General scan in flavor parameter space in the models with vector quark doublets and an enhancement in B X s γ process
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1 General scan in flavor parameter space in the models with vector quark doublets and an enhancement in B X s γ process Wang Wenyu Beijing University of Technology Hefei / 31
2 OUTLINE: 1 The problem 2 The Trick of diagonalization of vector quark doublet 3 B X s γ process in extension of the SM with one vector like quark doublet The standard model with vector like quarks Enhancement in b s transition 4 Summary 2/ 31
3 The problem: In the SM M U, M D for the up and down type fermions. and the so called CKM matrix Z U M UU U = M D U = diag.[m u,m c,m t ], Z D M DU D = M D D = diag.[m d,m s,m b ], V CKM = U U U D. Since M U, M D come from separate Yukawa couplings, we can always set one of the matrices diagonal, for example M U, and use the CKM matrix to get the Yukawa couplings Z D m d m s m b V CKM = Y D Y D 11 v Y 12 Dv Y D 21 v Y 22 Dv Y D Y31 Dv Y 32 Dv Y D 13 v 23 v 33 v 3/ 31
4 In the model witha vector doublet, 1 Q : 3, 2, Q : 3, 2, resulting bilinear term in the lagrangian M U = M D = M V Q Q. Y11 Uv Y 12 Uv Y 13 Uv Y21 Uv Y 22 Uv Y 23 Uv Y31 Uv Y 32 Uv Y 33 Uv M41 V M42 V M43 V, Y11 Dv Y 12 Dv Y 13 Dv Y21 Dv Y 22 Dv Y 23 Dv Y31 Dv 32 Dv 33 Dv M41 V M42 V M43 V Though this is just a numerical problem, when one treats the VLP contributions to the flavor physics seriously, diagonalization of quark matrices will be the first and important step. 4/ 31
5 The Trick of diagonalization of vector quark doublet diagonalization of N N matrix M U and M D : Z U M UU U = M D U, Z D M DU D = M D D in which MU D,MD D are the diagonal mass matrices for up and down type quark, respectively. Adding two matrices M U + M D = ( Z U MU D U CKMN + Z D MD D ) U D The left side of the equation is Y11 Uv 11 Dv 12 Uv 12 Dv 13 Uv 13 Dv M U1N + M D1N Y21 Uv 21 Dv 22 Uv 22 Dv 23 Uv 23 Dv M U2N + M D2N Y31 Uv 31 Dv 32 Uv 32 Dv 33 Uv 33 Dv U3N + M D3N M UNN + M DNN 5/ 31
6 We can denote the matrix in the form as ( MA M M U + M D = M UD = B M 0 M C ) in which M A, M B, M 0 are (N 1) (N 1), (N 1) 1 and 1 (N 1) matrices correspondingly. Input is (m u,m c,m t, m X ), (m d,m s,m b,,m Y ) and a matrix U CKMN ( ) U CKMN = U U U (UCKM ) D = 3 3 U NN U ud U us U ub = U cd U cs U cb U td U ts U tb U NN (U CKM ) 3 3 is not an ordinary CKM matrix V CKM which is non-unitary in this case. 6/ 31
7 In the similar way we denote U D as ( UDA U U D = DB U D0 U DNN ) ( MA U M UD U D = DA + M B U D0 M A U DB + M B U DNN M C U D0 M C U DNN = ( Z U MU D U CKMN + Z D MD D ) ) We can get the last line of U D simply by inputting M D U, MD D, U CKMN and random Z U, Z D : ( ZU M D U U CKMN + Z D M D D )last line = ( M C U D0 M C U DNN ) = M C U DN where U DN = ( U DN1 U DN2 U DNN ) is a unit vector in N dimension. 7/ 31
8 Next we use the unit vector to generate total U D. U DN 1 can be determined as U DN 1 = ( U DN2 UDN1 2 + U DN2 2 U DN1 0 0 ) UDN1 2 + U DN2 2 Then we use the first three elements of U DN and U DN 1 to generate U DN 2 : Normalize the algebraic complements of first line of the 3 3 matrix. Step by step, we can finally get (U D1, U D2,, U DN 1 ) and form a special U S D U D1 U DN 2 U DN 1 U DN = U D11 U D12 U D13 U D1N 0 U D(N 2)1 U D(N 2)2 U D(N 2)3 0 U D(N 1)1 U D(N 1)2 0 0 U DN1 U DN2 U DN3 U DNN 8/ 31
9 From above steps, we can see that (U D1, U D2,, U DN 1 ) can be rotated into any other orthogonal N 1 vectors to construct random matrix M A and M B, only U DN must be kept unchanged. Therefore, a general unitary matrix can be realized by timesing a unitary N N matrix U R, U D = U R U S D = ( URN ) U S D in which U RN 1 is a (N 1) (N 1) unitary matrix. We finish the work by U U M U M D = U CKMNU D = Z U M D U U U = Z D M D D U D 9/ 31
10 Summary of the method Step 1: Chose (m u,m c,m t,,m X,m d,m s,m b,,m Y ) and U CKMN and input random unitary matrices Z U and Z D ; Step 2: Determine the last line of matrix Z U M D U U CKMN + Z D M D D as M C ( UDN1 U DN2 U DNN ) and normalize it into a unit vector U DN. Step 3: Use the unit vector U DN to generate other N 1 to form a special U S D U S D = ( U D1 U DN 2 U DN 1 U DN ) T. Step 4: Generate a N 1 unitary matrix U RN 1 and a general U D is obtained by Step 5: Use these equations U D = U R U S D. U U = U CKMNU D, M U = Z U M D U U U, M D = Z D M D D U D, to get the inputs for the flavor physics. 10/ 31
11 11/ 31
12 One dimension well 12/ 31
13 Figure: Wave of one dimension well 13/ 31
14 B X s γ process in extension of the SM with one vector like quark doublet 14/ 31
15 Table: A simple extension of the standard model with one vector like quarks doublet SU(3), SU(2), U(1) «U Q = 3, 2, D L u R 3, 1, d R 3, 1, 1 3 «SU(3), SU(2), U(1) Vd V Q = 3, 2, V 1 u 6 R V ul 3, 1, V dl 3, 1, 3 The lagrangian for two quarks of the model is written as: L = Y d QHdR + Y u Q HuR + Y V u VQ H V ul + Y V d VQ H V dl +M Q V q Q + M u VuL u R + M d VdL d R + h.c., in which A B = ǫ ij A i B j. The mass matrices of up and down quarks in the basis of (u, c, t, V u ) and (d, s, b, V d ): M U = Yu 11 v Yu 12 v Yu 13 v Mu 1 Yu 21v Y u 22v Y u 23v M2 u Yu 31v Y u 32v Y u 33v M3 u MQ 1 M2 Q M3 Q Y V u v, M D = Yd 11v Y d 12v Y d 13v M1 d Yd 21v Y d 22v Y d 23v M2 d Yd 31v Y d 32v Y d 33v M3 d MQ 1 MQ 2 MQ 3 Y Vd v 15/ 31
16 These two matrices can be diagonalized by unitary matrices U and Z, Z u M UU u = diag.[m u,m c,m t,m X ], Z d M DU d = diag.[m d,m s,m b,m Y ]. Product of the two matrices is denoted as U CKM4 = U uu d, which is unitary 4 4 matrix. Feynman rules for the interaction of ū l d j W + i g 2 γ µ [ g W L (i,j)p L + g W R (i,j)p R], where g W L (i,j) = 3 m=1 Uu mi U m,j d, gr W (i,j) = Zu 4i Z 4j d, 16/ 31
17 Feynman rules for the interaction of ū l d j G + and d l d j Z in the Feynman gauge: g [ i g G L (i, j)p L + g G ] g R(i, j)p R, i 2 γ µ [ gl(i, Z j)p L + g Z ] R(i, j)p R, 2mW where g G L(i, j) = g G R (i, j) = 3 3 k,m=1 k,m=1 Y km u vzu ki U mj d Y mk d vz kj d Uu mi + Y V d vzu 4i U 4j d, g Z L (i, j) = 1 2cosθW [( sin2 θ W ) Y V u vz 4j d Ud 4i. gr(i, Z 1 j) = [ 2 2cosθW 3 sin2 θ W δ ij + Z 4i δ ij U 4i d d Z 4j d ]. ] U 4j d, The Yukawa terms can not be written into the simple form in the SM such as g G,SM L (i, 2) = m ui V is, g G,SM R (i, 3) = m b V ib. 17/ 31
18 Two points: The CKM matrix is got from the W + ū i d j vertex in Eq. (1) V ij CKM4 = 3 m=1 Uu mi U mj d = U ij CKM4 U u 4i U 4j d. which is non-unitary for that the indexes i,j range form 1 to 4, but the summation of index m is from 1 to 3. V ij CKM4 is also a 4 4 matrix of which the upper left elements (i,jν e 4) are physical measurable value of CKM matrix V as in the SM. The tail terms violate the gauge universality of fermions and cause tree-level FCNC processes induced by the processes such as b sl + l, then the constraints on the parameter space need to be explored. 18/ 31
19 Table: The CKM matrix elements constrained by the tree-level B decays. absolute value direct measurement from V ud ± nuclear beta decay V us ± semi-leptonic K-decay V ub ± semi-leptonic B-decay V cd ± semi-leptonic D-decay V cs ± (semi-)leptonic D-decay V cb ± semi-leptonic B-decay V tb 0.89 ± 0.07 (single) top-production In order to keep gauge universality of quarks, the tail terms in the Feynman rules must be much smaller than the SM like terms, namely Zu,d 4i 2 i=1,2,3, Uu,d 4i 2 i=1,2,3 sin 2 θ W 19/ 31
20 m X =1172GeV 14 M V (GeV) m Y (GeV) Figure: M V versus M Y under constraints Zu,d 4i 2 i=1,2,3, U4i u,d 2 i=1,2,3 < M V Increases as m Y growing up, it is much smaller than m X and m Y. The deviation from unitarity is suppressed by the ratio m/m X,Y where m denotes generically the standard quark masses, which is a typical result of VLP models. 20/ 31
21 Implication on B physcs: the Hamiltionian H eff = G F 2 V tsv tb 10 i=1 in which the operators in SM are: [C i (µ)o i (µ) + C i(µ)o i(µ)], O 1 = ( s i c j ) V A ( c j b i ) V A O 2 = ( sc) V A ( cb) V A O 3 = X ( sb) V A ( qq) V A q O 4 = X ( s i b j ) V A ( q j q i ) V A q O 5 = X ( sb) V A ( qq) V +A q O 6 = X ( s i b j ) V A ( q j q i ) V +A q O 7 = e 8π 2 m b s i σ µν (1 + γ 5 )b i F µν O 8 = g 8π 2 m b s i σ µν (1 + γ 5 )Tij a b jg a µν O 9 = ( sb) V A ( ll) V O 10 = ( sb) V A ( ll) A 21/ 31
22 New operators and the implication The chirality-flipped operators O i are obtained from O i by the replacement γ 5 γ 5 in quark current. CKM matrix is replaced by a 4 4 matrix. In our analysis we take a reasonable assumption that the deviation from unitary is not large. The effective coefficient C eff 9 (µ b ) have the same as the SM. The coefficient of operator O 2 = (sc) V +A(cb) V A, is proportional to the elements of quark mixing matrix Vu 4j or Ud 4i. C,eff 9 (µ b ) receives contributions mainly from the tree-level diagrams, 22/ 31
23 We concentrate on B X s γ b γ s W W b γ s W b γ s G b γ s G G b γ s W G b γ s G W Figure: Leading order Feynman diagram of B X s γ process. 23/ 31
24 C 7 (m W ) = 1 V tb V ts 4 [ gl W (i, 2)gW L (i, 3)A(x i) + gg L (i, 2)gG L (i, 3) m 2 x i B(x i ) u i i=1 + gg L (i, 2)gG R m ui m b where x i = m 2 u i /m 2 W. (i, 3) x i C(x i ) + gw L (i, 2)gG R (i, 3) D(x i ) m b + m u i gl W (i, 2)gR W (i, 3)E(x i ) + gg L (i, 2)gW R (i, 3) D(x i ) m b m b [ 3 ] gl G (4, 2) = UCKM ( M m m X m Q Ud m2 Zu 44 Mu m Ud 42 Z m4 ) u +, X m=1 ] ) + [ 3 gr G (4, 3) = UCKM ( M m m b m Q Uu m4 Zd 43 b m=1 Note that g G,SM L (i, 2) m ui = V is, g G,SM R (i, 3) = V ib m b + M m d Uu 44 Zd m3 ] 24/ 31
25 In the SM4 the term V 4b V4s satisfying the unitary constraint V ub V us + V cbv cs + V tbv ts + V 4bV 4s = 0 In the VLP models such relation does not exists. The suppression of Zd 43 (order of m/m X,Y ) are enhanced by terms with factor such as Y V uv m b, etc., resulting g G R (4,3) m b V 43 CKM4. In order to do the comparison We define two factors K 1 = gg R (4, 3)gG L (4, 2) m X m b V tb Vts K 2 = U43 U 42 V tb V ts = UV b U V s V tb V ts = ggv R b s ggv L m X m b V tb V ts 25/ 31
26 K K 1 and K K 2 C 7 (m W ) SM value m V (GeV) K 1 Figure: K 1, (red ) K 2 (green ) versus M V and enhancement of C 7 (m W ) case of Zu,d 4i 2 i=1,2,3, U4i u,d 2 i=1,2,3 < 10 4 (color online). We can see that though K 2 increase as M V increases, it is still much smaller than V tb Vts, implying that deviation of unitarity are negligible. However the factor K 1 can be enhanced up to order O(1) by the increase of M V. 26/ 31
27 In the numerical scan, we vary Z u,d and U u,d randomly, keeping the constraints of Vu,d 4i 2 i=1,2,3, U4i u,d 2 i=1,2,3, scan m X and m Y in the range of (1,2000)GeV. The branching ratio of B X s γ is normalized by the process B X c e ν e : Br(B X s γ) = Br ex (B X c e ν e ) V tsv tb 2 6α V cb 2 πf(z) [ C7 eff (µ b ) 2 + C,eff 7 (µ b ) 2 ]. We use the following bounds on the calculation Br ex (b ceν e ) = (10.72 ± 0.13) 10 2, Br ex (B X s γ) = (3.55 ± 0.24 ± 0.09) The numerical results show that the C,eff 7 (µ b ) is much smaller than C7 eff(µ b), therefore we do not present the formula of C,eff 7 (m W ) here. 27/ 31
28 0.02 x Br(B X s γ) σ upper bound M V (GeV) Figure: B X s γ prediction in random scan. Br(B X s γ) can be enhanced much greater than the experiment bound. Then the measurements of FCNC process can give a stringent constraint on the vector like quark model, especially when the masses of vector quark are much greater than the electro-weak scale. 28/ 31
29 Two remarks There is one point of view on the unitarity of the CKM matrix which is that the 3 3 ordinary quark mixing matrix is regarded as nearly unitary, deviation from unitarity is suppressed by heavy particle in the new physics beyond the SM. All the new physical effects should decouple from the flavor sector and what should be checked is that if 3 3 unitariry is consistent in all kinds of flavor processes. Another one is that the 3 3 ordinary quark mixing matrix elements are only extracted by experiments in the measurements of tree and loop level precesses. The unitarity should be checked, experiment measurements on the elements of matrix can be used as the constraints to the new physics beyond the SM. In the numerical analysis, the elements of CKM matrix are regarded as inputs. Thus what should be done is to scan the parameter space generally under these constraints, no prejudice should be imposed. 29/ 31
30 K 1 K m X (GeV) m X (GeV) Figure: Enhancement factor and deviation from unitarity versus m X, red are excluded by the bound of B X s γ measurement which the green are the survived points. We can see that deviation from unitarity are very small and almost irrelevant with m X since we are doing a general scan of Z u,d and U u,d. However as However, as m X increases up, Br(B X s γ) measurement will constrain the enhancement factor and then constrain the input parameter of m X. 30/ 31
31 Summary: We find a trick to deal with the scan in the model with vector doublets in which there exist bilinear terms in the lagrangian. Our scan method are exactly and the more efficient. Even the deviations from the unitarity of quark mixing matrix are small, the enhancement to rare B decay from VLPs are still significant. The enhanced effect is an important feature in the vector like particle model. 31/ 31
32 Summary: We find a trick to deal with the scan in the model with vector doublets in which there exist bilinear terms in the lagrangian. Our scan method are exactly and the more efficient. Even the deviations from the unitarity of quark mixing matrix are small, the enhancement to rare B decay from VLPs are still significant. The enhanced effect is an important feature in the vector like particle model. Thank you! 31/ 31
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