Progress on QCD Theory Beyond Three Loops J. Vermaseren

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 converging? 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 integrations, 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 background 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.

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. If done parametrically the construction of the program would be comparable to the construction of an L-1 massless propagator program or worse. 5-loop results

Baikov, Chetyrkin and Kühnused the global R-star to compute a number of 5-loop quantities like e+e− → hadrons and eventually last year the QCD beta-function (hence the color factors are numbers). This was a rather heroic effort using the 1/D expansion techniques of Baikov and a large cluster for 1.5 years. Currently Chetyrkin and our group are trying to do this for a general gauge group, using Forcer. The color problems are in this case not simple and require some fancy group theory: all operators have to be calculated separately with ’open’ color indices and then mixed and renormalized. The K → ggg operator with 4 open color indices has 7 channels at 3 loops. Because the method gives faster programs (currently by more than a factor 10 compared to the local R-star) it allows the inclusion of a power of the gauge parameter as a check. All major runs have been finished (about a week on a good computer with 32 cores), and now the mixings and renormalizations are sorted out. This should be finished later this year. In the mean time we used the local R-star to compute the 5-loop beta-function for a general gauge group. When the QCD values are substituted, the result agrees with BCK. Takes about 6 days on a single computer with 32 cores. 11 4 β = C − T n , 0 3 A 3 F f 34 20 β = C 2 − C T n − 4 C T n , 1 3 A 3 A F f F F f 2857 1415 205 β = C 3 − C 2 T n − C C T n + 2 C 2 T n 2 54 A 27 A F f 9 F A F f F F f 44 158 + C T 2 n 2 + C T 2 n 2 , 9 F F f 27 A F f ! abcd abcd ! 4 150653 44 dA dA 80 704 β3 = CA − ζ3 + − + ζ3 486 9 NA 9 3 39143 136 ! 7073 656 ! + C 3 T n − + ζ + C 2 C T n − ζ A F f 81 3 3 A F F f 243 9 3 ! abcd abcd ! 2 4204 352 dF dA 512 1664 + CA CF TF nf − + ζ3 + nf − ζ3 27 9 NA 9 3 7930 224 ! 1352 704 ! + 46 C 3 T n + C 2T 2n 2 + ζ + C 2 T 2 n 2 − ζ F F f A F f 81 9 3 F F f 27 9 3 ! abcd abcd ! 2 2 17152 448 dF dF 2 704 512 + CA CF TF nf + ζ3 + nf − + ζ3 243 9 NA 9 3 424 1232 + C T 3 n 3 + C T 3 n 3 , 243 A F f 243 F F f 8296235 1630 121 1045 ! β = C 5 − ζ + ζ − ζ 4 A 3888 81 3 6 4 9 5 abcd abcd ! dA dA 514 18716 15400 + CA − + ζ3 − 968 ζ4 − ζ5 NA 3 3 3 5048959 10505 583 ! + C 4 T n − + ζ − ζ + 1230 ζ A F f 972 81 3 3 4 5 8141995 902 8720 ! + C 3 C T n + 146 ζ + ζ − ζ A F F f 1944 3 3 4 3 5 548732 50581 484 12820 ! + C 2 C 2 T n − − ζ − ζ + ζ A F F f 81 27 3 3 4 3 5 5696 7480 ! 4157 ! + C C 3 T n 3717 + ζ − ζ − C 4 T n + 128 ζ A F F f 3 3 3 5 F F f 6 3 abcd abcd ! dA dA 904 20752 4000 + TF nf − ζ3 + 352 ζ4 + ζ5 NA 9 9 9 abcd abcd ! dF dA 11312 127736 67520 + CA nf − ζ3 + 2288 ζ4 + ζ5 NA 9 9 9 abcd abcd ! dF dA 1280 6400 + CF nf −320 + ζ3 + ζ5 NA 3 3 843067 18446 104 2200 ! + C 3 T 2 n 2 + ζ − ζ − ζ A F f 486 27 3 3 4 3 5 5701 26452 944 1600 ! + C 2 C T 2 n 2 + ζ − ζ + ζ A F F f 162 27 3 3 4 3 5 31583 28628 1144 4400 ! + C 2 C T 2 n 2 − ζ + ζ − ζ F A F f 18 27 3 3 4 3 5 5018 2144 4640 ! + C 3 T 2 n 2 − − ζ + ζ F F f 9 3 3 3 5 abcd abcd ! dF dA 2 3680 40160 1280 + TF nf − + ζ3 − 832 ζ4 − ζ5 NA 9 9 9 abcd abcd ! dF dF 2 7184 40336 2240 + CA nf − + ζ3 − 704 ζ4 + ζ5 NA 3 9 9 abcd abcd ! dF dF 2 4160 5120 12800 + CF nf + ζ3 − ζ5 NA 3 3 3 2077 9736 112 320 ! + C 2 T 3 n 3 − − ζ + ζ + ζ A F f 27 81 3 3 4 9 5 736 5680 224 ! + C C T 3 n 3 − − ζ + ζ A F F f 81 27 3 3 4 9922 7616 352 ! + C 2 T 3 n 3 − + ζ − ζ F F f 81 27 3 3 4 abcd abcd ! dF dF 3 3520 2624 1280 + TF nf − ζ3 + 256 ζ4 + ζ5 NA 9 3 3 916 640 ! 856 128 ! + C T 4 n 4 − ζ − C T 4 n 4 + ζ . A F f 243 81 3 F F f 243 27 3

For QCD the color objects are: 1 4 TF = 2, CA = 3, CF = 3 abcd abcd abcd abcd abcd abcd dA dA = 135, dF dA = 15, dF dF = 5 . NA 8 NA 16 NA 96 f 2 We deﬁne now β ≡ −β(as)/(as β0). This gives, as an expansion in terms of αs:

f 2 3 4 β(αs, nf =3) = 1 + 0.565884 αs + 0.453014 αs + 0.676967 αs + 0.580928 αs ,

f 2 3 4 β(αs, nf =4) = 1 + 0.490197 αs + 0.308790 αs + 0.485901 αs + 0.280601 αs ,

f 2 3 4 β(αs, nf =5) = 1 + 0.401347 αs + 0.149427 αs + 0.317223 αs + 0.080921 αs ,

f 2 3 4 β(αs, nf =6) = 1 + 0.295573 αs − 0.029401 αs + 0.177980 αs + 0.001555 αs . Thus far the series looks suﬃciently convergent. Note also that at new orders in perturbation theory there will be new Casimir operators entering. At 6 loops there will be 7 new ones: dabcdef dabcdef with the varieties AA, AF, FF. dabcd dabef dcdef with the varieties AAA, AAF, AFF, FFF. At 7 loops there will be another 6 and at 8 loops there will be yet another 27. Graphically the series looks like: 1.14 1.06

1.12 βN n LO / βNLO αs,N n LO / αs,NLO

1.1 1.04 n = 2 n = 2 1.08 n = 3 n = 3 1.02 1.06 n = 4 n = 4

1.04 1 1.02 2 nf = 4 nf = 4, fixed value at 40 GeV 1 2 3 4 0 0.1 0.2 0.3 0.4 0.5 1 10 10 2 10 10 αs µ Luthe, Marquard, Maier and Schröderare also computing the 5-loop beta-function. They use the ”all lines massive” method. It is rather similar to the way the 4-loop beta-function was first calculated: ie. the infrared divergences are regulated by a small mass. For the 4-loop beta-function this was the easiest way because the R-star method was not sufficiently developed at the time. From the viewpoint of resources it is quite demanding though when one uses a Laporta-style reduction program. In principle they finished running some time ago, but still have technical problems. Using a crossbreed of the R-star operation and the operator method for moments of the splitting functions we should be able to eventually obtain a few moments of the 5-loop splitting functions to match the moments of the coefficient functions that we have at the 4-loop level. It will give some more information about the convergence of the perturbative series. Conclusions

Using the Forcer program we have been able to obtain results at the 4-loop level that are far beyond what was available in the literature. Together with some supporting packages like the R-star program we can even calculate 5-loop results for UV divergent quantities. The programs are/will be available, but optimizing them for a particular calculation requires a good knowledge of what is happening inside. Better automatization techniques might allow the derivation of a 5-loop program, or 4-loop reduction programs with one or more extra parameters, or an all N program although that may be rough.