Summary and Outlook Davison E. Soper University of Oregon LoopFest V, June 2006
Parton Showers & jet matching R. Erbacher emphasized how important this issue is for CDF analyses. CDF has been using a somewhat handmade analysis based on Alpgen + Pythia. M. Moretti discussed it with respect to Alpgen. P. Skands reported that he is building it into future Pythia. S. Frixione discussed it in general. P. Stephens promised it for Herwig++. Z. Nagy explained new methods for shower Monte Carlos that incorporate this feature. So what is it?
Your basic shower in pictures Use an evolution variable t that represents softness. evolves from t = 0 (hard) towards t = (soft). Shower U(t 3, t 1 ) = N(t 3, t 1 ) + Red dots are splitting operators H. t3 t 1 dt 2 U(t 3, t 2 ) H(t 2 ) N(t 2, t 1 ). Yellow ovals are Sudakov exponentials. The new version of Pythia is organized this way.
Shower cross section Start with 2 2 cross section (green blob) and iterate the evolution equation, say, twice:
Deficiency of the standard shower Standard shower Small PT approx. M2 The shower approximation relies on small PT splitting. Maybe the exact matrix element would be better. But that lacks the Sudakov exponentials.
An improved version This is the essential idea of Catani, Krauss, Kuhn, and Webber. CKKW use the k T jet algorithm to define the ratio.
Two step calculation CKKW break evolution into 0 < t < t ini and t ini < t < t f.
CKKW use improved weighting for 0 < t < t ini Nth term has N + 1 jets at scale t ini. Last term term has > 4 jets at scale t ini, so is pretty small.
NLO Calculations and Parton Showers S. Frixione described putting Monte Carlo event generators together with an NLO calculation with particular reference to the Herwig offspring MC@NLO. First, let s consider why NLO calculations might need some help. Keeping the ILC in mind, consider e + + e jets. Calculation of the three jet fraction f 3 at NLO is fine. But consider distribution of masses M of the three jets, df 3 dm
2 e + e 3 jets: jet-mass distribution f -1 3 df 3 /dm [GeV-1 ] ( s = M Z ; µ = s/6, k T algorithm, y cut = 0.05) Standard NLO calculation of df 3 /dm gives nonsense. 0 NLO -2 0 5 10 15 20 M [GeV]
Adding showering gives a better answer Calculate df 3 /dm with combined NLO + PS + Had Compare to Pythia and to the pure NLO result. 1 10-1 10-2 10-3 10-4 e + e 3 jets: jet-mass distribution f -1 3 df 3 /dm [GeV-1 ] ( s = M Z ; µ = s/6, k T algorithm, y cut = 0.05) NLO + PS + Had Pythia NLO 0 5 10 15 20 M [GeV] Calculation: Kramer, Mrenna, DES. Code available on www.
What is behind this Shower splitting. I[shower] = d q 2 q Tr exp ( 0 d q 2 q 2 d l 2 q 2 l 2 1 π dx 0 1 π dφ 2π α s 0 dz α s 2π P g/q ( l 2, z) 2π P g/q ( q 2, x)/qr( q 2, x, φ) ).
Perturbative expansion I[Born] + I[real] + I[virtual] = d q 2 q Tr /q R 0 + 0 d q 2 q 2 1 π dx 0 π dφ 2π [ αs 2π P g/q ( q 2, x)/qr( q 2, x, φ) α ] s 2π P g/q ( q2, x) /q R 0. Subtract the NLO terms, which then remove divergences from the ordinary NLO graphs.
NLO Calculations for SUSY S. Martin has examined masses a general SUSY model. Model parameters relate rather directly to physics at a high momentum scale. We may expect to measure pole masses at LHC. Need to relate them precisely and systematically. Propagator integrals related to basis integrals I n. d ds I n = K nm I m + C n Integrate this numerically along a contour in s.
K. Kovarik reported on Supersymmetry Parameter Analysis. This is a set of conventions to analyze SUSY parameters in MSSM. Parameters in L and most other parameters given in DR scheme. Need to transform SM parameters from MS to DR. On-shell masses determined as pole masses. My comment: there is a danger of anarchy with no conventions and a danger of getting stuck with a convention that is not optimal.
D. Stöckinger reported on regularization of SUSY. Discuss dimensional reduction. (Gluon 4 rather than D components) Is DR mathematically consistent? Yes. Just don t insist on too many 4D identities. What happens to QCD factorization? Problem in calculations from gluon mismatch. Effectively new kind of parton.
Does DR preserve SUSY in all cases? Look at Slavnov-Taylor identies. OK at two loops: SUSY restoring counterterms vanish.
H. Rzehak described the MSSM Higgs sector at two loops. Describe Higgs sector in MSSM with complex parameters. Generate diagrams; apply tensor reduction; extract relevant terms with on-shell scheme. Soft SUSY breaking parameters can be complex. α t α s corrections can be substantial.
Loop calculations for effective field theories A. Hoang described top pair threshold physics. e + e t t to measure top mass and width. Theory issues for developing effective theory (NRQCD): multiple scales m t m t v m t v 2 Coulomb singularities Γ t Λ QCD. Γ t E t so top is off-shell. Improvements apparent from LL to NLL to NNLL. Improvements apparent for pure perturbation theory to effective theory.
A. Signer described top pair threshold physics. Approach is to go from high scale QCD to effective theory I and then to effective theory II ( pnrqcd instead of vnrqcd ). With two scales, estimate error by varying both. Take the top width into account. Sum (γ/m) n terms from self-energy insertions. Should also expand in δ (p 2 m 2 )/m 2 using effective field theory methods. Achieving δm t 100 MeV and δf max 3% needs further theoretical progress.
M. Neubert discussed Soft Collinear Effective Theory. This is a framework for dealing with problems with more than one hard scale. It was developed for B physics but is more generally useful. One can use SCET to discuss DIS for x 1. Divide theory to momentum modes: hard, hard-collinear, anti-collinear, softcollinear ( soft ) modes. (Here I worry a bit about missing something.) One uses SQET to extract matching coefficients, analogues to Wilson coefficients in operator product expansion. Summing logs is from applying renormalization group on the factors. Their analysis of x 1 DIS works in momentum space rather than Mellin moment space.
N. Kidonakis also discussed calculation of the coefficients from threshold (x 1) summation. The idea is that if you don t know the full α N s contribution to a cross section, you can use calculated α N s log k contributions as an approximation. B. Fuks described a calculation of P T summation for slepton production at LHC (pp W/Z + X l 1 + l 2 + X). One predicts the P T of the slepton pair. S. Pozzorini discussed Sudakov logarithms, log 2 (s/mw 2 ), in hadronhadron collisions. These logarithms can make electroweak corrections substantial for high transverse momentum electroweak processes at LHC. Summing these is not simple because we have to deal with a spontaneously broken gauge symmetry. Comment: this would be a challenge for SCET.
Differential NNLO Calculations For a high value cross section like Higgs production via gluongluon fusion, one wants a next-to-next-to-leading order calculation if it can be done. Calculation should be differential enough that we can implement cuts. (But inclusive enough so that the calculated cross section in IR safe.) Want a cross section for an arbitrary infrared safe observable σ[f ] = 3 n=1 d p 1 d p n dσ d p 1 d p n F n ({p} n ).
e + e 3 jets T. Gehrmann reported a partial calculation of three jet event shape variables in e + e annihilation. Two loop virtual diagrams already calculated. Use antenna subtraction scheme to remove singularities from real emission diagrams. Calculation organized as a parton level event generator.
pp to Higgs, W, or Z K. Melnikov described calculations for hadron collisions to produce simple and important final states: Higgs, W, or Z. The method: sector decomposition. {p 1, p 2, p 3 } {x 1, x 2,..., x 7 }, 0 < x i < 1. Extract the divergent pieces automatically by computer algebra.
Here is a (vastly) simplified version: I = 1 0 dx 1 x ɛ 1 1 dx 2 0 f(x 1, x 2 ) x 1 = 1 0 dx 1 x ɛ 1 1 dx 2 0 { f(0, x2 ) + f(x } 1, x 2 ) f(0, x 2 ) x 1 x 1 1 ɛ 1 0 dx 2 f(0, x 2 ) + 1 0 dx 1 1 0 dx 2 f(x 1, x 2 ) f(0, x 2 ) x 1. Need multiple remappings to disentangle the singularities. Complicated virtual diagrams would be harder.
An example that was simple enough to calculate using methods of a couple of years ago is the rapidity distribution of Z bosons to be produced at the LHC. This example illustrates the added precision of a NNLO result. At NLO the precision is not good enough to distinguish two similar parton distributions using the experimental result. With the NNLO result, we can distinguish the two possibilities.
Other NNLO calculations T. Becher reported on two loop calculations underway for the description of B X s γ. A. Mitov reported on calculations underway for NNLO timelike fragmentation of a b quark.
NLO calculations with many legs One loop virtual diagrams generally calculated by reducing them to known master integrals. To start with one has tensor integrals d D k where q i = k + r i. q µ 1 a 1 q µ r a r (q 2 1 m2 1 + i0) (q2 N m2 N + i0) Express these as a linear combination of scalar integrals of the form d D 1 k (q1 2 m2 1 + i0) (q2 N m2 N + i0) with various D and N values.
d D k 1 (q 2 1 m2 1 + i0) (q2 N m2 N + i0) In turn the scalar integrals are reduced to linear combinations of scalar master integrals that are known. Typically, need the inverse of matrices G ij = r i r j (i, j {1,..., N 1}) or of S ij = (r i r j ) 2 (i, j {1,..., N}). But G and S may have no inverse for certain choices of the external momenta. Furthermore G has no inverse for N 6 and S has no inverse for N 7. Thus one needs to be clever.
G. Heinrich described a method for reducing the integrals to a suitable set of (non-scalar) basis integrals that does not involve inverse G 1 or S 1. For some exceptional external momentum choices, the analytic evaluation of these integrals may require G 1 where det G is very small. In such cases, the method is to perform the integration numerically, with a suitable deformation of the (Feynman parameter) contour.
W. Giele described methods to reduce the integrals to known master integrals while avoiding problems from inverse matrices. One feature is the use of four of the external momenta as basis vectors for the space of loop momenta when there are enough external legs that four of the external momenta are linearly independent. In this way the one-loop amplitude for six gluons can be calculated.
Purely numerical approach C.f. talks by E. de Doncker and A. Daleo on numerical evaluation of multiloop diagrams. Can one perform loop integrals for one loop diagrams either in momentum space or after some sort of an integral transformation by Monte Carlo integration? If you have to expand the numerator function, that s seminumerical.
Existence proof Program for e + + e 3 jets at NLO D. E. Soper, Phys. Rev. Lett. 81 (1998) 2638 Caution: the method here runs into technical problems for more complicated cases. Possible basis for a purely numerical approach Subtraction scheme for eliminating the collinear and soft divergences, as in real emission diagrams. Z. Nagy and D. E. Soper, JHEP 0309 (2003) 055
Possible advantages with purely numerical approach Generality and flexibility.
Possible difficulties with purely numerical approach Lack of precision. Convergence is only 1/ time. But for NLO calculations, high precision is not needed. Too many graphs. Number of graphs (N verts )!. By itself, this is not a problem since one can just sample the graphs.
Too much cancellation. A result (109.14 ± 0.27) (107.14 ± 0.25) is not so nice. Cancellation is both among graphs and within graphs. Too singular. N j=1 1 (l Q j ) 2 + iɛ has lots of powers of something small in the denominator if N is large. (Similar singularities after an integral transformation.) Deforming integration contour saves you from 1/0, but may not be perfect.
There can be problems from double parton scattering singularities. This starts at N = 6.
Could a purely numerical approach work? It holds promise for a not too high number of legs. Zoltan Nagy and I are working on how high not too high is.
Outlook With the approach of the LHC era, powerful theoretical tools are urgently needed. The reports at this workshop indicate that powerful theoretical tools are, in fact, available. Furthermore, there are promising developments along several lines of attack. I am excited to witness such vigorous progress and look forward to developments in the next year.
Thanks to the organizers: Ulrich Baur, Sally Dawson, Lance Dixon, Michael Peskin, Doreen Wackeroth.