Neutral B mixing The Standard Model and Beyond E. Freeland, C. - PowerPoint PPT Presentation

Neutral B mixing The Standard Model and Beyond E. Freeland, C. Bouchard, C. Bernard, A.X. El-Khadra, E. Gamiz, A.S. Kronfeld, J. Laiho, and R.S. Van de Water for the Fermilab Lattice and MILC Collaborations

⌇ ⌇ ⌇ ⌇ ⌇ ⌇ Neutral B mixing V tq W ± b q ɤ ɤ ɤ ɤ ɤ ɤ ɤ u, c, t u, c, t ɤ 0 B 0 ɤ B ɤ q q ɤ ɤ ɤ ɤ ɤ ɤ b V ∗ W ∓ q tq B 0 In the Standard Model, mixing is suppressed q loop process, diagram with dominates, m t but is Cabibbo suppressed ⇒ “Relatively” easy for new physics to cause observable effects.

⌇ ⌇ ⌇ ⌇ Matrix Elements t, c, u 0 0 B 0 B 0 W ± B B W ∓ q q q q O i t, c, u 5 � H e ff = C i O i i =1 Operators are O 1 = (¯ b α γ µ Lq α ) (¯ b β γ µ Lq β ) O 2 = (¯ b α Lq α ) (¯ SM b β Lq β ) O 3 = (¯ b α Lq β ) (¯ BSM b β Lq α ) O 4 = (¯ b α Lq α ) (¯ b β Rq β ) O 5 = (¯ b α Lq β ) (¯ b β Rq α )

⌇ ⌇ ⌇ ⌇ Matrix Elements t, c, u 0 0 B 0 B 0 W ± B B W ∓ q q q q O i t, c, u 5 � H e ff = C i O i i =1 Operators are Common parametrization O 1 = (¯ b α γ µ Lq α ) (¯ b β γ µ Lq β ) � ¯ B 0 q |O i ( µ ) | B 0 q � ∝ f 2 B q B i ( µ ) O 2 = (¯ b α Lq α ) (¯ b β Lq β ) O 3 = (¯ b α Lq β ) (¯ b β Lq α ) O 4 = (¯ b α Lq α ) (¯ b β Rq β ) f 2 B i ( µ ) B q O 5 = (¯ b α Lq β ) (¯ b β Rq α )

Experiment and SM: ∆ M q � G 2 F M 2 � W S 0 � ¯ tq | 2 B 0 q |O 1 ( µ ) | B 0 ∆ M q = η B ( µ ) | V tb V ∗ q � 4 π 2 experiment known want need from lattice < 1% tq | 2 � ¯ | V tb V ∗ B 0 q |O 1 ( µ ) | B 0 q � Our ability to constrain , is limited by . “Tension” in the CKM matrix. Lenz et al., arXiv: 1203:0238; Laiho et al. PhysRevD. 81, 034503, and end-of-2011 update

Experiment and SM: ∆ M q 2 � ¯ 2 M B s B 0 s |O 1 ( µ ) | B 0 � � � � s � ∆ M s V ts V ts � � � � ξ 2 = d � ≡ � ¯ � � � � B 0 d |O 1 ( µ ) | B 0 ∆ M d V td V td M B d � � � � experiment want lattice

Experiment and SM: ∆ M q SU(3)-breaking ratio 2 � ¯ 2 M B s B 0 s |O 1 ( µ ) | B 0 � � � � s � ∆ M s V ts V ts � � � � ξ 2 = d � ≡ � ¯ � � � � B 0 d |O 1 ( µ ) | B 0 ∆ M d V td V td M B d � � � � lattice want experiment - Some (lattice) errors cancel in the ratio of matrix elements, - In CKM matrix fits, use of can aid in minimizing correlations ξ between lattice inputs.

Experiment and SM: ∆Γ q Recent experimental results are putting focus on . ∆Γ (Lenz et al., arXiv:1203.0238; Haisch, Moriond 2012 ) � � G 1 � ¯ q � + G 3 � ¯ B 0 q |O 1 ( µ ) | B 0 B 0 q |O 3 ( µ ) | B 0 ∆Γ q = q � cos φ q + O (1 /m b , α s ) also needed dominates Lenz, Nierste JHEP 0706:072, 2007 hep-ph/0612167 Beneke, Buchalla, Dunietz, PRD 54:4419, 1996, Erratum-ibid.D 83 119902 (2011); hep-ph/9605259v1

Experiment and SM: ∆Γ q Recent experimental results are putting focus on . ∆Γ (Lenz et al., arXiv:1203.0238; Haisch, Moriond 2012 ) � � G 1 � ¯ q � + G 3 � ¯ B 0 q |O 1 ( µ ) | B 0 B 0 q |O 3 ( µ ) | B 0 ∆Γ q = q � cos φ q + O (1 /m b , α s ) ∆Γ / ∆ M � O 3 � / � O 1 � yields O R ≡ O 2 + O 3 + (1 / 2) O 1 useful for estimating errors. 1 /m b

Experiment and BSM: ∆ M q 5 0 � C i ( µ ) � B 0 ∆ M q = q |O i ( µ ) |B q � i =1 need from lattice model experiment dependent < 1% Including BSM contributions, takes the generic form above. ∆ M q Lattice values of (matrix elements of) through are needed to O 1 O 5 check that a given BSM model is consistent with experiment.

Status of Lattice

Status of Lattice: O 1 FNAL-MILC arXiv: 1205.7013, ξ = 1 . 268(63) 5.0% submitted to PRD

Status of Lattice: O 1 HPQCD Gamiz et al ., Phys.Rev.D80:014503, 2009, arXiv:0902.1815 2.6% ξ = 1 . 258(33) � ˆ 6.8% B B d = 216(15) MeV f B d � ˆ B B s = 266(18) MeV 6.9% f B s RBC Albertus et al ., Phys.Rev.D82:014505, 2010, arXiv:1001.2023 11% ξ = 1 . 13(12) domain-wall test calculation; one, 0.11 fm, lattice spacing

Status of Lattice: O 1 HPQCD Gamiz et al ., Phys.Rev.D80:014503, 2009, arXiv:0902.1815 � ˆ 6.8% B B d = 216(15) MeV f B d FNAL-MILC 2012 � ˆ B B s = 266(18) MeV 6.9% f B s 2.6% ξ = 1 . 258(33) RBC 2010 RBC Albertus et al ., Phys.Rev.D82:014505, 2010, arXiv:1001.2023 HPQCD 2009 11% ξ = 1 . 13(12) domain-wall test calculation; one, 0.11 fm, lattice spacing 1 1.2 1.4 1.8 1.6

Status of Lattice: O 1 ... 5 Quenched : Two ensembles : O 2 , 3 Becirevic et al., JEHP 0204 (2002) 0250 E. Dalgic et al., PRD76:011501, 2007 Preliminary FNAL-MILC: Lattice 11 Proceedings (Dec 2011), arXiv:1112:5642 Estimated 9 to 6% on O 1 � ˆ f B q B B q ETMC also working on this.

Details of Our Calculation

Previous vs current analysis Ensembles more ensembles higher statistics smaller lattice spacing smaller light-quark mass Results full set of matrix elements bag parameters in conjunction with analysis f B (Ethan Neil’s talk) able to do all ratios and combinations Use complete ChiPT expression This alone improves the error on from 5.0% to 3.8%. ξ

Analysis Overview 0 B 0 B 0 B 0 B q q q q • Generate two- and three-points correlator data. • Fit 2pt+3pt correlators simultaneously for each meson. • Renormalize & match the matrix elements. • Do a chiral and continuum extrapolation for each matrix element.

Actions and Ensembles ~2000 configurations MILC (asqtad) gauge configurations ~1000 configurations • 2+1 asqtad sea quarks, ~500 configurations • tadpole improved gluons analyzed m l /m s • from 0.4 to 0.05 partially analyzed 0.5 gauge configurations arXiv: 1205.7013 ξ light valence quark 0.4 • staggered action • mass from > to 0.05 . 0.3 m s m s m l / m s 0.2 heavy valence quark 0.1 • improved Wilson action • Fermilab interpretation 0 0 0.02 0.04 0.08 0.1 0.12 0.14 0.06 lattice spacing in fm physical point

Actions and Ensembles ~2000 configurations MILC (asqtad) gauge configurations ~1000 configurations • 2+1 asqtad sea quarks, ~500 configurations • tadpole improved gluons analyzed m l /m s • from 0.4 to 0.05 partially analyzed 0.5 gauge configurations Lat11 light valence quark 0.4 • staggered action • mass from > to 0.05 . 0.3 m s m s m l / m s 0.2 heavy valence quark 0.1 • improved Wilson action • Fermilab interpretation 0 0 0.02 0.04 0.08 0.1 0.12 0.14 0.06 lattice spacing in fm physical point

Actions and Ensembles ~2000 configurations MILC (asqtad) gauge configurations ~1000 configurations • 2+1 asqtad sea quarks, ~500 configurations • tadpole improved gluons analyzed m l /m s • from 0.4 to 0.05 partially analyzed 0.5 gauge configurations Lat12 light valence quark 0.4 • staggered action • mass from > to 0.05 . 0.3 m s m s m l / m s 0.2 heavy valence quark 0.1 • improved Wilson action • Fermilab interpretation 0 0 0.02 0.04 0.08 0.1 0.12 0.14 0.06 lattice spacing in fm physical point

Three-points � m e − E m t + ( − 1) ( t +1) ( Z p m t � � C 2pt ( t ) = Z 2 m ) 2 e − E p m � � C 3pt ( t 1 , t 2 ) = Z m Z n � O � mn e − E m t 1 e − E n t 2 m,n mn ( − 1) t 1 e − E p + Z p m Z n � O � p m t 1 e − E n t 2 mn ( − 1) t 1 e − E m t 1 e − E p + Z m Z p n � O � p n t 2 � mn ( − 1) t 1 + t 2 e − E p m t 1 e − E p + Z p m Z p n � O � pp n t 2 ✦ simultaneous fit of two-point + three-point; ✦ constrains energies and two-point amplitudes ✦ use constrained curve fitting

Renormalization and Matching Operators mix under renormalization (even in the continuum). E.g. � O 1 � R = (1 + α s ζ 11 ) � O 1 � + α s ζ 12 � O 2 � ζ ij are calculated using 1-loop perturbation theory. We use the “V” scheme as implemented by Q. Mason et al. with 4-loop running. α s = α v (2 /a ) Q. Mason et al. [HPQCD Collaboration], Phys. Rev. Lett. 95 , 052002 (2005) hep-lat/0503005 T. van Ritbergen et al., Phys. Lett. B 400, 379 (1997) hep-ph/9701390

Data O 2 -0.5 -0.75 3 <O 2 > / M B -1 r 1 -1.25 -1.5 0 1 2 0.5 1.5 2 ( r 1 m π )

Data O 3 0.2 0.15 3 <O 3 > / M B 0.1 r 1 0.05 0 0 1 2 0.5 1.5 2 ( r 1 m π )

Status: ChiPT We use SU(3), partially-quenched, heavy-meson, staggered ChiPT continuum, PQ: Detmold and Lin, aXiv:0612028, hep-lat, 2006 Claude Bernard’s talk With staggered light quarks, matrix elements of wrong-spin operators appear in the ChiPT. Because the five matrix elements form a complete basis, � O 1 ... 5 � wrong-spin contributions can be written in terms of them. ✦ Mixing occurs: and � O 4 � � � O 5 � � O 1 � � � O 2 � � � O 3 � ✦ No new LEC’s are introduced.

Status: ChiPT E.g. � � W qb + W bq 0 q |O q 1 | B 0 + T q + Q q + ˜ T (a) + ˜ Q (a) � B q � = β 1 1 + q q 2 +(2 β 2 + 2 β 3 ) ˜ 3 ) ˜ T (b) Q (b) + (2 β ′ 2 + 2 β ′ q q O 1 +analytic terms wrong-spin contributions ✦ Mixing occurs. ✦ No new LEC’s are introduced. We will do a simultaneous fit for each set of mixed operators.

Status of Lattice: O 1 FNAL-MILC arXiv: 1205.7013, ξ = 1 . 268(63) 5.0% submitted to PRD