1. 𝐻𝑖𝑔𝑔𝑠 𝑑𝑒𝑐𝑎𝑦 𝑡𝑜 𝑡𝑤𝑜 𝑔𝑙𝑢𝑜𝑛𝑠
𝐸𝑙𝑒𝑓𝑡ℎ𝑒𝑟𝑖𝑜𝑠 𝑀𝑜𝑠𝑐ℎ𝑎𝑛𝑑𝑟𝑒𝑜𝑢

2. The decay rate
𝑑Γ= 1 2 𝑚 ℎ 𝑑 3 𝑝 2𝜋 𝐸 𝑝 ⋅ 𝑑 3 𝑘 2𝜋 𝐸 𝑘 𝑀 𝑚 ℎ →𝑝,𝑘 𝜋 4 𝛿 4 ( 𝑚 ℎ −𝑝−𝑘)

3. The decay rate
𝑑Γ= 1 2 𝑚 ℎ 𝑑 3 𝑝 2𝜋 𝐸 𝑝 ⋅ 𝑑 3 𝑘 2𝜋 𝐸 𝑘 𝑀 𝑚 ℎ →𝑝,𝑘 𝜋 4 𝛿 4 ( 𝑚 ℎ −𝑝−𝑘)

4. Lorentz Invariant Phase Space
The decay rate
𝑑Γ= 1 2 𝑚 ℎ 𝑑 3 𝑝 2𝜋 𝐸 𝑝 ⋅ 𝑑 3 𝑘 2𝜋 𝐸 𝑘 𝑀 𝑚 ℎ →𝑝,𝑘 𝜋 4 𝛿 4 ( 𝑚 ℎ −𝑝−𝑘)
Lorentz Invariant Phase Space

5. The decay rate
𝑑Γ= 1 2 𝑚 ℎ 𝑑 3 𝑝 2𝜋 𝐸 𝑝 ⋅ 𝑑 3 𝑘 2𝜋 𝐸 𝑘 𝑀 𝑚 ℎ →𝑝,𝑘 𝜋 4 𝛿 4 ( 𝑚 ℎ −𝑝−𝑘)

6. Decay amplitude. Describes the interactions during the decay.
The decay rate
𝑑Γ= 1 2 𝑚 ℎ 𝑑 3 𝑝 2𝜋 𝐸 𝑝 ⋅ 𝑑 3 𝑘 2𝜋 𝐸 𝑘 𝑀 𝑚 ℎ →𝑝,𝑘 𝜋 4 𝛿 4 ( 𝑚 ℎ −𝑝−𝑘)
Decay amplitude. Describes the interactions during the decay.

7. The Feynman diagrams

8. Gluons are massless. They don’t interact with the Higgs!
The Feynman diagrams
Gluons are massless. They don’t interact with the Higgs!

9. Interaction occurs through a fermion (quark) loop
The Feynman diagrams
Interaction occurs through a fermion (quark) loop

10. From the Feynman diagram to the decay amplitude𝑀=

11. From the Feynman diagram to the decay amplitude
𝑀=−𝑖 𝑚 𝑞 𝜐 (−1)

12. From the Feynman diagram to the decay amplitude
l + p l
l - k
𝑀=−𝑖 𝑚 𝑞 𝜐 − 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀

13. From the Feynman diagram to the decay amplitude
l + p l
l - k
𝑀=−𝑖 𝑚 𝑞 𝜐 − 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎

14. From the Feynman diagram to the decay amplitude
l + p l
l - k
𝑀=−𝑖 𝑚 𝑞 𝜐 − 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝑙 2 − 𝑚 𝑞 +𝑖𝜀

15. From the Feynman diagram to the decay amplitude
l + p l
l - k
𝑀=−𝑖 𝑚 𝑞 𝜐 − 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝑙 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏

16. From the Feynman diagram to the decay amplitude
l + p l
l - k
𝑀=−𝑖 𝑚 𝑞 𝜐 − 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝑙 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏 ⋅ 𝑖 𝑙+𝑝+ 𝑚 𝑞 𝑙+𝑝 2 − 𝑚 𝑞 +𝑖𝜀

17. From the Feynman diagram to the decay amplitude
l + p l
l - k
𝑀=−𝑖 𝑚 𝑞 𝜐 − 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝑙 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏 ⋅ 𝑖 𝑙+𝑝+ 𝑚 𝑞 𝑙+𝑝 2 − 𝑚 𝑞 +𝑖𝜀
×𝜖 𝜈 ∗𝜆 ′ 𝑘

18. From the Feynman diagram to the decay amplitude
l + p l
l - k
𝑀=−𝑖 𝑚 𝑞 𝜐 − 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝑙 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏 ⋅ 𝑖 𝑙+𝑝+ 𝑚 𝑞 𝑙+𝑝 2 − 𝑚 𝑞 +𝑖𝜀
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝

19. From the Feynman diagram to the decay amplitude
l + p l
l - k
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 ∫ 𝒅 𝟒 𝒍 𝟐𝝅 𝟒 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝑙 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏 ⋅ 𝑖 𝑙+𝑝+ 𝑚 𝑞 𝑙+𝑝 2 − 𝑚 𝑞 +𝑖𝜀
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝

20. From the Feynman diagram to the decay amplitude
l + p l
l - k
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 ∫ 𝒅 𝟒 𝒍 𝟐𝝅 𝟒 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝑙 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏 ⋅ 𝑖 𝑙+𝑝+ 𝑚 𝑞 𝑙+𝑝 2 − 𝑚 𝑞 +𝑖𝜀
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝

21. Calculating the decay amplitude
The numerator:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 ∫ 𝑑 4 𝑙 2𝜋 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝑙 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏 ⋅ 𝑖 𝑙+𝑝+ 𝑚 𝑞 𝑙+𝑝 2 − 𝑚 𝑞 +𝑖𝜀
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝

22. Calculating the decay amplitude
The numerator:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 ∫ 𝑑 4 𝑙 2𝜋 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝑙 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏 ⋅ 𝑖 𝑙+𝑝+ 𝑚 𝑞 𝑙+𝑝 2 − 𝑚 𝑞 +𝑖𝜀
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝
Contains 3×2×3=18 terms

23. Calculating the decay amplitude
The numerator:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 ∫ 𝑑 4 𝑙 2𝜋 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝑙−𝑘 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝑙 2 − 𝑚 𝑞 +𝑖𝜀 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏 ⋅ 𝑖 𝑙+𝑝+ 𝑚 𝑞 𝑙+𝑝 2 − 𝑚 𝑞 +𝑖𝜀
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝
Numerator algebra (trace technology)
Tr[𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝛾 ′ 𝑠 ] = 0
𝛾 𝜇 , 𝛾 𝜈 =4 𝑔 𝜇𝜈
𝑇𝑟 𝛾 𝜇 𝛾 𝜈 =2 𝑔 𝜇𝜈
𝑇𝑟 𝛾 𝜇 𝛾 𝜈 𝛾 𝜌 𝛾 𝜎 =4( 𝑔 𝜇𝜈 𝑔 𝜌𝜎 − 𝑔 𝜇𝜌 𝑔 𝜈𝜎 + 𝑔 𝜇𝜎 𝑔 𝜈𝜌 )

24. Calculating the decay amplitude
The denominator:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 ∫ 𝑑 4 𝑙 2𝜋 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝒍−𝒌 𝟐 − 𝒎 𝒒 +𝒊𝜺 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝒍 𝟐 − 𝒎 𝒒 +𝒊𝜺 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏 ⋅ 𝑖 𝑙+𝑝+ 𝑚 𝑞 𝒍+𝒑 𝟐 − 𝒎 𝒒 +𝒊𝜺
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝

25. Calculating the decay amplitude
The denominator:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 ∫ 𝑑 4 𝑙 2𝜋 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝒍−𝒌 𝟐 − 𝒎 𝒒 +𝒊𝜺 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝒍 𝟐 − 𝒎 𝒒 +𝒊𝜺 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏 ⋅ 𝑖 𝑙+𝑝+ 𝑚 𝑞 𝒍+𝒑 𝟐 − 𝒎 𝒒 +𝒊𝜺
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝
1 𝐴𝐵𝐶 = 1 𝒍−𝒌 𝟐 − 𝒎 𝒒 +𝒊𝜺 𝒍 𝟐 − 𝒎 𝒒 +𝒊𝜺 𝒍+𝒑 𝟐 − 𝒎 𝒒 +𝒊𝜺

26. Calculating the decay amplitude
The denominator:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 ∫ 𝑑 4 𝑙 2𝜋 𝑇𝑟 𝑖 𝑙−𝑘+ 𝑚 𝑞 𝒍−𝒌 𝟐 − 𝒎 𝒒 +𝒊𝜺 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑎 ⋅ 𝑖 𝑙+ 𝑚 𝑞 𝒍 𝟐 − 𝒎 𝒒 +𝒊𝜺 ⋅𝑖 𝑔 𝑠 𝛾 𝜈 𝑡 𝑏 ⋅ 𝑖 𝑙+𝑝+ 𝑚 𝑞 𝒍+𝒑 𝟐 − 𝒎 𝒒 +𝒊𝜺
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝
1 𝐴𝐵𝐶 = 1 𝒍−𝒌 𝟐 − 𝒎 𝒒 +𝒊𝜺 𝒍 𝟐 − 𝒎 𝒒 +𝒊𝜺 𝒍+𝒑 𝟐 − 𝒎 𝒒 +𝒊𝜺
We would like it to be in the form
1 𝐴𝐵𝐶 = 𝑙+𝛽 2 −Δ) 3

27. Feynman Parameterization
0 1 𝑑𝑥 1 𝑥𝐴+ 1−𝑥 𝐵 2 = 1 𝐴𝐵
observe that:
𝑑𝑥𝑑𝑦 𝑑𝑧 2𝛿(𝑥+𝑦+𝑧−1) 𝑥𝐴+𝑦𝐵+𝑧𝐶 3 = 1 𝐴𝐵𝐶
This generalizes to:

28. Feynman Parameterization
1 𝐴𝐵𝐶 = 1 𝒍−𝒌 𝟐 − 𝒎 𝒒 +𝒊𝜺 𝒍 𝟐 − 𝒎 𝒒 +𝒊𝜺 𝒍+𝒑 𝟐 − 𝒎 𝒒 +𝒊𝜺
1 𝐴𝐵𝐶 = 0 1 𝑑𝑥 0 1 𝑑𝑦 0 1 𝑑𝑧 2 𝛿(𝑥+𝑦+𝑧−1) 𝒍−𝒌 𝟐 − 𝒎 𝒒 +𝒊𝜺 𝑥+ 𝒍 𝟐 − 𝒎 𝒒 +𝒊𝜺 𝑦+ 𝒍+𝒑 𝟐 − 𝒎 𝒒 +𝒊𝜺 𝑧 3
1 𝐴𝐵𝐶 = 0 1 𝑑𝑥 0 1−𝑥 𝑑𝑧 [𝒍+ 𝒛𝒑−𝒙𝒌 ] 2 +2𝑥𝑧𝑘𝑝− 𝑚 𝑞 2 +𝑖𝜀 3 ℓ Δ

29. The loop integral becomes:
Feynman Parameterization
The loop integral becomes:
𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 { 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫 3 +
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3 }

30. The loop integral becomes:
Feynman Parameterization
The loop integral becomes:
𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 { 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫 3 +
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝜟 3 }

31. The loop integral becomes:
Dimensional Regularization
The loop integral becomes:
𝑑 4 ℓ 2𝜋 4 ⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 ℓ 2 +𝛥 3 → 𝑑 𝑑 ℓ 2𝜋 𝑑 ⋅ 𝟒 𝒅 −1 ℓ 𝟐 𝒈 𝝁𝝂 ℓ 2 +𝛥 𝑑→4

32. 𝑑 𝑑 ℓ: 𝑑−𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝐸𝑢𝑘𝑙𝑒𝑑𝑒𝑎𝑛 𝑠𝑝𝑎𝑐𝑒
Dimensional Regularization
The loop integral becomes:
𝑑 4 ℓ 2𝜋 4 ⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 ℓ 2 +𝛥 3 → 𝑑 𝑑 ℓ 2𝜋 𝑑 ⋅ 𝟒 𝒅 −1 ℓ 𝟐 𝒈 𝝁𝝂 ℓ 2 +𝛥 𝑑→4
Wick Rotation ℓ 𝐸 0 →𝑖 ℓ 0 :
𝑑 𝑑 ℓ: 𝑑−𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝐸𝑢𝑘𝑙𝑒𝑑𝑒𝑎𝑛 𝑠𝑝𝑎𝑐𝑒

33. 𝑑 𝑑 ℓ: 𝑑−𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝐸𝑢𝑘𝑙𝑒𝑑𝑒𝑎𝑛 𝑠𝑝𝑎𝑐𝑒
Dimensional Regularization
The loop integral becomes:
𝑑 4 ℓ 2𝜋 4 ⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 ℓ 2 +𝛥 3 → 𝑑 𝑑 ℓ 2𝜋 𝑑 ⋅ 𝟒 𝒅 −1 ℓ 𝟐 𝒈 𝝁𝝂 ℓ 2 +𝛥 𝑑→4
Wick Rotation ℓ 𝐸 0 →𝑖 ℓ 0 :
𝑑 𝑑 ℓ: 𝑑−𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝐸𝑢𝑘𝑙𝑒𝑑𝑒𝑎𝑛 𝑠𝑝𝑎𝑐𝑒
→𝑖 4 d −1 g μν d d ℓ E 2𝜋 𝑑 ⋅ ℓ 𝑬 𝟐 ℓ 𝐸 2 −𝛥 𝑑→4 =𝑖 4 d −1 g μν 𝑑 Ω 𝑑 d ℓ E 2𝜋 𝑑 ⋅ ℓ E 𝒅−𝟏 ℓ 𝑬 𝟐 ℓ 𝐸 2 −𝛥 𝑑→4

34. 𝑑 𝑑 ℓ: 𝑑−𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝐸𝑢𝑘𝑙𝑒𝑑𝑒𝑎𝑛 𝑠𝑝𝑎𝑐𝑒
Dimensional Regularization
The loop integral becomes:
𝑑 4 ℓ 2𝜋 4 ⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 ℓ 2 +𝛥 3 → 𝑑 𝑑 ℓ 2𝜋 𝑑 ⋅ 𝟒 𝒅 −1 ℓ 𝟐 𝒈 𝝁𝝂 ℓ 2 +𝛥 𝑑→4
Wick Rotation ℓ 𝐸 0 →𝑖 ℓ 0 :
𝑑 𝑑 ℓ: 𝑑−𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 𝐸𝑢𝑘𝑙𝑒𝑑𝑒𝑎𝑛 𝑠𝑝𝑎𝑐𝑒
→𝑖 4 d −1 g μν d d ℓ E 2𝜋 𝑑 ⋅ ℓ 𝑬 𝟐 ℓ 𝐸 2 −𝛥 𝑑→4 =𝑖 4 d −1 g μν 𝑑 Ω 𝑑 d ℓ E 2𝜋 𝑑 ⋅ ℓ E 𝒅−𝟏 ℓ 𝑬 𝟐 ℓ 𝐸 2 −𝛥 𝑑→4
𝑑 4 ℓ 2𝜋 4 ⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 ℓ 2 +𝛥 3 = 𝑖 16 𝜋 2

35. The loop integral : Calculation of M
𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 { 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫 3 +
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3 }

36. The loop integral : Is part of M: Calculation of M
𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 { 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫 3 +
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3 }
Is part of M:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝

37. The loop integral : Is part of M: Calculation of M
𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 { 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫 3 +
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3 }
Is part of M:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝

38. The loop integral : Is part of M: Calculation of M
𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 { 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫 3 +
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3 }
Is part of M:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝

39. The loop integral : Is part of M: Calculation of M Many terms vanish!
𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 { 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫 3 +
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3 }
Is part of M:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝
𝑝 𝜇 𝜖 𝜇 𝜆 =0 and 𝑘 𝜇 𝜖 𝜇 𝜆 ′ =0
Many terms vanish!

40. The loop integral : Is part of M: Calculation of M Many terms vanish!
𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 { 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫 3 +
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3 }
Is part of M:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 𝟒 𝒛 𝟐 −𝟐𝒛 𝒑 𝝂 𝒑 𝝁 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝝐 𝝁 ∗𝝀 𝒑
Many terms vanish!

41. The loop integral : Is part of M: Calculation of M Many terms vanish!
𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 { 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 4 𝑧 2 −2𝑧 𝑝 𝜈 𝑝 𝜇 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫 3 +
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3 }
Is part of M:
𝑀=−𝑖 𝑚 𝑞 𝜐 −1 𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 𝒅 𝟒 ℓ 2𝜋 4 ⋅𝑖 𝑔 𝑠 2 ⋅4 𝑚 𝑞 2⋅ 𝟒 ℓ 𝝁 ℓ 𝝂 − ℓ 𝟐 𝒈 𝝁𝝂 + 𝟒 𝒛 𝟐 −𝟐𝒛 𝒑 𝝂 𝒑 𝝁 + 4 𝑥 2 −2𝑥 𝑘 𝜈 𝑘 𝜇 ℓ 𝟐 +𝚫
2⋅ 1−4𝑥𝑧 𝑝 𝜈 𝑘 𝜇 + 2𝑥+2𝑧−1−4𝑧𝑥 𝑘 𝜈 𝑝 𝜇 + 𝑚 𝑞 2 − 1 2 𝑚 ℎ 2 +𝑥𝑧⋅ 𝑚 ℎ 2 𝑔 𝜇𝜈 ℓ 𝟐 +𝚫 3
×𝜖 𝜈 ∗𝜆 ′ 𝑘 𝝐 𝝁 ∗𝝀 𝒑
Many terms vanish!

42. Final expression for M
𝑀 𝑞 𝑎𝑏𝜆 𝜆 ′ =−𝑒⋅ 𝑔 𝑠 2 8 𝜋 2 𝑚 𝑊 𝑠𝑖𝑛 𝜃 𝑊 ⋅ 𝛿 𝑎𝑏 ⋅ 𝑚 ℎ 2 ⋅ 𝑔 𝜇𝜈 − 𝑝 𝜈 𝑘 𝜇 ⋅𝐼 𝑚 ℎ 2 𝑚 𝑞 2 ⋅ 𝜖 𝜈 ∗𝜆 ′ 𝑘 𝜖 𝜇 ∗𝜆 𝑝

43. Final expression for M
𝑀 𝑞 𝑎𝑏𝝀 𝝀 ′ =−𝑒⋅ 𝑔 𝑠 2 8 𝜋 2 𝑚 𝑊 𝑠𝑖𝑛 𝜃 𝑊 ⋅ 𝛿 𝑎𝑏 ⋅ 𝑚 ℎ 2 ⋅ 𝑔 𝜇𝜈 − 𝑝 𝜈 𝑘 𝜇 ⋅𝐼 𝑚 ℎ 2 𝑚 𝑞 2 ⋅ 𝜖 𝜈 ∗ 𝝀 ′ 𝑘 𝜖 𝜇 ∗𝝀 𝑝
For a specific polarization 𝝀 ′ of the k gluon and 𝝀 of the p gluon

44. Final expression for M
𝑀 𝑞 𝒂𝒃𝜆 𝜆 ′ =−𝑒⋅ 𝑔 𝑠 2 8 𝜋 2 𝑚 𝑊 𝑠𝑖𝑛 𝜃 𝑊 ⋅ 𝛿 𝒂𝒃 ⋅ 𝑚 ℎ 2 ⋅ 𝑔 𝜇𝜈 − 𝑝 𝜈 𝑘 𝜇 ⋅𝐼 𝑚 ℎ 2 𝑚 𝑞 2 ⋅ 𝜖 𝜈 ∗ 𝝀 ′ 𝑘 𝜖 𝜇 ∗𝝀 𝑝
For a specific polarization 𝜆 ′ of the k gluon and 𝜆 of the p gluon
For specific gluon colors a and b

45. Final expression for M
𝑀 𝒒 𝑎𝑏𝜆 𝜆 ′ =−𝑒⋅ 𝑔 𝑠 2 8 𝜋 2 𝑚 𝑊 𝑠𝑖𝑛 𝜃 𝑊 ⋅ 𝛿 𝑎𝑏 ⋅ 𝑚 ℎ 2 ⋅ 𝑔 𝜇𝜈 − 𝑝 𝜈 𝑘 𝜇 ⋅𝐼 𝑚 ℎ 2 𝑚 𝑞 2 ⋅ 𝜖 𝜈 ∗ 𝝀 ′ 𝑘 𝜖 𝜇 ∗𝝀 𝑝
For a specific polarization 𝜆 ′ of the k gluon and 𝜆 of the p gluon
For specific gluon colors a and b
For a specific quark q

46. For the decay rate Γ Remembering the decay rate:
𝑑Γ= 1 2 𝑚 ℎ 𝑑 3 𝑝 2𝜋 𝐸 𝑝 ⋅ 𝑑 3 𝑘 2𝜋 𝐸 𝑘 𝑴 𝒎 𝒉 →𝒑,𝒌 𝟐 2𝜋 4 𝛿 4 ( 𝑚 ℎ −𝑝−𝑘)
We need to sum over all different polarizations , over all different colors and over different quark species!
𝑀 𝒒 𝑎𝑏𝜆 𝜆 ′ =−𝑒⋅ 𝑔 𝑠 2 4 𝜋 2 𝑚 𝑊 𝑠𝑖𝑛 𝜃 𝑊 ⋅ 𝛿 𝑎𝑏 ⋅ 𝑚 ℎ 2 ⋅ 𝑔 𝜇𝜈 − 𝑝 𝜈 𝑘 𝜇 ⋅𝐼 𝑚 ℎ 2 𝑚 𝑞 2 ⋅ 𝜖 𝜈 ∗ 𝝀 ′ 𝑘 𝜖 𝜇 ∗𝝀 𝑝

47. For the decay rate Γ
𝑀 𝒒 𝑎𝑏𝜆 𝜆 ′ =−𝑒⋅ 𝑔 𝑠 2 4 𝜋 2 𝑚 𝑊 𝑠𝑖𝑛 𝜃 𝑊 ⋅ 𝛿 𝑎𝑏 ⋅ 𝑚 ℎ 2 ⋅ 𝑔 𝜇𝜈 − 𝑝 𝜈 𝑘 𝜇 ⋅𝐼 𝑚 ℎ 2 𝑚 𝑞 2 ⋅ 𝜖 𝜈 ∗ 𝝀 ′ 𝑘 𝜖 𝜇 ∗𝝀 𝑝

48. For the decay rate Γ
𝑀 𝒒 𝑎𝑏𝜆 𝜆 ′ =−𝑒⋅ 𝑔 𝑠 2 4 𝜋 2 𝑚 𝑊 𝑠𝑖𝑛 𝜃 𝑊 ⋅ 𝛿 𝑎𝑏 ⋅ 𝑚 ℎ 2 ⋅ 𝑔 𝜇𝜈 − 𝑝 𝜈 𝑘 𝜇 ⋅𝐼 𝑚 ℎ 2 𝑚 𝑞 2 ⋅ 𝜖 𝜈 ∗ 𝝀 ′ 𝑘 𝜖 𝜇 ∗𝝀 𝑝
Useful identity: 𝝀 𝝐 𝝂 ∗𝝀 𝒑 𝝐 𝝁 𝝀 𝒑 = 𝒈 𝝁𝝂

49. Useful identity: 𝝀 𝝐 𝝂 ∗𝝀 𝒑 𝝐 𝝁 𝝀 𝒑 = 𝒈 𝝁𝝂
For the decay rate Γ
𝑀 𝒒 𝑎𝑏𝜆 𝜆 ′ =−𝑒⋅ 𝑔 𝑠 2 4 𝜋 2 𝑚 𝑊 𝑠𝑖𝑛 𝜃 𝑊 ⋅ 𝛿 𝑎𝑏 ⋅ 𝑚 ℎ 2 ⋅ 𝑔 𝜇𝜈 − 𝑝 𝜈 𝑘 𝜇 ⋅𝐼 𝑚 ℎ 2 𝑚 𝑞 2 ⋅ 𝜖 𝜈 ∗ 𝝀 ′ 𝑘 𝜖 𝜇 ∗𝝀 𝑝
Useful identity: 𝝀 𝝐 𝝂 ∗𝝀 𝒑 𝝐 𝝁 𝝀 𝒑 = 𝒈 𝝁𝝂
𝑎,𝑏 𝛿 𝑎𝑏 2 =8

50. Useful identity: 𝝀 𝝐 𝝂 ∗𝝀 𝒑 𝝐 𝝁 𝝀 𝒑 = 𝒈 𝝁𝝂
For the decay rate Γ
𝑀 𝒒 𝑎𝑏𝜆 𝜆 ′ =−𝑒⋅ 𝑔 𝑠 2 4 𝜋 2 𝑚 𝑊 𝑠𝑖𝑛 𝜃 𝑊 ⋅ 𝛿 𝑎𝑏 ⋅ 𝑚 ℎ 2 ⋅ 𝑔 𝜇𝜈 − 𝑝 𝜈 𝑘 𝜇 ⋅𝐼 𝑚 ℎ 2 𝑚 𝑞 2 ⋅ 𝜖 𝜈 ∗ 𝝀 ′ 𝑘 𝜖 𝜇 ∗𝝀 𝑝
Useful identity: 𝝀 𝝐 𝝂 ∗𝝀 𝒑 𝝐 𝝁 𝝀 𝒑 = 𝒈 𝝁𝝂
𝑎,𝑏 𝛿 𝑎𝑏 2 =8
𝑀 2 = 𝑒 2 ⋅ 𝑔 𝑠 4 ⋅ 𝑚 ℎ 𝜋 4 𝑚 𝑊 2 sin 2 𝜃 𝑊 ⋅ 𝑞 𝐼 𝑚 ℎ 2 𝑚 𝑞

51. For the decay rate Γ Γ= 𝑀 2 16 𝑚 ℎ 𝜋
Γ= 𝑀 𝑚 ℎ 𝜋
Γ= 𝛼⋅ 𝛼 𝑠 2 ⋅ 𝑚 ℎ 3 4 𝜋 2 𝑚 𝑊 2 sin 2 𝜃 𝑊 𝑞 𝐼 𝑚 ℎ 2 𝑚 𝑞

52. For the decay rate Γ Γ= 𝑀 2 16 𝑚 ℎ 𝜋
Γ= 𝑀 𝑚 ℎ 𝜋
Γ= 𝛼⋅ 𝛼 𝑠 2 ⋅ 𝑚 ℎ 3 4 𝜋 2 𝑚 𝑊 2 sin 2 𝜃 𝑊 𝑞 𝐼 𝑚 ℎ 2 𝑚 𝑞
The form factor 𝐼 𝑚 ℎ 2 𝑚 𝑞 2 = 𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 1−4𝑥𝑧 1−xz 𝑚 ℎ 2 𝑚 𝑞 2

53. For the decay rate Γ
The form factor 𝐼 𝑚 ℎ 2 𝑚 𝑞 2 = 𝑥=0 1 𝑧=0 1−𝑧 𝑑𝑥𝑑𝑧 1−4𝑥𝑧 1−xz 𝑚 ℎ 2 𝑚 𝑞 2