Progress on QCD Theory Beyond Three Loops J. Vermaseren
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Progress on QCD theory beyond three loops J. Vermaseren ”We” means a topic dependent subset of the following people: Giulio Falcioni, Franz Herzog, Ben Ruijl, Takahiro Ueda, JV, Josh Davies, Sven Moch, Andreas Vogt Introduction After finding the Higgs it looks like the major discovery potential of the LHC is in precision physics. To match experimental and theoretical precisions we need to go beyond NNLO with QCD. Al- ready Higgs production has been computed at 3 loops (NNNLO). In principle needs matching splitting functions, ie 4 loops. Interesting from the theoretical viewpoint: To what number of loops will the series be con- verging? It seems better converging than it has a ”right to be”. Over the past few years the Karlsruhe group has managed to compute a number of 4/5-loop QCD quantities. At great cost. Also Velizhanin got a (very) few moments of the nonsinglet splitting functions at 4 loops. Recently we completed a program (Forcer) for the reduction of 4-loop massless propagator diagrams to master integrals in the style of the 3-loop Mincer program. Mincer was used for the computation of many 3-loop (and some 4-loop) quantities in QCD. Hence: L → L + 1. If possible it is an advantage to compute for arbitrary Yang-Mills theories with fermions. It shows more structure. New Casimir’s defy extrapolations. Example 1: 4-loop beta function: extrapolations within 1 sigma without quartic Casimirs. With them: off by 9 sigma. Example 2: 3-loop splitting functions: the approximations has assumed that the ’new’ dabc dabc terms would be small (or they were just forgotten). It turned out that for some things they were the leading terms. Forcer Use parametric reductions as in id lala(n1?{>0},n2?{>0},n3?{>1},n4?{>0},n5?{>0},n6?{>0},n7?{>0},n8?{>0} ,n9?{>0},n10?{<=0},n11?{<=0},n12?{<=0},n13?{<=0},n14?{<=0}) = -rat(1,(n3-1))*( +lala(n1,n2,n3,-1+n4,n5,n6,n7,n8,n9,n10,n11,n12,n13,n14)* rat(-(n3-1),1) +lala(n1,n2,n3-1,-1+n4,n5,n6,1+n7,n8,n9,n10,n11,n12,n13,n14)* rat(-n7,1) +lala(n1,n2,n3-1,n4,-1+n5,n6,1+n7,n8,n9,n10,n11,n12,n13,n14)* rat(n7,1) +lala(n1,n2,n3-1,n4,n5,n6,n7,n8,n9,n10,n11,n12,n13,n14)* rat(-2*ep-(n3-1)-2*n4-n7-n12-n14+4,1) ); Repeated application will either bring n3 down to one (in which case we continue with the next variable), or bring n4 or n5 down to zero, which is equivalent to removing a line. p1 p3 p2 QQ p p9 8 p7 p5 p6 p4 Topology name: lala, master. Numerators: 2 Q · p5, 2 Q · p2, 2 p1 · p4, 2 p1 · p5, 2 p2 · p4. This means that for each topology in principle at most 14 reduction statements are needed, unless there are shortcuts. For most topologies (417 out of 438) there are (one loop integra- tions, triangle/diamond rules). For the others it can happen that the best strategy is to use different sets of reductions, depending on the value of the parameters and hence there are more than 14 statements (see paper). Derivation of most of the program and rewriting from one topology to the next is done automatically. 21 topologies needed manual interference. The complete Forcer program takes more than 200000 lines. Because the majority, including the most error-prone parts, were generated automatically, the debugging was relatively easy. The program LiteRed by R. Lee also does parametric reductions and its 4-loop version is constructed fully automatically. For some reason it is not very fast and hence has not been used for any of the results mentioned here. Amount of work to create Forcer: about one year for 3 people. The automatic parts have been used also for generating the framework of the 5-loop program. Alas there are about 200 topologies that need manual interference. That will be very much work. But more than 30% of the diagrams for a 5-loop propagator could already be run. First 4-loop results, cq. warming up. The first thing to do when having such a powerful program is to rerun results from the literature as there are: • propagators and vertices with all powers of the gauge parameter. • the 4-loop beta-function, both the ’regular’ way and in the background gauge. The back- ground gauge is actually slightly faster. This could be run with all powers of the gauge parameter. Without gauge parameter it takes 44 minutes on my laptop (34 fu). • some moments of nonsinglet splitting functions. We used the same method as was used in the papers about the 3-loop splitting and coefficient functions. The literature used operators. • the Gross-Llewellyn Smith sum rule. All results agreed with the literature except for two. In both cases it turned out that our results were the correct ones. Moments The ‘simplest’ method to calculate moments of splitting functions and coefficient functions is by calculating total crosssections and taking derivatives with respect to the parton momentum. This is what was done for the 3-loop calculations. We managed to do this up to N=6 for the nonsinglet and to N=4 for the singlet. Estimates for one more moment (N=8, N=6 respectively) are that it would take 10 powerful computers with 24 cores each close to one year. These are expensive calculations because the number of diagrams is very large and the diagrams have many propagators and vertices. For the coefficient function we can compare the situation at three loops versus what we see at four loops with resummation results. 20 50 c(3) (N) c(4) (N) 2,ns 40 2,ns 15 all N 0 exp. N 0 − − + all N 1 30 + exp. N 1 10 MV(’09) exact 20 exact 5 10 0 0 ∗ ∗ nf = 4 ( 1/2000) nf = 4 ( 1/25000) -5 -10 0 5 10 15 20 0 5 10 15 20 N N The conceptually more complicated, but computationally friendlier way is to study operator vertices, although these will give us only splitting functions. At the moment we have already the quark operators with any number of gluons, allowing us to do calculations to 4 loops (and maybe more in the future). This has given us the nonsinglet moments to N = 16 at the 4-loop level. In the large nc limit we could go to N = 19. From that, combined with theoretical considerations we could make 2 a reconstruction of the general N formula. For the nf moments we could go to N > 40 to make a reconstruction. The literature shows results to N=4 (Velyzhanin, Karlsruhe group) 3 for the nonsinglet and of course the nf results by Gracey. Note that CPU(N)/CPU(N-1) is about 2.5 and our program took 17 hours on a single core for N=4. Since then the programs have been optimized further. 2.5 γ (3)± (N) 0.8 γ (3)± (N) 2 ns ns 0.6 1.5 nf = 3 exact 1 0.4 n = 4 large nc f 0.5 difference 0.2 large nc 0 0 0 coeff. of nf points: ± at even/odd N -0.5 0 5 10 15 20 25 0 5 10 15 20 25 N N From the all-N results we could read off the corresponding part of the cusp anomalous dimen- sion, which was shortly after confirmed by Henn, Lee, Smirnov2 and Steinhauser. The other operators are currently being constructed (gluon, ghost and gauge-breaking). Also the renormalization is not completely trivial. They would allow the computation of the other channels, like singlet etc. to values for N of at least 12. This should give at least a moderately accurate approximation of the splitting functions in the relevant x-space region. The purely gluonic operator is rather horrible. We need it with up to 6 gluons. Bose symmetry creates ng! permutations, and this is still independent of other structures. Fortunately, the more complicated the operator, the easier the topology and hence the integrations. R-star The divergent part of a diagram is a polynomial in the masses and external invariants. This means that if the only dimensionful parameter is Q (as in a massless propagator diagram), we can take a d’Alembertian w.r.t. Q to make the diagram dimensionless. Once dimensionless, the UV divergence does not depend of where the external lines attach and we can move them around in such a way that we have an integral that can be done. p After this the integral over p can be done. Problem: IR divergences! There are several methods to control the IR divergences: 1. global R-star Used by Chetyrkin and collaborators. Introduces two massive lines. Takes a relatively small amount of computer resources, but is conceptually rather complicated (many operators and operator mixings). 2. local R-star Singles out the UV and IR divergences. Needs more resources and is harder to program, but is conceptually easier. The literature shows examples, but usually for φ4 theory and in that case the full complexity of tensorial structures in the numerator is not treated. H+R have solved these problems in a recent paper. We have constructed a program for it. 3. all lines massive From the physics viewpoint the easiest method. The IR divergences are all absorbed by a mass parameter, but it needs reductions of vacuum bubbles with massive lines to master integrals. If done by generic reduction programs it is very demanding in computer resources.