Solving the B  K  Puzzle Cheng-Wei Chiang National Central University & Academia Sinica Cheng-Wei Chiang National Central University & Academia Sinica Based upon following works: PRD 69, (2004) [hep-ph/ ]; PLB 580, 186 (2004) [hep-ph/ ]; PRD 70, (2004) [hep-ph/ ]; PLB 598, 218 (2004) [hep-ph/ ]; hep-ph/ Third International Conference on Flavor Physics October 3- 8, 2005

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)2 Outline  The problem  Large color-suppressed amplitude solution  Large electroweak penguin solution --- FCNC Z 0 boson  The  K and K *  modes  Summary and outlook

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)3  The amplitudes of the four K  modes can be decomposed according to the flavor flow topology as follows (ignoring smaller amplitudes): Flavor Amplitudes of K  Modes

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)4  Two ratios of the BR’s of K  modes (charged and neutral):  To the leading order, R c and R n should be the same in the SM; corrections should be ~ O([(C 0 + P 0 EW ) / P 0 ] 2 ) ~ O(  2 ). K  Puzzle – Phase I 2.4  1.9  1.5  2.4  1.9   1.5 

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)5  Now the bigger puzzle is in two CPA’s of the K  modes:  In the SM, T 0 and C 0 have the same weak phase (  ) and a small relative strong phase  A CP (K +   ) and A CP (K ±  0 ) are expected to at least have the same sign. K  Puzzle – Phase II 3.6  ! establishing direct CPV in B system at 5.7  level

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)6  For puzzle phase-I only: underestimate of  0 detection efficiency, thus overestimating the BR’s of those corresponding modes. [Gronau and Rosner, PLB 572, 43 (2003)]  New mechanism in SM: large color-suppressed amplitude C from NLO vertex corrections. [Charng & Li, PRD 71, (2005); He & McKellar, hep-ph/ ]  Beyond SM: large electroweak penguin amplitude P EW from new physics. [Yoshikawa, JKPS 45, S479 (2004); Buras et al, PRL 92, (2004); NPB 697, 133 (2004); Baek, Hamel, London, Datta and Suprun, PRD 71, (2005); Hou, Nagashima and Soddu, hep-ph/ ] [see also talks by Baek, Kim, Nagashima, Oh, Yoshikawa and Yu] Possible Explanations

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)7 Large Color-Suppressed Amplitude

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)8  charmless V P modes,  。  。 ; charmless P P modes,  。   。 ; both consistent with constraints from other observables. Global Fits to VP and PP Modes VP PP [CWC, Gronau, Luo, Rosner, and Suprun, PRD 69, (2004); PRD 70, (2004)]

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)9  charmless V P modes,  。 ; charmless P P modes,  ~  。 ; both still consistent with constraints from other observables. Updated Global Fits PPVP

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)10  The ratio of | C / T | in  2 fits to available data in the   and K  modes ranges from 0.5 to >1.  In a fit to all the available PP data, the ratio is 0.89 (old)  0.77 (new).  They are larger than the naïve expectation (~ 0.25 – 0.3) in SM.  There is a large relative strong phase: arg[C / T ] ~ 90 。.  Mainly driven by the mysteriously large BR(  0  0 ). Large C ( T+C ) exp(i  ) ( T+C ) exp(-i  ) T exp(i  ) P Br ¼ Br

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)11 [Li, Mishima and Sanda, hep-ph/ ] [Li, Mishima and Sanda, hep-ph/ ]  | C | or | C 0 | is enhanced by a factor of 2 to 3 after including the vertex corrections at NLO.  Strong phase changes a lot. NLO Vertex Correction

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)12 PQCD Predictions

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)13 Large Electroweak Penguin Amplitude

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)14 [see also Valencia’s talk] [see also Valencia’s talk]  In most extensions beyond the SM, there are always extra heavy neutral Z 0 gauge bosons.  Properties of the new gauge boson, such as the mass and couplings, are model dependent.  In the gauge eigenbasis, the general Z 0 neutral-current Lagrangian is given by  In string models, it is possible to have family- nonuniversal Z 0 couplings to fermion fields due to different constructions for the three families. [Chaudhuri et al, NPB 456, 89 (1995)] Z 0 Boson

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)15  After flavor mixing, one obtains FCNC Z 0 interactions (non-diagonal) in the fermion mass eigenstates, which may lead to new CP-violating effects:  This may induce flavor-violating Z couplings if there is Z-Z 0 mixing.  In view of the fact that the  K data can be explained with a new EW penguin amplitude, we assume that the Z 0 mainly contributes to these operators and obtain  This is possible through an O(10 -3 ) mixing angle between Z and Z 0.  Here we only include the LH coupling for the Z 0 -b-s coupling. RH coupling can be included as well, at the price of more free parameters. Z 0 -Induced FCNC

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)16  The effective Hamiltonian of the anti-b → anti-s q anti-q transitions mediated by the Z ' is  Even though the operator is suppressed by the heavy Z 0 mass, they can compete with SM loop processes because of their tree-level nature. Low-Energy Effective Hamiltonian s s Z 0Z 0 Z 0Z 0 [Barger, CWC, Langacker and Lee, PLB 580, 186 (2004); 598, 218 (2004)]

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)17  To study the K  puzzle Buras et al introduce the ratio [Buras et al, PRL 92, (2004] [Buras et al, PRL 92, (2004)]  One should note that although c 7,8 play a less important role compared to c 9,10 within the SM, they can receive contributions from the Z ' such that we cannot neglect them.  In the analysis of Buras et al, it was implicitly assumed that new physics contributes dominantly to the (V – A) ­ (V–A) EW penguins.  In the analysis of Buras et al, it was implicitly assumed that new physics contributes dominantly to the (V – A) ­ (V – A) EW penguins.  As one of their conclusions under this assumption, S  K S will be greater than S  K S or even close to unity if one wants to explain the K  anomaly. Some Notations

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)18  Using the same hadronic inputs from   modes as given by Buras et al, we get two sets of solutions: (q,  ) = (1.61, –84 。 ) and (3.04, 83 。 )  (0.94, –85 。 ) and (2.37, 85 。 ), whereas they only take the small q solution. Solutions

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)19  Use the following variables to parameterize our model:  We obtain the solutions  It is possible to find solutions (except for (A L )) that account for both the K  and S  K S data because the contributions from the O 7,8 (from RH couplings at the Z 0 -q-qbar vertices) and O 9,10 operators interfered differently in these two sets of decay modes. Fitting S  K S Too

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)20 K *  and  K Decays

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)21  A distinction between the V P system and the P P system is that there are two types of amplitudes for each topology in the former case, depending upon whether the spectator quark in B ends up in the P or V meson in the final state.  If new physics appears in the P P system, it is likely to show up in the V P system too.  The flavor amplitude decompositions and data are given below: K *  and  K Modes [CWC, hep-ph/ ]

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)22  With r 1 ≡ |T 0 P / P 0 P | and r 2 ≡ |T 0 V / P 0 V |, we have:  With particular choices of r 1 and r 2, one may constrain the weak phase  without knowing the relative strong phase. Another Method for Constraining 

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)23  Hopefully, higher statistics in data can improve the bounds. (BR’s are measured at (5~10)% for K  and (10~20)% for K *  and  K. ) Result r 1 = 0.37:  ≥ 76  or   22 

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)24  Instead of treating r 1 and r 2 independently, one may employ the factorization assumption and get  This number can be compared with the result of 0.7 ± 0.1 obtained from a global fit. [CWC, Gronau, Luo, Rosner and Suprun, PRD 69, (2004)]  There are four parameters ( , r 1,  P, and  V ) for the four observables in the above-mentioned equations. Solving them exactly is possible and gives (up to discrete ambiguities of the phases and central values only)  = 69 。, r 1 = 0.19,  P = 170 。, and  V = 128 。.  (The above results are obtained by cheating because of the assumption |P 0 P | = |P 0 V |  (The above results are obtained by cheating because of the assumption |P 0 P | = |P 0 V |.) Combining the K *+  – and  – K + Modes

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)25  One can further consider the following observables:  As in the case of the K  system, each pair of R c and R n should be equal as long as the color-suppressed and electroweak penguin amplitudes are negligible, as expected in the SM. R c and R n Again

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)26  Note that we have the following two approximate sum rules: which are held only when the terms |C ' V(P) + P ' EW,V(P) | Re[T 0 * P(V) (C ' V(P) + P ' EW,V(P) )] are negligible in comparison with the dominant penguin contributions.  The first sum rule is satisfied by current data.  Using the second sum rule, one can deduce from current data that BR(  + K 0 ) = (8.8 ± 4.1) £ 10 -6, consistent with the current upper bounds.  In particular, BR(  + K 0 ) thus obtained and the measured BR(K * 0  + ) are about the same, which is an indication of the equality |P ' P | = |P ' V |. Sum Rules

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)27  1.5  discrepancy between R c and R n ; 3.6  discrepancy between the CPA’s of K +   and K +  0.  Nonuniversal Z’FCNC and provide new CP-violating sources for low-energy physics.  Nonuniversal Z’ can induce FCNC and provide new CP-violating sources for low-energy physics.  BR’s and CPA’s of K  and  K S modes can be explained in this model.  R c and R n for K*  agree, but with a large uncertainty. For the  K mode, a direct experimental comparison is not yet available because there is no data for BR(  + K 0 ). Employing the relation |P 0 P | = |P 0 V | for the  + K 0 mode, the current data show an approximate agreement between R c and R n.  If the puzzles in the K  system is due to new (short-distance) physics, we also expect deviations in the K*  and  K systems. This does not seem to be the case according to the current data.  A precise determination of the rates of the  + K 0 decay will be very helpful in checking the R c -R n relations and the |P’ P |-|P’ V | equality. Summary and Outlook - I

C.W. ChiangSolving the B to K pi Puzzle (10/6/2005)28  A true solution to the puzzle is still in the hand of experimentalists. Summary and Outlook - II