1. CKM06 A. Soni Φ 3 (γ) from charmless modes using U-spin Amarjit Soni HET, BNL (soni@bnl.gov)

2. CKM06 A. Soni Outline Introduction Application to B +- : 4 Sets of γ’s Missing modes & theo expectations Application to (B 0,B S ): 7 Additional sets of γ’s Numerical results from B+- Numerical results from B0,Bs & overlap with B+ Handle(s) on systematic errors Summary & Outlook

3. CKM06 A. Soni Introduction & Motivation

4. CKM06 A. Soni Use of flavor symmeries QCD complicates extraction Specifically for γ in the long run “BDK” methods extremely clean, given enough # of B’s, can exract γ to O(0.1%) However,BDK methods are rather insensitive to NP QCD respects flavor symmetries…exploit these to get γ with much less #of B’s but expect systematic errors O(few%)……due sym. breaking. But now method is sensitive to NP as, unlike for BDK, penguin graphs are important. In measuring UT angles REDUNDANCY is of CRUCIAL IMPORT

5. CKM06 A. Soni Use of Flavor Symm in UT (sample) Talk based on AS+ D. Suprun, hep- ph/0511012; hep-ph/0609089 Uspin also used extensively by R. Fleischer; see e.g. PLB459,306(’99);EPJC10,299(’99) ForSU3useagesee:Chiang,Gronau,Rosner,Sup run, many papers A celebrated use of isopsin: Gronau & London, PRL’91

6. CKM06 A. Soni U-spin Basics

7. CKM06 A. Soni

9. Effective Hamiltonian & Uspin

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11. Contributing diagrams HUGE ADVANTAGE of U-SPIN USE IS THAT DIAGRAMMATIC APPROACH IS NOT NEEDED….{KEY difference with heretofore use of SU3 (CGRS for γ)} THEREFORE NOTHING IS IGNORED and ONLY ONE APPROXIMATION INVOLVED IN use of U-spin ( CONTRAST with SU3 usage for γ) Also sharp contrast in fact with isospin use for α For correspondence note, tree, penguins (QCD & EW) and annihlation contribute NOTE also each transforms as a U-spin Doublet, ΔU=1/2

12. CKM06 A. Soni Manifest U-spin symmetry in graphical topologies This crucial advantage of U-spin also emphasized by Flescher,see PLB’99; EPJC’99

13. CKM06 A. Soni Contrast with isospin in use for α extraction ALSO some EWP operators are LXR and not LXL

14. CKM06 A. Soni Unknowns & Observables (B+-) For B+- there are 4 subsets: P0 P+-; P0 V+-; V0 P+-; V0 V+- Each set has 8 modes for B+ and 8 for B- Each set has 12 unknowns (one of which is φ 3 (γ)) and 16 observables …therefore soluble THUS from B+- you can get 4 extractions of γ ; so method has built in indication of syst. error due Uspin…THIS IS THE ONLY SOURCE

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17. Missing modes for B+- Two modes Φπ+-; K*0 K+- Br not known….. Even tighter bounds will help For other 4 sets improved Br’s will of course continually help improve statistics

18. CKM06 A. Soni Neutral B,B s decays

19. CKM06 A. Soni Neutral B,B s decays:Available data sets for solving for φ 3 (γ) There are now 7 subsets each of which can be used for γ : P - P +, V - V +, P - V +, V - P +, P 0 P 0, V 0 V 0, V 0 P 0 This is in addition to 4 extractions possible from charged B’s Spread in these many (11!) values should give one indication of U-spin breaking systematics…(that is the ONLY systematic) CONTRAST these 11 modes (for γ with Uspin) with 3 modes (for α with isospin)

20. CKM06 A. Soni Specific modes: unknown parameters & data points For each of the 4 charged cases {P - P +, V - V +, P - V +, V - P + ) there are 6 relevant decay modes, e.g. B 0 -> π- π+, K- K+, π- K+ & cc Bs -> K- K+, π- π+, K- π+ These involve 8 unknown params (including γ) but provide 12 data points; therefore soluble… Similarly for VV, VP, PV …each has 10 data pts. B0 by itself is never enough in any given category (unlike B+-).

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25. other cracks at systematic error U-spin method allows model independent determination in the symmetry limit. However, systematic error due to symmtery breaking is (as always) difficult to determine reliably. I. Since method allows several determination of gamma, from the various subsets, the spread will give an indication of the systematics II. In the Su(3) method (CGRS), Errors determined by fK=fpi is O(1-2 degrees)…relevant also to U-spin

26. CKM06 A. Soni Summary &Outlook U-spin symmetry allows use of B+- &(B0,Bs) for model independent exraction of φ 3 (γ) (upto sym. breaking corrections which are hard to reliably estimate). Several subsets of data can be used in 2-body modes; each gives γ Method is sensitive to NP (unlike B->DK) Crucial advantage of the method is it does not use diagrammatic description; so none is neglected. Irreducible theory error i.e. systematic probably O(few degrees) Current data (~10 8 B’s) gives Δ φ 3 ~ 8 deg. Outlook: with ~2 X10 9 should be able to reduce Δ φ 3 ~O(few deg.) i.e. systematic error