1

1 Theory of Decays

There could be no fairer destiny for any physical theory that that it should point the way to a more comprehensive theory in which it lives on as a limiting case

Albert Einstein

This section will review the underlying theory and consequences of the different pion decay channels.

Although this work is mostly interested in the π → lνl(γ) decays it will would incomplete without a small discussion of decays as it is the biggest background in the PEN experiment. Muon decays alone can also be used to search for Beyond the . It would also be an injustice if the pion was not mentioned since this work is the successor to the PiBeta experiment which looked at this particular channel in great detail.

+ + 1.1 Michel and Electronic Decay: π → l (γ)νl

All decay processes can be understood through Fermi’s gold rules, where the probability of a process to

2 transition from state i to f, Γfi is proportional to | h f |Hint|ii | ρ where ρ represents the density of states, a

2 kinematics factor that imposes and conservation, and | h f |Hint|i | is related to the inter- action itself. This term can be transformed into the square of a matrix element by normalizing the initial and final state wavefunctions in a manner that adheres to Lorentz invariance The differential decay rate for

π → lν is

1 1 d3 p d3 p dΓ = |M|2 l ν δ4(q − p − p ) 3 3 l ν 2mπ 2El2Eν (2π) (2π)

Where mπ is the pion , q, pl, and pν are the four momenta of the pion, , and , and M is the matrix element of the interaction. This matrix element would originate from the Lagrangian (or

Hamiltonian, whichever you prefer). The type of interaction which occurs determines the decay rate of a + + 1.1 Michel and Electronic Decay: π → l (γ)νl 2 particular process. To get a sense of the type of interactionss that could be involved one must go back to the

Fermi era where β− decays were first being studied in detail. By constructing the Lagrangian for the beta decay, it is possible to see what type of interactions occur.

−G h i F ¯ ¯ λ ¯ ¯ λ √ ψpγλψnψeγ ψνe + ψnγλψpψνe γ ψe 2

The expresson on the left in side the brackets represent the process n→pe¯νe, beta decay. The second expression is capture. This Lagrangian was constructed in analogy to electrodynamics.

However, Fermii, Gammow, and Teller [12] argued that a general method to construct the Lagranian is to have a linear combination of all different types of Lorentz invariant terms, and therefore a matrix element in general, would have the following form

G Z F ¯ ¯ M = √ Σi CiψpOiψnψlOiψν¯e 2

Where i is summed over all types of operators, scalar, pseudoscalar, vector, axial-vector (pseudo-vector) and tensor. Lee and Yang [16] were the first to propose that is not a conserved quantity which was confirmed by a Wu et al. [23]. Since parity conservation was no longer considered to be necessarily conserved, the lagrangian doesn’t need to be invariant under parity transformation. β− decays revealed that

νe are left handed andν ¯e are right handed. This meant that the scalar, pseudoscalar and tensor term had to be very small (or zero). So a matrix element with vector and axial vector interaction would be expected and generalized as 1 γ (C + C γ ) 2 λ V A 5

Further From Moller scattering, results had shown that β decays are longitudainally polarized with helicity

−(−v)/c for and +(−v)/c for , this means that CV = −CA, which leads to the expected interaction for weak decays as 1 γ (1 − γ ) 2 λ 5

It was pointed out by Feynman and Gell-Mann and independently by Sudarshan and Marshak [22] that proposed that all charged weak processes are described by an effective lagrangian that has a weak current, + + 1.1 Michel and Electronic Decay: π → l (γ)νl 3

Jµ that coupled to itself, giving a lagrangian of

1 GF µ µ L = − √ (Jµ J † +J † Jµ) 2 2

Now the weak current itself is a sum of two portions, Jµ = jlµ + jhµ, a leptonic portion which is the V−A interaction that has been mentioned, and the hadronic portion, which is similar to the V−A interaction, but acting on . For the pion decay, the hadronic portions takes the form

jhµ = D¯ Cγµ(1 − γ5)U

Once it was realized that an intermediate vector was required in order to maintain unitarity the full interaction can be realized igµν M = j j 2 2 πµ lν MW − q

Where the pion is the and the lepton will either be the muon or , and MW is the mass of the intermediating W boson. For the case of low momentum transer, that is much less than mass of the

W boson (80 GeV) ,the fermi theory can be recovered and the matrix element becomes

iG M = √ h0|V − A|πi u¯lγµ(1 − γ5)vνl 2

Since the first part of the matrix element invovles the pseudoscalar pion to the vacuum using an vector term, this term should vanish, describing a matrix element

−iG M = √ h0|A|πi u¯lγµ(1 − γ5)vνl 2

The first term is a little difficult to describe, but it should be a Lorentz four vector, and since the only four vector available is the momentum transfer qµ, one can write

µ h0|A|πi = i fπq

Which produces

−iG µ M = √ fπq u¯lγµ(1 − γ5)vνl 2 + + 1.1 Michel and Electronic Decay: π → l (γ)νl 4

After accounting for polarizations, the square of the matrix element is

2 2 2 2 |M| = 4G fπ ml pl pν

When all is said and done and the phase space factor (density of states) is accounted for (see appendix B), the rate of the decay for a particular channel is

G2 f 2m2 Γ = F π l (m2 − m2)2 3 π l 8πmπ

Here it seen that the the decay rate depends on the square of the lepton mass, in particular, by looking at the ratio of the decay rates, otherwise known as the branching ratio, is given as

Γ(π → eν) m2 (m2 − m2)2 = e π e ≈ 1.283 × 10−4 2 2 2 2 Γ(π → µν) mµ (mπ − mµ)

Notice, that the ratio of the squared appears in the branching ratio. This term is extremely small,

≈ 2.5 × 10−5 This very small term is a manifestation of the V − A of the . It is this term that so highly supresses the π → eν decay channel. A deeper dive into the would show that these types of decays prefer to pick out left-handed helicity states for and right-handed helicity states for anti-fermions. This is apparent in a term that appears in the feynman rules (matrix element calculation) as 1 − γ5, which is the helicity projection operator for massless . Now since the positron and muon are not massless, their helicities are mixed according their masses. This wouldn’t seem like such a problem since both the positron and neutrino are light ( and helicity become the same in the massless, high energy limit), however, consider a pion decaying at rest. In this case the essentially massless neutrino will decay as a left-handed (no problem so far). But, because has to be conserved and the pion is a spin 0 particle, then the positron must emerge in the opposite direction of the neutrino with a left-handed helicity state as well. This conflicts with the weak interaction which demands the positron to be right-handed. It can be shown that the probability of picking out a left handed state from this type of interaction is ∼ (1 − vl/c). Now when looking at the ratio again,

Γ(π → eν) 1 − v /c m2 ∼ e ∼ e 2 Γ(π → µν) 1 − vµ/c mµ + + 1.1 Michel and Electronic Decay: π → l (γ)νl 5

We see the origin of the term that supresses the signal channel by so much comes from the probability of picking the proper helicity of this interaction. Also note that if the positron were a , then the velocity emerging from the interaction would be c, and thus the branching ratio would be 0, (ie the

π → eν would never occur).Thus far, only tree level calculations have been discussed. The branching ratio is not complete unless radiative and loop corrections are accounted for. These corrections for a pion decay process will depend on the mass of the resulting lepton and they originate from the emission of a via Inner Bremsstrahlung, and the virtual emision and reabsorption of . Classically, Bremstrahulung radiation is emitted whenever a charged particle is accelerated or deceleratated (old school physicists call this breaking radiation). Whenever a charged particle scatters in a medium, this means the directon changes, ie the velocity changes. This means that acceleration has occurred and therefore a photon must be emitted.

Now a positron cann t radiate a photon unless it exchanges a soft photon with the nucleus that is near. Once the corrections are put into the branching ratio, the branching ratio becomes

Γ(π → eν) m2 (m2 − m2)2 = e π e (1 + δ)(1 + ) = 1.233 × 10−4 2 2 2 2 Γ(π → µν) mµ (mπ − mµ)

  where δ = −3α/π ln mµ/me and  = −.92α/π. Now in leading up to this ratio, we cheated. An assumption was made, often refered to as lepton universality. The idea behind this assumption originates from the coupling of the W boson to the muon or electron. The assumption was that the coupling is the same for these particles, (this is the standard model hypothesis as pointed out earlier) To include the possibility of violation of lepton universality and combining the radiative corrections, we obtain

 2 !2 !2 2 2 Γ(π → eν) g m 1 − me/mµ RSM = = e e (1 + δR) πe2 2 Γ(π → µν) gµ mµ  2 2 1 − me/mπ where here the radiative corrections and loop corrections have been combined into δR, and lepton universal- ity holds if the ratio of the coupling between the W boson and the two leptons are the same.This implies the constants ge and gµ are the same and therefore lepton universality holds if this ratio is 1. According to the + + 1.2 Muon Decay and Radiative Muon Decay: µ → e νeν¯µ(γ) 6 recent standard model calculations of the branching ratio, which assumes lepton universality

  −4 (1.2352 ± 0.0005) × 10 [17]   RSM =  −4 πe2 (1.2354 ± 0.0002) × 10 [11]    − (1.2352 ± 0.0001) × 10 4 [6]

Now as mentioned, there is no distinction between the radiative and “non-radiative” events, the decay in question has a photon but the photon is mostly infrared and therefore undetectable. However, the radiative decay does allow some interesting physics to be probed which will be reviewed later on. For a more in depth treatment on radiative decay, the works of Michael Vitz, Pete Alonzi, and Max Bychkov are available

+ + 1.2 Muon Decay and Radiative Muon Decay: µ → e νeν¯µ(γ)

Historically, muon decay, sometimes called Michel decay, has played an important role in verifying the

V−A relationship in the Standard Model. Understanding the muon decay will be important for this work because the main background of the experiment will involve muon decays. The muon decay is a pure leptonic weak decay, that is there is no structure of the parent or daughter particle that must be accounted for when calculating decay rates. Starting with the Matrix element Where

GF λ M = √ v¯νγλ(1 − γ5)¯veγ (1 − γ5)uν 2

Following similar steps to the pion decay. 2 5 G mµ Γ = F 192π3

Now once the decay rate, and therefore the lifetime (1/Γ)is known, arguably that is all one would need to know for the main purpose of the experiment. However, a pion decay at rest is essentially a muon birth since this is the predominant decay channel for the pion. And muon decay physics may also be used to study physics beyond the standard model. Including all possible interactions that may occur due to Lorentz + + 1.2 Muon Decay and Radiative Muon Decay: µ → e νeν¯µ(γ) 7 invariant, a general matrix element for muon decay may be written as

GF 0 M = √ Σu¯eOiuµu¯2Oi(Ci + Ci γ5)v1 2

Using the general interaction, instead of a simple V−A, a more general decay rate is computed

2 5 " #! G mµ D 4 4 dΓ = F 2 12(1 − ) + ρ(8 − 6) ∓ P ξ cos θ 4(1 −  + δ(8 − 6) dd cos θ/2 192π3 16 3 µ 3

Once this is done, the may be defined as

2 2 2 ρ = (3gA + 3gV + 6gT )/D

2 2 2 2 η = (gS − gP + 2gA − 2gV )/D

2 ξ = (6gS gP cos φSP − 8gAgV cos φAV + 14gT cos φTT )/D

2 δ = (−6gAgV cos φAV + 6gT cos φTT )/D where

2 2 2 2 2 D = gS + gP + 4gv + 6gT j + 4gA

2 2 0 2 gi = |Ci| + |Ci | and

∗ 0 0 ∗ cos φi j = Re(Ci C j + CiC j )

0 0 0 For an amplitude withon only V and A components that is Cs = Cs = CT = CT = CP = CP = 0, and in particular a V− A law, the standard model Michel parameters takes the the form:

2 D = 16|CV |

3 ρ = δ = 4

η = 0 + + 1.3 Radiative Pion: π → l γνl 8

ξ = −1

These parameters may be tested experimentally, in particular η and ρ may be measured by the momentum spectrum of electrons in the decay of unpolarized . ξ and δ may be measured in the asymmetry in the decay of polarized muons, where a departure of ξ, for example, would signify the existence of a right- handed (V+A) current.

In addition to muon decay, radiative muons can probe standard model expectations. For this type of decay, a new parameter,η ¯, emerges when the decay of a muon is accompanied by a hard photon. The defini-

1 tion ofη ¯ = A (2gT +gP +gS ), where A is a parameter that is measured in experiment. Measuring the value of η¯ is equivalanet to a measurement of the transverse polarization of the electron in the muon decay. Standard

Model expectations suggestη ¯ = 0 Deviations fromη ¯ would suggest the presence of non V−A physics in weak interactions.

+ + 1.3 Radiative Pion: π → l γνl

+ + + + It was stated that the only difference between π → l γνl and π → l (γ)νl is simply the parenthesis. The parenthesis is used to indicate that the photon emitted is a soft (low energy) photon. Once again, there is no such process as a non radiative pion electronic decay and that most of the decays emit photons that are just too low of energy to be detected. However, for those that are high enough to be detected, there is something new that can be learned. The Feynman diagrams which described this decay mode was too simplified. The positron emitting a photon is indicative of Inner Bremsstrahulung process. This type of process is described in any quantum field theory book see Peskin [21]. The pion is also a charged particle, so treating it as a point particle will produce a Pion Bremsstrahulung process. However, the pion is in fact a composite particle composed of charge quarks. Which means a photon can emanate from the structure of the pion which brings about structure dependent terms in the amplitude of this process. In particular it is governed by a vector ad an axial vector form factor. The photon is emitted from intermediate states that emerge from strong interactions, so hadronic states. The matrix element contribution can therefore be broken into the structure dependent and inner bremsstrahlung contributions, where the inner bremsstrahlung differential + 0 + 1.4 PiBeta: π → π + e νe 9 decay rate is proportional to the the “non-radiative” decay rate. This means that this particular contribution is helicity supressed therefore the structure dependent term is accessible and may give information on the structure of the pion.

ν γ e u W+ π+{ V A d e+

In writing the total differential decay rate of this process as a sum of the structure depednent and inner bremsstrahulung, it can be shown that

 !2  d2Γ α  1 m2 FV h i FV  = Γ IB(x, y) + π (1 + γ)2SD+(x, y) + (1 − γ)2SD−(x, y) + (1 + γ)F(x, y) + (1 − γ)G(x, y) π→eν  2  dxdy 2π 4 me fπ fπ

Where F(x, y), G(x, y), SD+(x, y), and SD−(x, y) are functions that depend on the kinematic terms x and y

FA with x = 2Eγ/mπ and y = 2Ee/mπ and γ = FV . In short, the pion structure form factors may be probed with radiative pion decay.

+ 0 + 1.4 PiBeta: π → π + e νe

THe pion beta decay is a vector transition between two spin 0 particles, the π+ and the π0 whcih of course belong to the same isospin triplet (π+, π0,π−). What a vector transition means is that the V component of the hadronic current will give a nonzero matrix element for this decay. Axial vector contributions vanish here due to the intrinsic parity of the two particles and their spin. The matrix element for the transition is given by

0 µ 5 M = iGF cos θc(p + p )µu¯νγ (1 − γ )ve

Where p is the 4-momentum of the π+ and p0 is the 4-momentum of the π0. Now since the π0 is produced approximately at rest. The matrix element can be written as

0 5 M ∼ 2iGF cos θCmπu¯νγ (1 − γ )νe [14] + 0 + 1.4 PiBeta: π → π + e νe 10

√ Using chiral basis, the leptonic contrubitons ammounts to −4 EνEe cos(θc/2) which ultimately leads to

2 2 2 2 |M| = 32GF cos θcmπEνEe(1 + cos θeν

Once the phase space factors are included the decay rate is given by

2 2 !3 G |Vud| ∆ Γ = F 1 − ∆5 f (, ∆)(1 + γ) 3 30π 2mπ+

2 me  Where ∆ = m + − m 0 ,  = π π ∆   √   √  9 15 1 + 1 −   3 ∆2  and f = 1 −  1 −  − 42 + 2 ln  √  −     2  2 2  7 (mπ+ + mπ0 )

This decay rate provides a test for CKM unitarity This decay rate can also provide a check on the conserved vector current hypothesis.For a more detailed discussion on the theory behind the pion beta decay, see [15], REFERENCES 11

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