exact solution of the minimally coupled Dirac equation.

3. β-decay in the presence of the intense laser field

Assuming the electron to be in a Volkov state, the scattering matrix element for the β-decayis given by (see equation (4) and [3])

Sfi = gV√2

∫d4x[ψpγμ(1 − λγ5)ψn][ψVolkov

e γ μ(1 − γ5)ψν], (11)

where ψVolkove is the electron Volkov state given by (8). ψi is the wavefunction of the ith particle

and the subscripts n, p, e and ν denote the neutron, proton, electron and neutrino, respectively.The λ = 1.26 is the ratio of the axial vector coupling constant gA to the vector coupling constantgV = 1.41×10−49 erg cm3. The γ are the usual Dirac matrices [16]. Typically, the initial statefor the neutron is given by

ψn(y) =√

M

Enun e−iPn·y, (12)

where Pn is the initial 4-momentum of the neutron given by Pn = (En, Pn). The final protonstate is similar to (12) but with the subscript p instead of n in all of the factors. The neutrinowavefunction

ψν(x) =√

1

2Eν (2π)uν e−ipν ·x. (13)

The neutrino wavefunction (13) is normalized such that

uν = limm−→0

√1

2mUν , (14)

where Uν is a Dirac solution; also

uν · uν = 0 (15)

and

u†ν · uν = 2Eυ. (16)

Multiplying the scattering matrix (11) by its complex conjugate and then averaging overthe initial spin and summing over the final spin only, we obtain1

2

∑∣∣S f i

∣∣2 = g2V

2

m

(2π)62EeEυ

MPMN

(2π)68EpEn∫d4x

∫d4x′ei{(pe−pν )·(x−x′ )+Se−S

′e+(pp−pn )·(x−x′ )} · T , (17)

where the trace factor T is given by

T =∑si,s f

{[ue

(1 + e

2n · peγ · Aγ · n

)γ μ (1 − γ5) uν

]

× ·[

uνγα(1 − γ5)

(1 + e

2n · peγ · nγ · A′

)ue

]}·∑

Si,S f

{[upγμ (1 − λγ5) un] · [unγα (1 − λγ5) up]}. (18)

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J. Phys. G: Nucl. Part. Phys. 39 (2012) 065103 S Rashid

The sums over the spin can be reduced to traces in the usual manner [16] so that (18) becomes

T = T1 · T2, (19)

where

T1 = Tr

[(pe · γ + m

2m

)(1 + e

2n · peγ · Aγ · n

)γ μ(1 − γ5)(pυ · γ )γ α

×(1 − γ5)

(1 + e

2n · peγ · nγ · A′

)](20)

and

T2 = Tr

[(Pp · γ + Mp

2Mp

)γμ (1 − λγ5)

(Pn · γ + Mn

2Mn

)γα (1 − λγ5)

]. (21)

Using the trace theorems [16] and cancelling the opposing terms, we reduce the first trace T1

of equation (20) to [16]

T1 = 1

mTr

[γ · peγ

μγ · pνγα − γ · peγ

μγ · pνγαγ 5 + e(A′ − A)

2n · pe

(γ · peγμγ · pνγ

α − γ · peγμγ · pνγ

αγ 5) · (γ · nγ · ε) + e2AA′

4(n · pe)2

(γ · peγ · εγ · nγ μγ · pνγα − γ · peγ · εγ · nγ μγ · pνγ

αγ 5) · (γ · nγ · ε)

].

(22)

When the radiation field is small, such that the Kibble parameter K � 1, the quiver energyof the electron is small compared to the electron’s rest mass energy1. Then, we obtain thefollowing approximation for T1 after doing some of the traces:

T1 = 1

m

[4 (pe

μ pνα + pν

μ peα − gμν pe · pν ) − Tr

(γ · peγ

μγ · pνγαγ 5

)](K � 1). (23)

But when the laser field is so intense that the Kibble parameter K � 1, the quiver energy ofthe electron is large compared to the electron’s rest mass energy. Conversely, we obtain thefollowing approximation for T1 after doing the required traces:

T1 = e2AA′

2m(n · pe)2{4(nμ pν

α + pνμnα − gμνn · pν )

−Tr[γ · peγ · εγ · nγ μγ · pνγαγ 5γ · nγ · ε]}, (K � 1). (24)

Similarly, using the trace theorems [16] and canceling the opposing terms, we reduce thesecond trace T2 of equation (21) to [14]

T2 = 4[PPμPNα

+ PPαPNμ

+ gμα(MPMN − PP · PN )] − λTr(γ · PPγμγ5γ · PNγα)

+ 4λ2[PPμPNα

+ PPαPNμ

− gμαPP · PN] − 4λ2MPMN gμβ gβμ. (25)

In the approximations for T1 and T2, we have ignored those traces, which due to their asymmetrywill give us zero for T = T1 · T2. Hence, for T = T1 · T2 there will arise the following productof the two traces:

− Tr(γ · peγ

μγ · pνγαγ 5

) · −λTr(γ · PPγμγ5γ · PNγα

)= −32λ [pe · PN pν · PP − pe · PP pν · PN] , (26)

1 The Kibble parameter K is defined as the square root of the ratio of the electron’s quiver energy Eq in the laserfield to its rest mass energy mc2, where the quiver energy is understood to be the electron’s average oscillating kineticenergy over one cycle.

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J. Phys. G: Nucl. Part. Phys. 39 (2012) 065103 S Rashid

which will survive. Finally, T of equation (19) will become (under the approximation K � 1)

T = 32

m

[(1 + λ + λ2)(pe · PP pν · PN ) + (1 − λ + λ2)(pν · PP pe · PN )

−(1 − λ2)(pe · pνPN · PP)]

, (K � 1). (27)

Also, we are assuming the neutron to be stationary and the proton to be non-relativistic so that

pe · PP ≈ EeMP; pν · PP ≈ EνMP; pe · PN ≈ EeMN;pν · PN ≈ EνMN; PN · PP ≈ MPMN; pe · pν ≈ EeEν . (28)

Hence,

T = T1 · T2 =(

32

m

)(EeEνMPMN )(1 + 3λ2). (29)

Collecting all the terms and factors, we obtain

1

2

∑∣∣S f i

∣∣2 = g2V

2

m

(2π)62EeEυ

1

(2π)68EpEn

(32

m

)(EeEνMPMN )

(1 + 3λ2

) ·

×∫

d4x∫

d4x ei{(pe−pν )·(x−x′ )+Se−S′e+(pp−pn )·(x−x′ )}. (30)

Using the theorem [14, 17]

eZ(s−1/s) =+∞∑

p=−∞spJp(2Z), (31)

where Jp is the Bessel function of order p, we obtain after doing the double 4-integrals inequation (30) the following expression:

1

2

∑|S f i|2 = g2

V

2

m

(2π)62EeEυ

1

(2π)68EpEn

(32

m

)(EeEνMPMN )(1 + 3λ2)·

+∞∑l1,l2,l3,l4=−∞

(2π)8δ4{p∗e + Pp − (pν + PN ) + l1k} · δ4{p∗

e + Pp

−(pν + PN ) + l2k} · Jl1−2l2 (eμ1) · Jl2 (e2μ2) · Jl3−2l4 (eμ1) · Jl4 (e

2μ2), (32)

where

μ1 = A0

k0

(ε · pe

n · pe

)(33)

and

μ0 = 2k0μ2 = A20

4

(1

n · pe

). (34)

These arguments of the Bessel functions are quite different from those previously calculatedby previous authors [13, 14] for processes where the lepton exists both before and after theinteraction. The first line of equation (32) is exactly the same as obtained for the nuclear β-decay of the neutron when no radiation is present. In the next section, we use the above-derivedexpression to find the decay rate of the neutron.

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J. Phys. G: Nucl. Part. Phys. 39 (2012) 065103 S Rashid

4. Decay rate of the neutron

Using expression (32) of the last section, we can calculate the decay rate for the neutron to be(where p∗

e is the quasi-4-momentum of the electron)

dω =∫

1

2

∑|S f i|2 · d3 p∗

ed3 pν

(2π)4δ4(0)· (2π)3δ3Pp = g2

V

(2π)12(1 + 3λ2)

∫ +∞∑−∞l1..l4

(2π)8δ4

×{p∗e + Pp − (pν + PN ) + l1k} · δ4{p∗

e + Pp − (pν + PN ) + l2k}Jl1−2l2 (eμ1)

·Jl2 (e2μ2) · Jl3−2l4 (eμ1) · Jl4 (e

2μ2) · d3 p∗ed3 pν

(2π)4δ4(0)· (2π)3d3Pp. (35)

After cancellations, the decay rate becomes

dω = g2V

(2π)5(1 + 3λ2)

∫ +∞∑l1,l2,l3,l4=−∞

δ4{p∗e + Pp − (pν + PN ) + l1k}Jl1−2l2 (eμ1)

·Jl2 (e2μ2) · Jl3−2l4 (eμ1) · Jl4 (e

2μ2) · d3 ped3 pνd3Pp. (36)

Performing the Pp integral, we obtain∫δ4{p∗

e + Pp − (pν + PN ) + l1k} d3Pp

=∫

δ3{p∗e + Pp − pν + l1k} · δ{MN − E∗

e − Mp − Eν − l1ω} d3Pp

= δ{Q − E∗e − Eν − l1ω}, (37)

where we have neglected the kinetic energy of the outgoing proton and defined the decayenergy Q as

Q = MN − Mp. (38)

In spherical polar coordinates, we have

d3 p∗ed3 pν = d e

(E∗2

e − m2e

) 12 E∗

e dE∗e · d νE2

ν dEν . (39)

Substituting (37), (38) and (39) back into equation (36) and doing the Eν integral with theδ − function, we obtain

dω = g2V

(2π)5(1 + 3λ2)

∫ +∞∑l1,l2,l3,l4=−∞

Jl1−2l2 (eμ1) · Jl2 (e2μ2)Jl3−2l4 (eμ1)Jl4 (e

2μ2)·

d e(E∗2e − m2

e )12 E∗

e · d ν(Q − E∗e )2dE∗

e . (40)

The types of laser which have high intensity have the following deficiencies if we try touse them to observe the change in the decay rate of the neutron: first, they are intense for a shortduration compared to the lifetime of the free neutron (∼ 900 s). Second, the interaction regionfor the decay is large compared to the laser intense region. Hence, we introduce approximationsin the above expression so that we can calculate the laser intensity effect on the decay ratewhich can be detected using the present laser systems. We consider the laser intensity to below enough so that e2μ2 ≈ 0. Moreover, if the Kibble parameter is very small (K � 1), thenthe summation in equation (40) reduces to a single term of the square of the Bessel function.The decay rate then becomes (assuming no photon is absorbed or emitted)

dω = g2V

(2π)5(1 + 3λ2)

∫ ∫ π

0(4π) · J2

0 (eμ1) · (E∗2e − m2

e )12

×E∗e (Q − E∗

e )2dE∗e (2π) sin θdθ, (41)

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J. Phys. G: Nucl. Part. Phys. 39 (2012) 065103 S Rashid

where

eμ1 = eA0

k0

( −pe cos θ

Ee − n · pe

). (42)

Expanding the Bessel function factor J20 (eμ1) into a series given by

J20 (u) =

∞∑k=o

(−1)k(2k)!

4k(k!)4u2k (43)

and retaining only the first two terms, we obtain after integration over θ

dω = g2V

2π3(1 + 3λ2)

∫ (E∗2

e − m2e

) 12 E∗

e

(Q − E∗

e

)2

·F(1, E∗

e

)[1 +

(eA0

k0

)2

· (E∗2e − m2

e )

6E∗e

2

]dE∗

e , (44)

where we have introduced the Fermi function F(1, E∗

e

)for the coulomb interaction [18]

between the final states of the electron and the proton. Defining new variables

ε = E∗e

me; ε0 = Q

me= 2.5312 (45)

and doing the integration over E∗e with the above substitutions, we obtain the following neutron

decay rate ω:

ω = g2V · me

2π3(1 + 3λ2) ·

[1.6369 − 0.1514

(eA0

k0

)2]. (46)

Substituting the values of all the constants into equation (46), we have

ω =(

1

972s−1

)[1 − 0.0925

(eA0

k0

)2]. (47)

Equation (47) gives us the final result for the β-decay rate of the neutron in the presence ofthe intense laser field of amplitude A0, where the Kibble parameter is very small (K � 1).

In the expression, we have introduced by approximation the 2% radiative correction to matchwith the experimental neutron lifetime.

5. Results

By inverting the expression for the neutron decay rate given by equation (47), we obtain theaverage neutron lifetime as

TnA = Tn0

[1

1 − 0.0925( eA0

k0

)2

], (48)

where TnA is the average lifetime of the neutron in the presence of the intense laser field andTn0 is the average lifetime of the neutron in the absence of the intense laser field. Figure 2shows how the lifetime TnA varies with the intensity of the laser field.

Recent measurements [19] show that the error in the measurement of the neutron’s lifetimeτn = 885.7 s is approximately 1 s. According to equation (48),

e2 A20

k20

� 0.012, (49)

so as to obtain a 0.11% increase in comparison to the non-altered lifetime needed to exceedthe cited lifetime uncertainty threshold. Hence, the corresponding level of the laser intensity

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J. Phys. G: Nucl. Part. Phys. 39 (2012) 065103 S Rashid

0 2 4 6 8 100

2

4

6

8

10

e2 A02

k02

TnA

Tn0

Figure 2. Variation of the average lifetime of the neutron as a function of laser intensity for thecase of small Kibble parameter.

would be sufficient for an experimental verification of the variation of neutron’s lifetime. Thelaser intensity in absolute terms is given by the equation

e A0

k0= 2eλ0

√I√

π c3· λ0

2π= 0.11, (50)

where I is the intensity of the laser field. This gives us the minimum laser intensity requiredin the neutron path as

I = 1 × 1010W cm−2, (51)

where we have assumed the laser field being produced by a CO2 laser at λ0 ∼ 10 μm. Suchan experimental verification should be readily feasible with the present day laser systems.

The neutron decay profile follows an exponential decay curve. When the laser is turnedon for �t seconds, the decay profile would still remain exponential for this duration but wouldfollow a longer lifetime (shallower) exponential decay curve. After the laser field is turned off,the decay profile would return back to the previous exponential decay curve but shifted to theright along the time axis.

6. Conclusion

We summarize that in the presence of an intense laser radiation, the lifetime of a free neutronis increased with higher intensity. This is due to the presence of a final charged lepton inthe interaction process. We predict that a change of lifetime of a free neutron would becomeobservable when the laser intensity is of the order 1010 W cm−2 which would put the lifetimeextension in single percentage points above the normal lifetime making it comparable to theerror in the measurement of neutron lifetime [19]. The characteristic non-linearity of thedescribed change however would only be revealed by testing our prediction in at least threepoints covering the intensity range of 1010 to 1012 W cm−2.

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J. Phys. G: Nucl. Part. Phys. 39 (2012) 065103 S Rashid

Acknowledgment

The author thanks Aleksej Mialitsin for discussions and technical assistance.

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