Progress in Sudakov resummations

Transcription

1 Progress in Sudakov resummations Lorenzo Magnea Università di Torino I.N.F.N. Torino HERA LHC Workshop - March 27, 2004 Abstract Some recent developments in the field of soft gluon resummations are briefly reviewed. Progress is being made: the scope and precision of the resummations are being extended, by including new classes of terms, or implementing new observables; detailed phenomenological studies are appearing; models of power corrections arising from resummations are being constructed in a variety of contexts. Resummed QCD predictions with modeling of power corrections are becoming the standard benchmark for sufficiently inclusive observables. CERN, 27/03/2004

2 Outline Resummed Sudakov logs Two examples. A two slides introduction. More logs for the old observables Subleading logs, suppressed logs, no logs. Enhanced subleading logs. More observables for the old logs Resummation farms. New logs The Non-Global movement. Joint logs Joint resummation. Perspective CERN, 27/03/2004 1

3 E. Gardi and J. Rathsmann, hep-ph/ CERN, 27/03/2004 2

4 66 < Q < 116 GeV CDF A. Kulesza, G. Sterman and W. Vogelsang, hep-ph/ CERN, 27/03/2004 3

5 Basics Electroweak annihilation processes receive large QCD corrections due to the emission of soft or collinear gluons. Threshold logarithms, log(1 Q 2 /ŝ) Transverse momentum logarithms log(p 2 t /Q2 ). ψ PSfrag replacements H U H ψ To the graphical representation corresponds a (re)factorization ω(n, ɛ) = H DY 2 ψ(n, ɛ) 2 U(N) + O(1/N). A typical result takes the form (here for the Drell-Yan cross section in the MS scheme) 2 bω MS (N) = exp 4 Z dz zn 1 1 < Z (1 z) 2 Q 2 1 z : 2 2 dµ Q 2 µ 2 A α s (µ 2 ) D α s (1 z) 2 Q 2 = «1 ; + F MS (α s) 5 + O. N CERN, 27/03/2004 4

6 Features Non trivial. Soft gluon resummation reorganizes perturbation theory in a predictive way. For threshold resummation, let L = log N. Then X k α k s 2kX p 2 c kp L p exp 4 X k α k s k+1 X p 3 d kp L p 5. Predictive. Resummation extends the range of perturbative methods. Fixed order: α s L 2 << 1. NLL resummation: α s << 1 suffices. Scale dependence is reduced. Widespread. NLL soft gluon resummations exist for most inclusive cross sections of interest at colliders (NNLL now available for processes which are electroweak at tree level). Non perturbative aspects of QCD become accessible. Integrals in the exponent run into the Landau pole. A variety of regularizations have been proposed Principal value/cutoff (G. Korchemsky and G. Sterman, E. Gardi) Regular IR coupling (Y. Dokshitzer, G. Marchesini and B. Webber) Dimensional regularization (LM) Minimal prescription, (Catani et al.) * Result: answer ambiguous by a power suppressed amount. * Conclusion: power suppressed, non PT contribution must exist, with a matching ambiguity. * Phenomenology: models of power corrections built using information from resummations. CERN, 27/03/2004 5

7 More logs... Three loops are now available for the nonsinglet splitting function (S. Moch, J. Vermaseren and A. Vogt, hep-ph/ ). A 3 = 16 C F C 2 A ζ ζ «5 ζ CF 2 n f «+ 16 C F C A n f ζ ζ C F n 2 f 55 1 « ζ 3 A 3 is a universal coefficient. It was accurately estimated numerically from approximate calculations (A. Vogt, hep-ph/ ). It has a small numerical effect on tested cross sections.. «Do suppressed logs exponentiate? In the MS scheme γ (n) ns (N) = A n(ln N + γ e ) B n C n ln N N «1 + O N C 1 = 0, C 2 = 4C F A 1, C 3 = 8C F A 2.. * (More) evidence for exponentiation at (ln N)/N level. Note: impact of (ln N)/N exponentiation can be sizeable. (ln N) 0 terms do! (T. Eynck, E. Laenen and LM, hep-ph/ ). For Drell-Yan in the DIS scheme bω DIS (N) = (Γ(Q2, ɛ)) 2 «ψ r(n, ɛ) 2 «U r(n, ɛ) 1 1 Γ( Q 2 χr(n, ɛ), ɛ) 4 Vr 2(N, ɛ) Jr 2 + O. (N, ɛ) N Similarly for all processes which are electroweak at tree level. CERN, 27/03/2004 6

8 Enhanced logs and power corrections It is possible to combine renormalon methods and Sudakov resummation to construct models of power corrections. One method is dressed gluon exponentiation (E. Gardi, hep-ph/ ). Step 1: Compute the characteristic function F(k 2, ω) of the dispersive method (integrated probability for virtual gluon emission) for the observable at hand, in the Sudakov limit. (Integrated with and appropriate coupling this corresponds to the resummation of bubble graphs ). Step 2: Dress the virtual gluon by turning to a Borel representation of the coupling and integrating over gluon virtuality (Note: use gluon bremsstrahlung coupling to achieve NLL accuracy). Step 3: Use dressed gluon distribution as kernel of exponentiation. «d σ ln = dν DGE Z 0 dω dσ e νω 1 dω SDG, Step 4: Singularities of Borel representation of the exponent suggests pattern of exponentiated power corrections construct shape function. CERN, 27/03/2004 7

9 Some results Features of DGE NLL Sudakov resummation reproduced. All subleading logs computed in the large n f limit. Factorial growth of subleading logs detected: Definite prescription for resummed PT at power accuracy. Phenomenology of thrust, jet masses; also DIS, Drell-Yan, fragmentation. DIS at large x Korchemsky et al., hep-ph/ propose a non perturbative factorization formula for DIS structure functions. Fa(N, Q 2 ) = Ha(Q 2 ) J P T a Q2 Nµ 2 f 1 A q(n, µ 2 f ) JNP a NΛ 2! 1 Q 2 + O N The renormalon ambiguities in structure functions cancel between twists. At large x twist-4 dominated by twist-2-like configurations. This supports ultraviolet dominance of power corrections at large x (M. Beneke, V. Braun and LM, hep-ph/ ). DGE suggests E. Gardi and R. Roberts hep-ph/ J NP 2 NΛ 2! 2 NΛ Q 2 = exp 4 ω 2! NΛ 2!2 3 1 Q 2 ω 2 Q 2 5 Phenomenological consequences: Sudakov resummation necessary, correlations between α s and higher twist. «. CERN, 27/03/2004 8

10 Resummation in Classical Times CERN, 27/03/2004 9

11 Resummation in Modern times CERN, 27/03/

12 More observables... Automated resummation procedures are being developed for a vast class of observables and processes, including hadronic collisions (A. Banfi, G. Salam and G. Zanderighi, hep-ph/ ). Observables: with up to 4 hard partons, must vanish when a softer parton becomes collinear to a hard one. «a kt V ({p i }, k) = d i e b i η g i (φ). Q Example: a = d i = g i = 1, b i = 0 thrust. Requirements Recursive IRC safety: slightly stronger than conventional IRC safety, it requires that the observable behave uniformly under the addition of a hierarchy of soft/collinear partons. Continuous globality: the observable must be sensitive to emissions in the whole phase space without discontinuities, to avoid non-global logs. NLL Master Equation nx ln Σ(v) = i=1 +» C i R i (a, b i ) + v R i v f `d «log v i, g i + Bi T a + b i 2 n X i f i (x i, v a+b i µ 2 f ) ln i=1 f i (x i, µ 2 f ) + ln Phenomenology: in progress.» S T ««log v + ln ˆFnum `R a i. CERN, 27/03/

13 ... and yet more... New results applying conventional methods to specific situations G. Corcella and A. Mitov, hep-ph/ : heavy quark production in charged current DIS. M. Cacciari, G. Corcella and A. Mitov, hep-ph/ : bottom fragmentation in top decays. N. Kidonakis, hep-ph/ : application to SUSY processes. C. Berger and G. Sterman, hep-ph/ : scaling for power corrections in a class of event shapes. A. Banfi and M. Dasgupta, hep-ph/ : dijet rates with symmetric E T cuts. M. Cacciari and E. Gardi, hep-ph/ : resummation and power corrections for heavy quark fragmentation.... Theory developments. S. Catani et al., hep-ph/ : method for Sudakov resummation in multi-hard-parton processes. S. Forte and G. Ridolfi, hep-ph/ : alternative derivation of Sudakov resummation with RG methods. CERN, 27/03/

14 New Logs... Thou shall not cut up your phase space! Consider radiation into a fixed angular region Ω, in the presence of a hard event at scale Q. Measure cross section for radiation into Ω to carry energy E < Q Ω << Q get α s log(q Ω /Q). Primary radiation: hard partons emit gluons into Ω. Standard soft gluon techniques apply. Secondary radiation: a primary semihard gluon carrying energy Q Ω into Ω emits softer gluons into Ω. With no restriction on radiation into Ω, get log(q Ω / Q Ω ) log(q Ω /Q) (M. Dasgupta and G. Salam, ). CERN, 27/03/

15 The Non-Global movement The rise of non-global logarithms has triggered considerable theoretical activity. Two approaches have been considered. Define observables that minimize the impact of non-global logs and apply standard tecniques. R. Appleby and M. Seymour, hep-ph/ : rapidity gap events at HERA. Constrain final state by clustering algorithm. C. Berger, T. Kucs, G. Sterman, hep-ph/ : event-shape energy-flow correlations. Constrain final state by focusing on two-jet limit. Resum non-global logarithms. A. Banfi, G. Marchesini and G. Smye, : leading non-global logs obey an evolution equation, valid at large N c, and can be resummed. Y. Dokshitzer and G. Marchesini, hep-ph/ : in event-shape energy-flow correlations leading non-global logs factorize and exponentiate. G. Marchesini and A. Mueller hep-ph/ : intriguing connection with BFKL dynamics, for a somewhat exotic observable. H. Weigert, hep-ph/ : analogy with small-x dynamics pursued beyond the large N c limit. CERN, 27/03/

16 Joint resummation Phenomenology requires applying resummation techniques to more differential distributions. More soft logarithms appear. Resummed logs in differential distributions may leave nonlogarithmic but large remainders in integrated distributions. Sudakov resummation techniques can treat simultaneously p t and threshold logarithms (E. Laenen, G. Sterman and W. Vogelsang, hep-ph/ ). For weak boson production dσ res AB dq 2 dq 2 T = X a σ (0) a Z C N dn 2πi τ N Z d 2 b (2π) 2 ei Q T b C a/a (Q, b, N, µ, µ F ) exp [ Eaā (N, b, Q, µ) ] Cā/B (Q, b, N, µ, µ F ). E aā is similar to the Sudakov exponent for p t resummation Z Q 2 dk 2 Eaā (N, b, Q, µ) = t Q 2 /χ 2 k t 2 " Aa (αs (k t )) ln C a/a act as generalized parton distributions. Q2! # k t 2 + Ba (αs (k t )). Landau pole is handled with minimal prescription. Phenomenology is well under way: electroweak annihilation (Kulesza, Sterman, Vogelsang); prompt photon (Laenen, Sterman, Vogelsang); heavy quark production (Banfi, Laenen). CERN, 27/03/

17 Perspective Sudakov resummations are a very active and rapidly progressing field of study in QCD. They are necessary for phenomenological analysis of data in a variety of processes. They provide a window into nonperturbative contributions to high energy cross-sections, which can be parametrized by shape functions. They are becoming a practical tool, directly applicable to many measurable cross sections, not only fully inclusive ones. more differential cross sections can be resummed, for example via joint resummation. realistic cuts begin to be implemented, and the associated non-global logs can also be resummed. Impact on PDF s? Most cross sections used in the extraction of PDF s are known in resummed form, to high accuracy (NLL, NNLL), often with a QCD-motivated parametrization of power corrections. Evolution kernels are also known in resummed form. Fully resummed observables require resummed parton distributions. Is it achievable? CERN, 27/03/

18 LOG OUT CERN, 27/03/