General scan in flavor parameter space in the models with vector quark doublets and an enhancement in $B\to X_s\gamma$ process

In the models with vector like quark doublets, the mass matrices of up and down type quarks are related. Precise diagonalization for the mass matrices became an obstacle in the numerical studies. In this work we propose a diagonalization method at first. As its application, in the standard model with one vector like quark doublet we present quark mass spectrum, Feynman rules for the calculation of $B\to X_s\gamma$. We find that i) under the constraints of the CKM matrix measurements, the mass parameters in the bilinear term are constrained to a small value by the small deviation from unitarity; ii) compared with the fourth generation extension of the standard model, there is an enhancement to $B\to X_s\gamma$ process in the contribution of vector like quark, resulting a non-decoupling effect in such models.


I. INTRODUCTION
Though the standard model (SM) has been verified to be correct times by times, many new physics beyond standard model are proposed to solve both experimental and aesthetical problems, such as neutrino masses, µ anomalous magnetic movement problem or hierarchy problem, etc. Many new models introduce vector like particles (VLP) [1] whose right handed and left handed components transform in the same way under the weak SU(2)×U(1) gauge group. The extension is acceptable because the anomalies generated by the VLPs cancel automatically, and vector quarks can be heavy naturally. VLPs also arise in some grand unification theories. For example, in order to explain the little hierarchy problem between the traditional GUT scale and string scale, a testable flipped SU(5) × U(1) X model are proposed in Ref. [2] in which the TeV-scale VLPs were introduced [3]. Such kind of models can be constructed from the free fermionic string constructions at the Kac-Moody level one [4,5] and from the local F-theory model [2,6].
However when we do the flavor physics with doublet VLPs in these models [7,8], a problem always appears when we are dealing with the mass spectrum of quarks and leptons.
Let us start with the SM in which all fermion masses come from the Yukawa couplings.
After the spontaneously gauge symmetry breaking, we can get two separate mass matrices 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 for the calculation in flavor physics. Note that v is the vacuum expectation value (VEV) of the Higgs, and Z D is a random unitary matrix.
Such a trick can not be used in case of the participation of a vector doublet, namely Q with gauge charge 3, 2, 1 6 andQ with gauge charge3, 2, − 1 6 , resulting bilinear term in the lagrangian It is clear that in the model, there are the same input parameters in the matrices M U , M D The mass matrices for up and down type quarks are related to each other. Therefore, we can not set one of the matrices diagonal and the CKM matrix can not be got easily. The shooting method is always used to treat such an obstacle. Random M U and M D are generated to meet the requirements after diagonlization: the mass of eigen state and the measurements of elements of CKM matrix. However this is too much time consuming, and precise solution for diagonalization is almost unavailable. Although 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.
In this paper, we will first propose a general method to solve the obstacle in models with vector like quark doublets. As its application, we will study rare B decay B → X s γ in the SM with one vector like quark doublet. The paper is organized as follows. We show the detail of the trick in Section 2. The simple application to B → X s γ process, including quark mass spectrum, Feynman rules and the Wilson coefficients, as well as the numerical analysis for calculation of B → X s γ is shown in Section 3. A summary is given in Section 4.

II. THE TRICK OF DIAGONALIZATION OF VECTOR QUARK DOUBLET
Firstly, we address the problem clearly on how to deal with the diagonalization of N × N matrix M U and M D : in which M D U , M D D are the diagonal mass matrices for up and down type quark, respectively. Note that N should be greater than 3 and the first three elements in the matices should be the three generations of quark multiplates in the SM, other elments with N > 3 are the new multiplates introduced in new physics beyond the SM. Then we have The last line of the two matrices has the same parameters except the last elements.
Considering that there are some same parameters in M U and M D , we find that a very simple way is to add two matrices in Eq. (6) The left side of the equation is Obviously, the mass inputs from bilinear terms vanish. We can denote the matrix in the form as To prepare for the diagonalization, we chose the diagonal mass matrix elements of quarks (m u , m c , m t , · · · m X ), (m d , m s , m b , · · · , m Y ) and a matrix U CKMN , which are determined partly by experimental measurements as input parameters Note that above Z U , Z D , U U , U D are unitary matrices, but (U CKM ) 3×3 is not an ordinary CKM matrix V CKM which is non-unitary in this case. Detailed dicussion will be shown in the following section.
What we need to do for the next is to generate a unitary matrix U D . In the similar way we denote U D as Both sides of Eq. (7) times the matrix U D , we can get From above equation, we can get the last line of U D simply by inputting M D U , M D D , U CKMN and random Z U , Z D : where is a unit vector in N dimension.
Next we use the unit vector to generate total U D . Since M A and M B are random matrix, U D can be random too. The unit vector U DN −1 of U D can be determined as It is clear that the vector is orthogonal to U DN and normalized to 1. 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 , · · ·, 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 We finish the work by At this stage, we would like to summarize our method here • Step 1: Chose (m u , m c , m t , · · · , m X , m d , m s , m b , · · · , m Y ) and U CKMN and generate random unitary matrices Z U and Z D as the inputs for the model; and normalize it into a unit vector U DN .
• Step 3: Use the unit vector U DN to generate other N − 1 unitary vectors (U D1 , U D2 , · · ·, U DN −1 ), and form a special U S then, a general U D is obtained by • Step 5: Use these equations to get the inputs for the flavor physics.
We can see that by this trick we can skip the inputs of the bilinear mass terms M V N i . In physical analysis, the mass of eigen states m X, Y in the VLP models are inputs freely. Z U and Z D can be generated randomly, U U and U D can also be scanned the most generally if we vary U R randomly. Thus the method can do the most general scan in the parameter space of mass matrices in the models with VLPs for the numerical studies, which will be shown in the following section.

QUARK DOUBLET
A. The standard model with vector like quarks As an application of the method, in this section we study the VLP contribution to B → X s γ in a very simple VLP extension of SM for the demonstration. In the Tab. I, we list the gauge symmetry of the matter multiplates in which the first two queues show the quarks in the SM and the last two queues show the VLPs with the anti-gauge symmetry.
Note that we ignore partners of the last two queues whose gauge symmetry are exactly the same as the first two queues of the SM. As talked in the introduction, these VLPs can be heavy naturally. Since gauge symmetry of Higgs H = (h + , h 0 ) T is (1, 2, 1/2), the lagrangian for two quarks of the model is written as: The first line of the lagrangian is Yukawa terms, the second line is the bilinear terms. Note that Y u , Y d are 3 × 3 matrix, without the bilinear terms, the model will be almost the same as the fourth generation standard model (SM4).
After the electro-weak symmetry breaking, we can get the mass matrices of up and down quarks in the basis of (u, c, t, V u ) and (d, s, b, V d ): where v is the VEV for H. The first three elements of last line of the matrices have the same parameter, making the scan of the parameter space very difficult. These two matrices can be diagonalized by unitary matrices U and Z, Product of the two matrices is denoted as which is unitary 4 × 4 matrix. We stress that the trick we introduced in the above section seems to just give us a numerical tool for quark masses and some quark mixing matrices, but it is important in studying the flavor physics in such models.
For studying VLP contributions to B → X s γ, we now present the Feynman rules for the interaction ofū l d j χ + , χ = W, G andd l d j Z in the Feynman gauge which read: Note that U(1) EM interaction is not changed by the VLPs, thus the vertices of photon and quarks are still the same as those in the SM. From above mass matrices and Feynman rules, we can see that the model has two points to be explored: • The CKM matrix is got from the W +ū i d j vertex in Eq. (32) 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. This is the key difference between VLP models and the SM4. Nevertheless, the loop-level flavor change neutral current (FCNC) will be changed by the Yukawa interactions, then the prediction of process B → X s γ may be changed significantly.
• The last terms in Eqs.(32)-(36), which we call 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.
where the effective operators O i are same as those in the SM defined in Ref. [9]. The chirality-flipped operators O ′ i are obtained from O i by the replacement γ 5 → −γ 5 in quark current [7]. We calculate the Wilson coefficient C 7 at matching scale m W . The leading order Feynman diagrams are shown in FiG. 1

and C 7 reads
(40) In the model with three generation quarks, the CKM matrix unitarity is already used in the calculations of the loop-level FCNC induced rare B decays. For consistency, in numerical analysis the constraints on CKM matrix element are not from processes occurred at loop level, such as rare B decays, but from tree-level processes shown in Table II [10,11]. Since there are no tree-level measurements of V td , V ts now, we use above inputs and the unitarity to get 3 × 3 unitary matrix at first. The method is that we scan (V ud , V us , V ub ) randomly (keeping |V ud | 2 +|V us | 2 +|V ub | 2 = 1 ) in range listed in Table II, then we define two parameters α, β and solve them by the equations (V td , V ts , V tb ) are got by the unitarity relation with (V ud , V us , V ub ) and (V cd , V cs , V cb ). After that we times the 3 × 3 unitary matrix with three matrices · · · · · · · · · · · · cos θ 4i · · · sin θ 4i · · · · · · 1 · · · · · · − sin θ 4i · · · cos θ 4i Thus in the numerical studies we require Note that though these elements are greater than λ 3 (parameter in the Wolfenstein parameterization [12]), they are much smaller than the product of V † CKM3 V CKM3 (almost equals 1), thus the requirements are suitable for indicating the contraints from the deviation from unitarity.
Since the scanning in the parameter space is freely, we set m X = 1172GeV (mass of top quark plus 1000 GeV) and scan m Y in the range of (4.2, 1004)GeV (mass of bottom quark plus 1000 GeV), and Z u,d , U u,d randomly (ignoring the CP phases). M V is defined by The result for M V versus M Y is shown in the FIG. 2 which checks the mass input of vector doublet. We can see that M V increases as m Y growing up. However M V is much smaller than m X and m Y . Small mixings lead to parameter M Q which determine the mixing between SM quarks and vector like quarks are also suppressed. This is in agreement with that 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. [13][14][15][16][17] The second task is to check the VLP contribution to B → X s γ. We find the Wilson  Fig. 1) In order to show the enhancement clearly, we define two factors in which K 2 denotes the deviation from the unitarity of 3 × 3 CKM matrix, while K 1 shows the enhancement of the contribution from vector like particles. K 1 is in fact got from the coefficient of first term in the second line of analytical expression of C 7 (m W ) in Eq. (39) when i = 4. It will be changed into exactly K 2 in case of the SM4. Note that other terms with g G R (4, 3)/m b can give enhancement too, we chose factor K 1 for a typical demonstration since it seems that it will be suppressed by m Since m X ≃ Y V u v, Z 44 u , U 44 u ≃ 1, one can easily obtain that the suppression of Z 43 d (order of m/m X,Y ) in Eq. (48) are enhanced by terms with factor such as Y V u v m b , etc., resulting Thus the term V 4b V * 4s satisfying the unitary constraint is enhanced greatly by heavy VLPs, then the factor leads the enhancement to C 7 . This is different from those in the SM4 in which the contribution from the fourth generation can be neglected.
In the numerical scan, we vary Z u,d and U u,d randomly, keeping the constraints of ratio of B → X s γ is normalized by the process B → X c eν e : Here z = mc m b , and f (z) = 1 − 8z 2 + 8z 6 − z 8 − 24z 4 ln z is the phase-space factor in the semi-leptonic B decay. The method of running of the operators from m W scale to µ b scale can be found in Ref. [7]. We use the following bounds on the calculation [10] Br ex (b → ceν e ) = (10.72 ± 0.13) × 10 −2 , Br ex (B → X s γ) = (3.55 ± 0.24 ± 0.09) × 10 −4 .
The numerical results show that the C ′,eff 7 (µ b ) is much smaller than C eff 7 (µ b ), therefore we do not present the formula of C ′,eff 7 (m W ) here. The branching ratio as a function of m V is shown in FIG. 4, from which we can see that 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. A few remarks should be addressed: • 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. In other word, one admits that the extended CKM matrix elements exist, they approach to zero while mass scale of the new physics approaches to infinity. 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 point of view is that, as in the SM case, the 3×3 ordinary quark mixing matrix  versus m X . From the right panel 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 we see from the left panel, 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 . In all, the enhancement can be summarized as that when mass of vector like particle increases up, it will increase the mass parameter m V thus give an enhancement factor under very small deviation from unitarity. This should be a special point when we do the study on the vector like quark models.

IV. SUMMARY
In the model with vector doublets, there exist bilinear terms in the lagrangian, making the general scan of the Yukawa coupling very difficult. In this paper, we show a trick to deal with the scan. Our scan method are exactly and the more efficient. We use the trick to study a very simple extension of the SM with vector like quarks. We studied one of the most important rare B decay B → X s γ process in which we found that 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. In this work we just show the scan method, the key point of the enhancement and how stringent constraints on the parameter space from B → X s γ measurements. What should be done includes models like extension of the SM with VLPs, two higgs doublets models [18] or supersymmetry models [19]. Such effect should be checked in all kinds of rare decays such as inclusive process b → sℓ + ℓ − and exclusive processes B s → µ + µ − , B s → ℓ + ℓ − γ and BB mixing et. al. The detailed studies on the parameter space including other rare B decays and new models will appear in our future work.