Lattice calculation of BSM B-parameters using improved staggered - PowerPoint PPT Presentation

Lattice calculation of BSM B-parameters using improved staggered fermions in N f = 2 + 1 unquenched QCD Boram Yoon Los Alamos National Laboratory Oct 1, 2013 1 / 49

Let me introduce my self • Boram Yoon • Profile – Ph. D. in Physics (Feb, 2013) Seoul National University, Korea (Adv: Prof. Weonjong Lee) – Los Alamos National Lab (Aug, 2013) • Research Interests – Lattice Gauge Theory (QCD) – Chiral Perturbation Theory – Data Analysis – High Performance Computing • Projects in LANL – Neutron Electric Dipole Moments (nEDM) – Illuminating the Origin of the Nucleon Spin 2 / 49

Lattice calculation of BSM B-parameters using improved staggered fermions in N f = 2 + 1 unquenched QCD Boram Yoon Los Alamos National Laboratory Oct 1, 2013 3 / 49

Collaboration • Seoul National University – Yong-Chull Jang, Hwancheol Jeong, Jangho Kim, Jongjeong Kim, Kwangwoo Kim, Seonghee Kim, Weonjong Lee, Jaehoon Leem, Boram Yoon • University of Washington – Stephen R. Sharpe • Brookhaven National Laboratory – Hyung-Jin Kim, Chulwoo Jung • Korea Institute of Science and Technology Information – Taegil Bae 4 / 49

Beyond the Standard Model B-parameters Motivation & Background 5 / 49

Neutral Kaon System • Flavor eigenstates 0 = ( sd ) K 0 = ( sd ) , K • CP eigenstates 1 2( K 0 ± K 0 ) , √ K ± = CP | K ± � = ±| K ± � • Mass eigenstate K S = K + + ǫK − K L = K − + ǫK + 10 − 3 � � 1 + | ǫ | 2 , 1 + | ǫ | 2 , | ǫ | ≈ O � � • Preferable decays into pion states K S → 2 π ( via K + , CP even ) K L → 3 π ( via K − , CP odd ) 6 / 49

Direct / Indirect CP Violation • CP violating K L → ππ can occur in two ways: • K − (CP odd) → ππ (CP even) : Direct CPV � A [ K L → ( ππ ) 2 ] � 1 A [ K L → ( ππ ) 2 ] ε ′ K = √ A [ K S → ( ππ ) 2 ] − ε K A [ K S → ( ππ ) 0 ] 2 • ǫK + (CP even) → ππ (CP even) : Indirect CPV 1 � A [ K L → ( ππ ) 0 ] � ε K = √ A [ K S → ( ππ ) 0 ] 2 0 mixing K L can have small CP even component via K 0 − K 7 / 49

0 Mixing in the Standard Model K 0 − K • Arises from the ∆ S = 2 , sd → sd FCNC • Responsible for indirect CPV and ∆ M K ≡ M K L − M K S • Dominated by the following box diagrams: 8 / 49

0 Mixing in the Standard Model K 0 − K • Integrating out heavy particles, the box diagram can be replaced by a local, four-quark operator = G 2 F M 2 H ∆ S =2 16 π 2 F 0 Q 1 + h.c. W eff Q 1 = [¯ sγ µ (1 − γ 5 ) d ][¯ sγ µ (1 − γ 5 ) d ] 9 / 49

Kaon Bag Parameter – B K • In the SM, indirect CPV can be predicted as follows ε K ∼ known factors × V CKM × ˆ B K • ˆ B K is the RG invariant form of B K 0 | [¯ sγ µ (1 − γ 5 ) d ] | K 0 � B K = � K sγ µ (1 − γ 5 ) d ][¯ 0 | sγ µ γ 5 d | 0 �� 0 | sγ µ γ 5 d | K 0 � 8 3 � K • B K contains all the non-perturbative QCD contribution for ε K , can be calculated from lattice simulations 10 / 49

Experiment vs SM prediction of ε K • There are two methods (exclusive, inclusive) to determine V cb , whose results are somewhat different • SM prediction of ε K deviates from the experimental value about 3 σ for exclusive V cb channel (Y. Jang & W. Lee, 2012) 11 / 49

0 Mixing BSM Contribution to K 0 − K • In the Standard Model, only the “left–left” form contributes to the 0 mixing box diagram K 0 − K 0 | [¯ sγ µ (1 − γ 5 ) d ] | K 0 � � K sγ µ (1 − γ 5 ) d ][¯ • Considering BSM physics, integrating out heavy particles (e.g. squarks and gluinos in supersymmetric models) leads to new operators with Dirac structures other than “left–left” h, k, l, m ∈ { L, R } 12 / 49

BSM Operators • Considering BSM, generic effective Hamiltonian is s a γ µ (1 − γ 5 ) d a ][¯ s b γ µ (1 − γ 5 ) d b ] Q 1 = [¯ s a (1 − γ 5 ) d a ][¯ s b (1 − γ 5 ) d b ] Q 2 = [¯ 5 � H ∆ S =2 s a σ µν (1 − γ 5 ) d a ][¯ s b σ µν (1 − γ 5 ) d b ] = C i Q i Q 3 = [¯ eff i =1 s a (1 − γ 5 ) d a ][¯ s b (1 + γ 5 ) d b ] Q 4 = [¯ s a γ µ (1 − γ 5 ) d a ][¯ s b γ µ (1 + γ 5 ) d b ] Q 5 = [¯ • Once a BSM physics is chosen, C i are determined 0 | Q i | K 0 � , • If we know � K we can calculate ε K estimated by the BSM physics • Comparing with experiments, we can give constraints on the BSM physics 13 / 49

BSM B-parameters • BSM B-parameters � ¯ K 0 | Q i | K 0 � B i = N i � ¯ K 0 | ¯ sγ 5 d | 0 �� 0 | ¯ sγ 5 d | K 0 � s a (1 − γ 5 ) d a ][¯ s b (1 − γ 5 ) d b ] Q 2 = [¯ s a σ µν (1 − γ 5 ) d a ][¯ s b σ µν (1 − γ 5 ) d b ] Q 3 = [¯ s a (1 − γ 5 ) d a ][¯ s b (1 + γ 5 ) d b ] Q 4 = [¯ s a γ µ (1 − γ 5 ) d a ][¯ s b γ µ (1 + γ 5 ) d b ] Q 5 = [¯ ( N 2 , N 3 , N 4 , N 5 ) = (5 / 3 , 4 , − 2 , 4 / 3) • In the lattice calculation, forming dimensionless ratio reduces statistical and systematic error • Chiral perturbation expression is simpler 14 / 49

Lattice QCD 15 / 49

Lattice QCD • Non-perturbative approach to understand QCD • Formulated on discretized Euclidean space-time – Hypercubic lattice – Lattice spacing “ a ” – Quark fields placed on sites – Gauge fields on the links between sites; U µ 16 / 49

Lattice QCD • Expectation value �O ( U, q, ¯ q ) � � � = 1 � � � q ) e − S g [ U ] − � f ¯ q f D [ U ]+ m f q f D ¯ q D q D U O ( U, q, ¯ Z   = 1 �  O ( U, ( D [ U ] + m f ) − 1 ) e − S g [ U ] � � � D U det D [ U ] + m f  Z f • Integrating over the q and ¯ q gives � � determinant of Dirac operator, det D [ U ] + m f and quark propagators, ( D [ U ] + m f ) − 1 17 / 49

Lattice QCD • Expectation value �O ( U, q, ¯ q ) �   = 1 �  O ( U, ( D [ U ] + m f ) − 1 ) e − S g [ U ] � � � D U det D [ U ] + m f  Z f • Numerical Integration By generating random samples of gauge links, U µ according to the probability distribution, one can perform the integration using the Monte Carlo method dx f ( x ) p X ( x ) ≃ 1 � � � f ( X ) � = f ( x i ) N i where x i are random samples of X 18 / 49

Lattice QCD • Use numerical method (Monte Carlo simulation) to calculate integral �O� = 1 � q D q D U O e − S D ¯ Z • “ Lattice action ” is needed to simulate in discretized space-time S [ U, ¯ q, q ] = S G [ U ] + S F [ U, ¯ q, q ] • In this work, we use “ Staggered fermion ” for the lattice fermion – The fastest lattice fermion action – Suffered from “ taste symmetry breaking ”, but manageable 19 / 49

Beyond the Standard Model B-parameters Lattice Calculation of B-parameters & Data Analysis 20 / 49

Physical Results from Unphysical Simulations • Chiral Extrapolation – In the lattice simulation, the smaller quark mass requires the exponentially larger computational cost ⇒ Use light quark masses larger than physical light quark mass, and extrapolate to the physical light quark mass using chiral perturbation theory – Tuning the strange quark mass to precise physical quark mass is not practical ⇒ Extrapolate to the physical strange quark mass • Continuum Extrapolation – Simulation is done with finite lattice spacing ( a � 0 . 045 fm) ⇒ Extrapolate to continuum limit, a = 0 21 / 49

Data Analysis Strategy 1. Calculate raw data Calculate BSM B-parameters for different quark mass combinations ( m x , m y ) 2. Chiral fitting X-fit: Fix strange quark mass, extrapolate m x → m phys d Y-fit: Extrapolate m y → m phys s 3. RG Evolution Obatin results at 2 GeV and 3 GeV from µ = 1 /a 4. Continuum extrapolation Repeat [1–3] for different lattices and extrapolate to a = 0 22 / 49

Analysis Data Lattices generated with the N f = 2 + 1 improved “asqtad” staggered action by the MILC collaboration a (fm) am l /am s size 1 /a (GeV) ens × meas ID 28 3 × 96 0 . 09 0 . 0062 / 0 . 031 2 . 3 995 × 9 F1 28 3 × 96 0 . 09 0 . 0093 / 0 . 031 2 . 3 949 × 9 F2 40 3 × 96 0 . 09 0 . 0031 / 0 . 031 2 . 3 959 × 9 F3 28 3 × 96 0 . 09 0 . 0124 / 0 . 031 2 . 3 1995 × 9 F4 32 3 × 96 0 . 09 0 . 00465 / 0 . 031 2 . 3 651 × 9 F5 48 3 × 144 0 . 06 0 . 0036 / 0 . 018 3 . 4 749 × 9 S1 48 3 × 144 0 . 06 0 . 0072 / 0 . 018 3 . 4 593 × 9 S2 56 3 × 144 0 . 06 0 . 0025 / 0 . 018 3 . 4 799 × 9 S3 48 3 × 144 0 . 06 0 . 0054 / 0 . 018 3 . 4 582 × 9 S4 64 3 × 192 0 . 045 0 . 0028 / 0 . 014 4 . 5 747 × 1 U1 23 / 49

Operator Matching • To find continuum (NDR with MS) results from those regularized on the lattice, “operator matching” is needed • We use one-loop matching factors (J. Kim, W. Lee and S. Sharpe, 2011) • Matching scale µ = 1 /a g 2 � � O Cont z ij O Lat d Lat ik O Lat = − i j k (4 π ) 2 j ∈ ( A ) k ∈ ( B ) g 2 � � − γ ij log( µa ) + d Cont − d Lat z ij = b ij + ij − C F I MF T ij ij (4 π ) 2 24 / 49

Calculation of BSM B-parameters � ¯ K 0 | Q i | K 0 � � W ( t 1 ) Q i ( t ) W ( t 2 ) � B i = → N i � ¯ N ′ i � W ( t 1 ) P ( t ) � � P ( t ) W ( t 2 ) � K 0 | ¯ sγ 5 d | 0 �� 0 | ¯ sγ 5 d | K 0 � • B 2 calculated on F1 ( a = 0 . 09 fm) • Valence quark : m x = 1 10 m s m y = m s 25 / 49