Lattice QCD & the search for BSM physics in beauty Matthew Wingate DAMTP, University of Cambridge
Outline ✤ Quark flavour Peering through the glue to study electroweak ✦ symmetry breaking ✤ Lattice QCD Uniting the gauge theory, statistical physics, and ✦ effective field theory
Quark flavour ✤ Discovery era & flavour ✤ High precision in flavour ✤ Rare decays
Quark flavour in the SM e + W + ν e ✤ Only weak interactions change quark flavor � u � � t � � c d � � u d � s � b � ✤ Flavor mixing d � V ud V us V ub d s � V cd V cs V cb s = b � V td V ts V tb b ✤ V is the CKM matrix. Unitarity + “rephasing” implies 4 free SM parameters (one of them a CP-violating phase)
CKM matrix from Higgs couplings � u ′ i � Q i u i d i RH SU(2) singlets LH SU(2) doublets L = d ′ i R R L Interact with gauge bosons in covariant derivative ¯ R + ¯ Q i D Q i u i D u i d i D d i L i / R i / R i / L quark = L + ¯ R J µ, + u ′ i L γ µ d ′ i Gives rise to weak current weak = ¯ L The coupling to the Higgs field is not apparently diagonal in generation √ � � La ǫ ab φ † λ ij d ¯ L φ d j u ¯ b u j Q i R + λ ij Q i L quark ,φ = − 2 R + h . c . Fields may be transformed to find mass eigenstates � � � d ¯ m i d i L d i R + m i u i L u i � � L quark ,φ | vev = − u ¯ R + h . c . i Showing the weak current allows mixing between generations J µ, + L γ µ V ij CKM d j u i weak = ¯ L
Physics Beyond the Standard Model ✤ Standard Model shortcomings: Higgs mass fine- tuning, dark matter, CP asymmetry & M/AM ✤ Direct production: BSM spectrum ✤ Indirect searches: BSM couplings ✤ Complementary approaches
Complementarity: top quark Indirect Direct FIG. 13. The � 2 curves for the standard model fit to the elec- FIG. 12. W mass and top-quark mass measurements from the troweak precision measurements from LEP, SLD, CDF, and Fermilab collider experiments (CDF and D0). The top-mass D0 ( W mass only) and neutrino-scattering experiments as a values are from the full Tevatron data sets, with an integrated luminosity of � 100 pb � 1 . The W mass values are derived function of M top for three different Higgs-mass values span- from analyses of the first 15–20 pb � 1 only. The lines are stan- ning the interval 60 GeV/ c 2 � M Higgs � 1000 GeV/ c 2 . The num- dard model predictions for four different Higgs masses (Flat- ber of degrees of freedom is 14 (LEP Collaborations, 1995). tum, 1996). from Campagnari and Franklin, Rev. Mod. Phys. 69 , 137 (1997)
Complementarity: Higgs boson Indirect inference Now out of date! Direct exclusion
Complementarity in BSM searches Indirect constraints on CKM params Direct measurements (please?)
Peering through the glue Model builder: Illustration from I. Shipsey, Nature 427, 591 (2004) Lattice theorist Experimentalist:
Snapshot of recent work (Q2, 2011) f B → π ( q 2 ) f B , f B s B B d , B B s + ETM, PoS(LAT2009); HPQCD, PRD 76 (2007); HPQCD, PRD 73 (2006); HPQCD, PRL 92 (2004); RBC-UKQCD, PoS(LAT2007); FNAL/MILC, PRD 79 (2009) 054507; FNAL/MILC, PoS(LAT2008); HPQCD, PRD 80 (2009); FNAL/MILC, PRD 80 (2010) HPQCD, PRD 80 (2009) RBC-UKQCD, PRD 82 (2010) F B → D ∗ (1) F B → D (1) FNAL/MILC, NPB Proc Suppl (2005) FNAL/MILC, PRD 79 (2009) 014506 f K → π ˆ (0) B K f π , f K + NPLQCD, PRD 75 (2007); RBC-UKQCD, PRL 100 (2008); JLQCD, PRD 77 (2008); HPQCD, PRL 100 (2008); ETM, PRD 80 (2009); HPQCD, PRD 73 (2006); QCDSF, PoS(LAT2007); RBC-UKQCD, EPJ C69 (2010) RBC-UKQCD, PRL 100 (2008); PACS-CS, PoS(LAT2008); Aubin et al., PRD 81 (2010) PACS-CS, PRD 79 (2009); RBC-UKQCD, PRD 78 (2008); Aubin et al., PoS(LAT2008); MILC, PoS(CD09); MILC, RMP 82 (2010); JLQCD/TWQCD, PoS(LAT2009); ETM, JHEP 07 (2009); BMW, PRD 82 (2010)
b ➙ s is rare in the SM W t b s s b W W ν t γ, Z For energies ≪ m W � � 10 − G F � � � H eff = √ V tb V ∗ C i ( µ ) Q i ( µ ) ts 2 i =1 Wilson Local coefficients operators b s b s g 2 G F = √ 8 m 2 2 W γ � �
Dominant operators Decays SM operators e B → K ∗ γ s i σ µ ν (1 + γ 5 ) b i F µ ν Q 7 γ = 8 π 2 m b ¯ B s → φ γ e B → ( ρ / ω ) γ s b ) V − A (¯ = 8 π 2 (¯ ℓ ℓ ) V Q 9 V B → K ( ∗ ) ℓ + ℓ − B s → φ ℓ + ℓ − Q 2 = (¯ s c ) V − A (¯ c b ) V − A Λ b → Λ γ Λ b → Λ ℓ + ℓ −
Long distance effects Phenomenological calculations necessary Charmonium resonances b s Khodjamirian, et al, PLB 402 (1997) Low q 2 Khodjamirian, et al, arXiv:1006.4945 Large recoil c c Buchalla & Isidori, NPB 525 (1998) High q 2 Grinstein & Pirjol, PRD 62 (2000), PRD 70 (2004) Low recoil Beylich, Buchalla, Feldmann, arXiv:1101.5118 γ, Z Weak annihilation s, d b doubly Cabibbo-suppressed W K ∗ B ρ u u Ball, Jones, Zwicky, PRD 75 (2007) γ
Regions of applicability B → X s ℓ + ℓ − large recoil J/ ψ ψ ′ ✤ Short distance effects dominate at low q 2 ✤ Short distance effects dominate at high q 2 (Grinstein-Pirjol, Beylich-Buchalla- Feldmann) low recoil q 2 (GeV 2 ) Plot from E Lunghi’s CKM2008 talk
Latest from LHC b B 0 → K ∗ 0 µ + µ − and B 0 s → φ µ + µ − di ff erential branching fractions Parkinson, Moriond QCD, March 2012 LHCb(1 . 0 fb − 1 ) : B 0 → K ∗ 0 µ + µ − : 900 ± 34 signal events Theory Binned theory LHCb 1.5 ] 2 LHCb /GeV Preliminary 4 c � 1 -7 [10 2 q /d 0.5 BF d 0 0 5 10 15 20 2 2 4 q [GeV / c ] s → φ µ + µ − branching fraction reported at Moriond EW Measurement of the B 0 s → φ µ + µ − : 77 ± 10 signal events LHCb(1 . 0 fb − 1 ) : B 0 s → φ µ + µ − ) = (0 . 778 ± 0 . 097( stat ) ± 0 . 061( syst ) ± 0 . 278( B )) × 10 − 6 [preliminary] B ( B 0 The most precise measurements to-date and are consistent with SM expectations [4] Chris Parkinson Rare Beauty and Charm Decays at LHCb 12 / 22
Lattice QCD ✤ Field theory as statistical mechanics ✤ Mending errors ✤ [Decisions, decisions] ✤ Work in progress (rare B decay form factors)
Lattice QCD in a nutshell ✤ QCD Lagrangian � � µν F a,µν − � L = − 1 γ µ ( ∂ µ − igA a µ t a ) + m q 4 F a q ψ q ψ q = L g − ψQψ Quarks on sites ✤ Break spacetime up into a grid ✤ Maintains gauge invariance Glue on links ✤ Regulates the QFT nonperturbatively ✤ Breaking of Lorentz and translational symmetries scales like the lattice spacing a p ( p =2, usually)
Lattice QCD in a nutshell QFT : Imaginary-time path integral � J ( z � ) J ( z ) � = 1 � [ dψ ][ d ¯ ψ ][ dU ] J ( z � ) J ( z ) e − S E Z SFT : Sum over all microstates � J ( z � ) J ( z ) � = 1 � J ( z � ) J ( z ) e − βH � Z Tr Use the same numerical methods! Monte Carlo Calculation : Find and use field “configurations” which dominate the integral/sum
Lattice QCD in a nutshell Gluonic expectation values 1 � ψ ][ dU ] Θ[ U ] e − S g [ U ] − ¯ [ dψ ][ d ¯ ψQ [ U ] ψ � Θ � = Z 1 � [ dU ] Θ[ U ] det Q [ U ] e − S g [ U ] = Z Fermionic expectation values Probability weight � � [ dU ] δ ζ Γ δ ζQ − 1 [ U ] ζ det Q [ U ] e − S g [ U ] δζ e − ¯ � ¯ � ψ Γ ψ � = δ ¯ � � ζ, ¯ ζ → 0 Determinant in probability weight difficult Quenched approximation 1) Requires nonlocal updating; 2) Matrix becomes singular Set det Q = 1 Partial quenching = different mass for valence than for sea Q − 1 det Q
Lattice QCD progress ✤ Effects of light sea u+d+s quarks important ✤ Much progress using staggered quarks (+ 4th root hypothesis) ✤ Single set of lattice inputs (quark masses) ✤ [MILC Collab’n lattices] C. Davies, et al ., PRL 92 (2004)
Systematic errors Source of error Controllable limit Theory Chiral pert. th. Lattice volume L ≫ 1 /m π Brute force Lattice spacing Symanzik EFT a ≪ 1 / Λ QCD NRQCD, HQET m Q ≫ 1 /a Heavy quark mass m Q < 1 /a Extra-fine, extra-improvement Fermilab m Q ≈ 1 /a Light quark mass Chiral pert. th. m π ≪ m ρ , 4 πf π
Choice of discretizations ✤ Gluon field: improved actions, w/ various criteria (perturbative/nonperturbative Symanzik, RG) ✤ Light quarks: staggered, Wilson (clover), domain- wall, overlap, twisted-mass, ... ✤ Heavy quarks: static, nonrelativistic, relativistic (Fermilab (perturbative/nonperturbative), extrapolated light quarks)
HPQCD approach with Stefan Meinel, Zhaofeng Liu, Eike Müller, A. Hart, R. Horgan ✤ NRQCD formulation to calculate QCD dynamics of physically heavy b quark ✤ Improved staggered light quarks ✤ Matching to MSbar scheme in pert. th. (Müller, Hart, Horgan, PRD 83 , 2011) ✤ Can work in lattice frame boosted relative to B (Horgan et al. , PRD 80 , 2009) ✤ Stat. and EFT errors mandate working at low recoil ✤ N f = 2 + 1 (MILC) configurations. No unquenched calculations of B ➙ V form factors published yet.
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