Hadronic production of a Higgs boson in association with two jets at ...

1.1. The Standard Model of Particle Physics 8bubble diagram shown in Fig 1.1. Application of the Feynman rules and reductionof intermediate tensor integrals [16] would ultimately lead to the following sort ofterm,∫I2 4D =d 4 l(2π) 4 1(l 2 )(l − P) 2 (1.21)This diagram diverges as l → ∞ (known as UV divergence) and to expose thesingularity structure of the integral we wish to regularise the integral at intermediatestages. By far the most popular method of regularisation is that of dimensionalregularisation, first proposed by ’t Hooft and Veltman [17]. In this approach onealters the number of spacetime dimensions to 4 − 2ǫ. Singularities then revealthemselves as inverse powers of ǫ. This method has numerous advantages, includingmaintaining gauge-invariance and regularising both UV (l → ∞) and IR (l →0) singularities at the same time. This point needs some clarification since thesesingularities arise from different sources, a UV singularity occurs when the powersof l in the numerator dominate as l → ∞. To regularise these divergences wewould wish to define ǫ > 0. Clearly the situation is reversed for IR singularities,where the denominator dominates and we would wish that ǫ < 0. However, inpractical calculations one can define ǫ > 0, reguarlise and renormalise (which willbe explained shortly) the UV singularities and then analytically continue to ǫ < 0,which regulates the remaining IR singularities.How one treats external particles is up to the discretion of the calculator, andseveral schemes exist and are related to each other by predictable quantities. In thisthesis unless stated we will work in the four-dimensional helicity scheme (FDH). Thisallows us to keep external particles strictly in four-dimensions, whilst only the loopmomenta (and the metric) are D-dimensional. The t’Hooft-Veltman scheme [17]defines γ µ in d dimensions with γ 5 = iγ 0 γ 1 γ 2 γ 3 defined such that it anticommuteswith γ µ for µ ∈ {0, 1, 2, 3} and commutes with γ µ for all other µ.Of course one does not expect to predict infinite cross sections, and this certainlyis not what is observed at colliders! Ultimately we wish to remove the singularities inǫ and there are systematic ways of doing this. Ultraviolet divergences can be removedby a process known as renormalisation. Basically the physical quantities written in