Cusp and Collinear Anomalous Dimensions in Four-Loop QCD from Form Factors

We calculate the complete quark and gluon cusp anomalous dimensions in four-loop massless QCD analytically from first principles. In addition, we determine the complete matter dependence of the quark and gluon collinear anomalous dimensions. Our approach is to Laurent expand four-loop quark and gluon form factors in the parameter of dimensional regularization. We employ finite field and syzygy techniques to reduce the relevant Feynman integrals to a basis of finite integrals, and subsequently evaluate the basis integrals directly from their standard parametric representations.


The Basic Quark and Gluon Form Factors
In this talk, I focus on the basic qqγ * and ggH vertices consider perturbative expansion, e.g. at 1-loop: they provide virtual corrections to Drell-Yan and Higgs production at N4LO after UV renormalization, the amplitudes contain soft and collinear poles in leading poles through to 1/ 2 predicted by cusp anomalous dimension Γcusp

Evolution Equation for the Form Factor
The evolution of the form factor F can be written as [Magnea, Sterman 1990] Q 2 ∂ ∂Q 2 ln F (αs, Q 2 /µ 2 , ) = 1 2 K (αs, ) + 1 2 G (Q 2 /µ 2 , αs, ) where µ is the renormalization scale K contains all infrared poles in G contains all dependence on Q 2 and is finite for → 0 renormalization scale dependence must cancel between K and G :

Representation Dependence
Cusp anomalous dimension Γ r cusp is representation r dependent (different for quarks and gluons) IR poles of general scattering amplitudes follow simple T i T j pattern observed through to 2 loops [Catani '98], considered also for higher loops ("dipole conjecture") [Becher,Neubert '09;Gardi,Magnea '09] Fitting this framework, the cusp anomalous dimension was hoped to fulfill quadratic Casimir scaling Γ g cusp = ?
Proposal for singularity resolution [AvM,Panzer,Schabinger '14] observation: always possible to decompose wrt basis of finite integrals . basis consists of standard Feynman integrals, but in shifted dimensions with additional dots (propagators taken to higher powers)

Practical algorithm for basis construction
Algorithm: construction of finite basis systematic scan for finite integrals with dim-shifts and dots (with Reduze 2) IBP + dimensional recurrence for actual basis change remarks: computationally expensive part shifted to IBP solver efficient, easy to automate any dim-shift good, e.g. shifts by [Tarasov '96], [Lee '10] see [Bern,Dixon,Kosower '93] for dim-shifted one-loop pentagon Analytical integration @ 4-loops [AvM,Panzer,Schabinger '15] a non-planar 12-line topology @ 4-loops:  would need to compute 10 terms of epsilon expansion (weight 9) to obtain cusp (weight 6) :( Choosing a Good Finite Basis good basis choice avoids substantial "weight drops" in our example, using a suitable finite integral basis including we need only 1 term (weight 6) for the cusp (weight 6) in many cases, our choice of basis allows us to avoid the calculation of complicated topologies altogether (see later) integration-by-parts (IBP) identities in dimensional regularisation, integral over total derivative vanishes: problems of above construction: introduces many auxiliary integrals with additional dots and/or numerators sparse but still rather coupled system of equations  hom.   Bitoun,Bogner,Klausen,Panzer '17]: define (twisted) Mellin Transform Feynman integrals are Mellin transforms: with ν = (ν 1 , . . . , ν N ) andĨ (ν) = Γ[(L + 1)d/2 − ν]I (ν)

Properties of Mellin transform
new in this talk: annihilators beyond linear order determine c 0 (x 1 , . . . , x N ), . . . via syzygy equations: Syzygies generate linear relations for Feynman integrals: such that i f i s i = 0, then s is called a syzygy if s is a syzygy, then s · g is a syzygy for any polynomial g the (infinite) set of syzygies for f is a syzygy module  Results: 4 loop cusp anomalous dimensions in QCD [Henn,Korchemsky,Mistlberger '19]