Calculating Branching Ratio of W
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Yuldashev Jasur Oraliq
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Calculating Branching Ratio of W − boson decay Yuldashev Jasurbek December 5, 2020 Deriving decay width for two body decay Particle differencial decay width for the particle with arbitrary spin: dΓ = 1
a + 1
1 2E a |M | 2 dΦ (1) where J
a is the spin of decaying particle and dΦ is phase space: dΦ = (2π) 4 δ 4 (p f − p i ) n Y l=1 d 3 p i (2π)
3 2E l (2) For the two body decay, we need to calculate two-body phase space. Γ = 1
a + 1
2π 4 2E a |M |
2 Z δ(E i − E
1 − E
2 )δ 3 (~ p i − ~ p 1 − ~ p 2 ) d 3 p 1 (2π) 3 2E 1 d 3 p 2 (2π)
3 2E 2 (3) Because the integral is Lorentz invariant (i.e. frame independent) it can be evaluated in any frame we choose. The C.o.M. frame is most convenient. In the C.o.M frame E i = E a and
p i = 0 Γ = 1 2J a + 1
1 8π 2 E a |M | 2 Z δ(E a − E
1 − E
2 )δ 3 (~ p 1 + ~ p 2 ) d 3 p 1 2E 1 d 3 p 2 2E 2 (4)
Integrating over ~ p 2 using the δ - function gives following result Z δ 3 (~ p 1 + ~
p 2 )dp 3 1 = 1 ⇒ Γ = 1 2J a + 1
1 8π 2 E a |M | 2 Z δ(E a − E
1 − E
2 ) d 3 p 1 4E 1 E 2 (5)
Writing d 3 p 1 = p
2 1 dp 1 sin(θ)dθdφ = p 2 1
1 dΩ and observing that E 2 2
2 1 + m 2 2 because p 2 = −p 1 , the last equation can be expressed as Γ = 1
a + 1
1 32π
2 E a |M | 2 Z δ(E a − q p 2 1 + m
2 2 − q p 2 1 + m
2 1 ) p 2 1 dp 1 dΩ E 1 E 2 (6)
1 By using following denotations: g(p
1 ) =
p 2 1 E 1 E 2 f (p
1 ) = E
a − q p 2 1 + m 2 2 − q p 2 1 + m 2 1 we rewrite Eq.6 as follows Γ = 1 2J a + 1
1 32π
2 E a |M | 2 Z g(p 1 )δ(f (p 1 ))dp
1 dΩ (7) where f (p 1 ) imposes energy conservation and g(p 1 ) = 0 determines the C.o.M momenta of the two decay products. We need an expression for the delta function of a function δ(f (x)) Z x
x 1 δ(f (x)) df dx dx = df dx x 0 Z x 2 x 1 δ(f (x))dx = ( 1 if x 1 < x 0
2 0
(8) By rearranging and expressing the RHS of the Eq.7 as a delta function one gets following expression: Z x 2 x 1 δ(f (x))dx = 1 |df /dx| x 0 Z x 2 x 1 δ(x − x
0 )dx
⇒ δ(f (x)) = df dx
x 0 δ(x − x 0 ) (9) Using Eq.9 g(p 1 ) and δ(f (p 1 )) can be integrated Z g(p
1 )δ(f (p
1 ))dp
1 = 1 |df /dp 1 | p ∗ Z g(p 1 )δ(p 1 − p
∗ )dp
1 = g(p ∗ 1 ) |df /dp 1 | p ∗ df dp 1 = − p 1 pp 2 1 + m 2 1 − p 1 pp 2 1 + m 2 2 = − p 1 E 1 − p 1 E 2 = −p 1 E 1 + E
2 E 1 E 2 (10) Finally we take following expression Γ =
1 2J a + 1 1 32π 2 E a |M | 2 Z E 1 E 2 p ∗ (E 1 + E 2 ) p ∗2 E 1 E 2 dΩ Γ = 1 2J a + 1
1 8πE
a |M |
2 p ∗ (E 1 + E 2 ) (11) From energy conservation E 1 + E 2 = E
a = m
a and f (p
∗ ) = 0
m a = q p ∗2 + m 2 2 − q p ∗2 + m
2 1 p ∗ = 1 2m a p [m 2 a − (m 1 + m 2 ) 2 ][m 2 a − (m 1 − m 2 ) 2 ] 2
Partial and Total width of W boson For the W boson decay masses of the final states (m 1 and m
2 ) can be neglacted campared to W boson mass (m a ) and taking into account J a for W bosons Eq.11 can be written as follows Γ = 1
a + 1
1 64π
2 m a |M | 2 Z dΩ = 1 48πm a |M |
2 (12)
|M | 2 = 1 g 2 − g
αβ + k α k β m 2 W ! 1 4 T r ˆ p e γ α ˆ p ν γ β (1 + γ
5 ) 2 = 1 2 g 2 4(p e p ν ) = g 2 m 2 W (13) By using g 2 2m 2 W = 4G √ 2 , one can write decay width as follows Γ(W −
− ν l ) = g 2 m W 48π = Gm 3 W 6 √ 2π (14)
There are 3 poossible lepton channels e − ν e , µν
µ , τ ν
τ but only 2 quark channels u ¯ d 0
s 0 .The quark channels also involve the color quantum number N C = 3 and the mixing factor d 0 i = V ij d j relating weak and mass eigenstates. Γ(W
− → ¯
u i d j ) = N
C |V ij | 2 Gm 3 W 6 √ 2π = N C 2 Gm 3 W 6 √ 2π (15) Here the unitarity of the CKM matrix is used that is |V ud | 2 +|V
us | 2 +|V ub | 2 = |V
cd | 2 +|V cs | 2 +|V
cb | 2 = 1 Branching ratio is ratio between the partial and the total width . The total width for W decay:
Γ W = 3Γ(W − → l
− ν l ) + Γ(W − → ¯ u i d j ) From the last two equations Br(W − → l − ν l ) = Γ(W
− → l
− ν l ) Γ W = Gm 3 W 6 √ 2π 3 Gm 3 W 6 √ 2π + N C 2 Gm 3 W 6 √ 2π = 1 3 + 2N
C = 1 9 = 0.111
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