Topics in Heavy Particle Effective Theories

Topics in Heavy Particle Effective Theories Thesis by Matthew P. Dorsten In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2006 (Defended May 25, 2006) ii c 2006 Matthew P. Dorsten All Rights Reserved iii Acknowledgments Many thanks to my advisor, Mark Wise, for his guidance, physics-related and otherwise. I would also like to thank my collaborators on the physics projects described here: Christian Bauer, Sean Fleming, Sonny Mantry, Mike Salem, and Iain Stewart. I acknowledge the support of a Graduate Research Fellowship from the National Science Foundation and the Department of Energy under Grant No. DE-FG03-92ER40701. Above all I must thank Rebecca, who helped me get through the whole thing. iv Abstract This thesis gives several applications of effective field theory to processes involving heavy particles. The first is a standard application of heavy quark effective theory to exclusive B decays. It involves two new sum rules discovered by Le Yaouanc et al. by applying the operator product expansion to the nonforward matrix element of a time-ordered product of b c currents. They lead to the constraints σ2 > 5ρ2/4 and σ2 > 3(ρ2)2/5 + 4ρ2/5 → on the curvature of the B¯ D(∗) Isgur-Wise function, both of which imply the absolute → lower bound σ2 > 15/16 when combined with the Uraltsev bound ρ2 > 3/4 on the slope. This thesis calculates order αs corrections to these bounds, increasing the accuracy of the resultant constraints on the physical form factors. The second application involves matching SCETI onto SCETII at one loop. Keeping the external fermions off their mass shell does not regulate all IR divergences in both theories. The work described here gives a new prescription to regulate infrared divergences in SCET. Using this regulator, we show that soft and collinear modes in SCETII are sufficient to reproduce all the infrared divergences of SCETI . We explain the relationship between IR regulators and an additional mode proposed for SCETII . Next we consider tt¯production at large energies. The production process is characterized by three disparate energy scales: the center-of-mass energy, √s; the mass, m; and the decay width, Γ; with √s m Γ. At the scale √s we match onto massive soft-collinear ≫ ≫ effective theory (SCET). The SCET current is run from √s to m, thereby summing Sudakov logarithms of the form logn(m/√s), where n = 2, 1. At the scale m, the top quark mass is integrated out by matching SCET jet functions onto a boosted version of heavy quark effective theory (bHQET). The jet functions in bHQET are then run from m to Γ, summing powers of single logarithms of the ratio m/Γ. Under certain assumptions factorization formulas can be derived for differential distributions in processes involving highly energetic jets, such as jet energy distributions. As a v final topic, we show how to test these assumptions using semileptonic or radiative decays of heavy mesons, by relating the jet P + distribution derived under these assumptions to other differential distributions in these decays, which are better understood. vi Contents Acknowledgments iii Abstract iv 1 Introduction 1 1.1 Constraining exclusive B decaymeasurements. 1 1.2 RegulatingSCETintheinfrared . ... 2 1.3 Precisiontopquarkphysics . ... 2 1.4 JetdefinitionsinSCET ............................. 3 1.5 Planofthethesis................................. 3 2 Heavy quark effective theory 5 2.1 Basics ....................................... 5 2.2 Leading-order in the heavy quark expansion . ....... 7 2.3 Lagrangian beyond leading order in 1/mQ ................... 8 3 Perturbative corrections to sum rules 10 3.1 Sum rules and V ................................ 10 | cb| 3.2 Derivation of the generic sum rule . 12 3.3 Vector and axial vector sum rules . 17 3.4 Physicalbounds ................................. 22 3.5 Summary ..................................... 27 4 Soft-collinear effective theory 29 4.1 Basics ....................................... 29 4.2 Lagrangian .................................... 31 4.3 Labeloperators.................................. 34 vii 4.4 Gaugeinvariance ................................. 35 4.5 Collinear-ultrasoft decoupling . ....... 38 5 Infrared regulators in SCET 40 5.1 Introduction.................................... 40 5.2 Matching from SCETI onto SCETII ...................... 42 5.3 InfraredregulatorsinSCET. 47 5.3.1 ProblemswithknownIRregulators . 47 5.3.2 AnewregulatorforSCET.. .. .. .. .. .. .. 50 5.4 Summary ..................................... 54 6 Summing logarithms in high-energy unstable heavy fermion pair production 56 6.1 Introduction.................................... 56 6.2 QCDatoneloop ................................. 57 6.3 SCETI atoneloop ................................ 59 6.4 SCETI crosssection ............................... 61 6.4.1 Ultrasoft-collinear factorization . ....... 62 6.4.2 Final state invariant mass constraints . ..... 64 6.4.3 Specifying jet momenta and SCETI jetfunctions . 66 6.4.4 Computing SCETI jetfunctions .................... 68 6.5 BoostedHQET.................................. 70 6.6 Summary ..................................... 74 7 Testing jet definitions in SCET 76 7.1 TheproblemofdefiningjetsinSCET . 76 7.2 Review of radiative and semileptonic B decayinSCET ........... 78 7.3 Standardobservables. 79 7.4 Jet P + distribution................................ 82 7.5 Summary ..................................... 85 8 Conclusions 87 Bibliography 90 viii List of Figures 3.1 The cuts of Tfi in the complex ε plane. The depicted contour picks up only contri- butions from the left-hand cut, which corresponds to physical states with a charm quark. The states given by the right-hand cut do not contribute here. ....... 13 3.2 Diagrams contributing to the order αs corrections to the sum rules. The squares indicate insertions of the currents Ji and Jf , respectively. The current Ji inserts ′ momentum q, while the current Jf carries away momentum q sufficient to leave the final b-quark with velocity vf . The velocity-labeled quark fields are those of the heavy quark effective theory. ................................ 15 3.3 One-loop renormalization of the leading operator in the operator product expansion of ¯ ′ Tfi. The blob indicates an insertion of this operator, bvf Γf (1 + /v )Γibvi . The external lines are bottom quarks in the heavy quark effective theory. ............. 16 3.4 Dispersive constraints on D derivatives combined with the corrected sum rule bounds F derived here at ∆ = 2 GeV. The interior of the ellipse is the region allowed by the dis- persion relations. Including the curvature bounds, given by the area above the dashed curves, further restricts the allowed region to the shaded area. The darker region is obtained by also including the Bjorken and Voloshin bounds. Both perturbative and nonperturbative corrections are included. ...................... 28 5.1 Diagrams in SCETI contributing to the matching. The solid square denotes an insertion of the heavy-light current. ...... 43 5.2 Diagrams in SCETII contributing to the matching. 44 5.3 Contribution of the additional SCETII modeproposedin Refs.[40]. 45 6.1 Tree-level current (a) and the one-loop correction (b) inQCD. ........ 58 6.2 One-loop vertex correction in SCET. ...... 59 6.3 Forward scattering amplitudes. ..... 69 ix 7.1 Kinematics of the current in the effective theory. The heavy quark bv has momentum mbv + k, the gauge boson has q and the total momentum of the usoft Wilson line Yn is l. ............................. 79 1 Chapter 1 Introduction The main theme of this thesis is QCD. This is a theory we know a lot about indirectly. It is difficult to calculate things in hadronic physics, however. This is because of the theory’s nonperturbative nature. Effective field theory is a useful way of making do with this situation. It is often used to parametrize unknown parameters and relate them to other unknown parameters through general symmetry arguments. This is a surprisingly powerful concept. It is often possible to eliminate some physics we do not understand by taking its effects from somewhere else. In this way we get around our lack of understanding. 1.1 Constraining exclusive B decay measurements This thesis gives several applications of the ideas of effective field theory to processes involving heavy particles, that is, heavy relative to the other energy scales involved. The first is a standard application of heavy quark effective theory (HQET) to exclusive B decays. In this case, the bottom quark mass is heavy relative to the other scale in the problem, ΛQCD. This is a very successful theory and will be discussed more fully in the next chapter. For now, suffice it to say that it is a self-contained theory just like any other and that sum rules can be derived by relating exclusive matrix elements with the inclusive result of the optical theorem. The third chapter of this thesis discusses two new sum rules that were discovered in this way by applying the operator product expansion to the nonforward matrix element of a time-ordered product of b c currents in the heavy quark limit of QCD. → They lead to the constraints σ2 > 5ρ2/4 and σ2 > 3(ρ2)2/5 + 4ρ2/5 on the curvature of the B¯ D(∗) Isgur-Wise function, which is basically just a parametrized matrix element. → These constraints imply the absolute lower bound σ2 > 15/16 when combined with the 2 Uraltsev bound ρ2 > 3/4 on the slope of the form factor. That chapter calculates order αs corrections to these bounds, increasing

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