Hadronic production of a Higgs boson in association with two jets at ...
Hadronic production of a Higgs boson in association with two jets at ...
Hadronic production of a Higgs boson in association with two jets at ...
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2.4. Sp<strong>in</strong>or <strong>in</strong>tegr<strong>at</strong>ion 48The explicit value <strong>of</strong> this <strong>in</strong>tegral is not <strong>of</strong> <strong>in</strong>terest here (it be<strong>in</strong>g dependent onwhether the external momenta Ki 2 are positive or neg<strong>at</strong>ive), the po<strong>in</strong>t <strong>of</strong> <strong>in</strong>terest isth<strong>at</strong> it <strong>in</strong>tegr<strong>at</strong>es to a logarithmic function. In fact, all basis <strong>in</strong>tegrals other thanthe scalar bubble <strong>in</strong>tegr<strong>at</strong>e <strong>in</strong> the double cut regime to a logarithmic function <strong>of</strong> thek<strong>in</strong>em<strong>at</strong>ics. This is a very useful observ<strong>at</strong>ion, s<strong>in</strong>ce if one can separ<strong>at</strong>e the <strong>in</strong>tegral<strong>in</strong>to pieces which <strong>in</strong>tegr<strong>at</strong>e to logarithmic funcitons, and those th<strong>at</strong> do not, thenthe task <strong>of</strong> extract<strong>in</strong>g the coefficient <strong>of</strong> the scalar bubble becomes simpler. It wasshown <strong>in</strong> [123] th<strong>at</strong> one can always separ<strong>at</strong>e these contributions and determ<strong>in</strong>e thecoefficient <strong>of</strong> the scalar bubble us<strong>in</strong>g the holomorphic anomaly eq. (2.56). S<strong>in</strong>ce theorig<strong>in</strong>al paper, this approach has been extended to D dimensions [133–135], andclosed forms for all basis <strong>in</strong>tegrals appear <strong>in</strong> the double cuts have been provided[136,137].In this thesis we used a more recent variant <strong>of</strong> sp<strong>in</strong>or <strong>in</strong>tegr<strong>at</strong>ion [132], which usesStokes’ Theorem r<strong>at</strong>her than the holomorphic anomaly to perform the <strong>in</strong>tegr<strong>at</strong>ions<strong>in</strong> λ. This is described <strong>in</strong> the follow<strong>in</strong>g section.2.4.2 Sp<strong>in</strong>or <strong>in</strong>tegr<strong>at</strong>ion via Stokes’ TheoremIn this section we describe the applic<strong>at</strong>ion <strong>of</strong> sp<strong>in</strong>or <strong>in</strong>tegr<strong>at</strong>ion via Stokes’ Theorem[132]. Firstly <strong>of</strong> course, we start <strong>with</strong> the double cut measure,∫∫ ∞d 4 l 1 δ(l 2 1)δ((l 1 − P) 2 ) = t dt δ(t − P ) 2 ∫ 〈l dl〉[ldl]〈l|P |l] 〈l|P |l]0(2.62)Here as <strong>in</strong> eq. (2.49) we have rescaled l 1 (the loop momenta appear<strong>in</strong>g <strong>in</strong> the treeproducts) as l µ 1 = t〈l|γµ |l]/2, which is equivalent to the follow<strong>in</strong>g sp<strong>in</strong>or shifts,|l 1 〉 = √ t|l〉 and |l 1 ] = √ t|l]. (2.63)At first glance there appears <strong>in</strong> eq. (2.62) an <strong>in</strong>verse factor <strong>of</strong> 〈l|P |l] rel<strong>at</strong>ive toeq. (2.49). This however is expected, s<strong>in</strong>ce <strong>in</strong> eq. (2.49) the second δ function(which is <strong>of</strong> the form δ(f(t)) has not been applied, whereas <strong>in</strong> the above equ<strong>at</strong>ionthe δ function is <strong>of</strong> the form δ(t − a).To proceed one notices th<strong>at</strong> <strong>with</strong> <strong>two</strong> free parameters one can write the loopmomentum <strong>in</strong> terms <strong>of</strong> <strong>two</strong> vectors p µ and η µ , such th<strong>at</strong> the sum <strong>of</strong> p and η is equal