Rare top decay $t \to c \bar{l}l$ as a probe of new physics

Abstract

The rare top decay $t \to c \bar{l}l$, which involves flavor violation, is studied as a possible probe of new physics. This decay is analyzed with one of the simplest Standard Model extensions with additional gauge symmetry formalism. The considered extension is the Left–Right Symmetric Model, including a new neutral gauge boson Z′ that allows one to obtain the decay at tree level through Flavor-Changing Neutral Currents (FCNC) couplings. The neutral gauge boson couplings are considered diagonal but family non-universal in order to induce these FCNC. We find $\mathit{BR}(t \to c\bar{l}l)\sim10^{-13}$ for the range 1 TeV≤M _Z′≤3 TeV.

Appendix A: Analytical expressions and numerical values

Below we give the complete analytic formulas for the family non-universal parameters in three considered scenarios. We also present the analytic expression of the integrals I _1,…,5.

1.1 A.1 Family non-universal couplings

The general expressions for the B _i coefficients are given by

$$ B_{1}=2 \bigl( \bigl \vert B_{L}^u\bigr \vert ^{2}\bigl \vert B_{L}^l\bigr \vert ^{2}+\bigl \vert B_{R}^u\bigr \vert ^{2}\bigl \vert B_{R}^l\bigr \vert ^{2} \bigr), $$

(A.1)

$$ B_{2}=2 \bigl( \bigl \vert B_{L}^u\bigr \vert ^{2}\bigl \vert B_{R}^l\bigr \vert ^{2}+\bigl \vert B_{L}^l\bigr \vert ^{2}\bigl \vert B_{R}^u\bigr \vert ^{2} \bigr), $$

(A.2)

$$ B_{3}=\operatorname{Re} \bigl( B_{L}^l B_{R}^{u\ast } \bigr) \bigl( \bigl \vert B_{L}^u \bigr \vert ^{2}+\bigl \vert B_{R}^u\bigr \vert ^{2} \bigr), $$

(A.3)

$$ B_{4}=\operatorname{Re} \bigl( B_{L}^u B_{R}^{l\ast } \bigr) \bigl( \bigl \vert B_{L}^l \bigr \vert ^{2}+\bigl \vert B_{R}^l\bigr \vert ^{2} \bigr), $$

(A.4)

and

$$ B_{5}=2 \bigl[ \operatorname{Re} \bigl( B_{L}^u B_{L}^l B_{R}^{u\ast} B_{R}^{l\ast } \bigr) + \operatorname{Re} \bigl( B_{L}^u B_{L}^{l\ast }B_{R}^u B_{R}^{l\ast } \bigr) \bigr] . $$

(A.5)

The matrix elements for the scenario 1 are

$$ \bigl[ B_{L}^{u} \bigr]_{32}=Q_{L}^{u} ( x-1 ) V_{tb}V_{cb}^{\ast }, $$

(A.6)

$$ \bigl[ B_{R}^{u} \bigr]_{32}=0, $$

(A.7)

$$ \bigl[ B_{L,R}^{l} \bigr]_{33}=Q_{L,R}^{l}. $$

(A.8)

Then, we write the expressions of the B _1,…,5 in terms of the Wolfenstein parameter, taking V _tb =1 and V _cb =Aλ ²,

$$ B_{1}=2 \bigl( Q_{L}^{u}Q_{L}^{l} \bigr)^{2} ( x-1 )^{2}A^{2}\lambda^{4}, $$

(A.9)

$$ B_{2}=2 \bigl( Q_{L}^{u}Q_{R}^{l} \bigr)^{2} ( x-1 )^{2}A^{2}\lambda^{4}, $$

(A.10)

$$ B_{3}=0, $$

(A.11)

$$ B_{4}=Q_{R}^{l}Q_{L}^{u} ( x-1 ) A\lambda^{2} \bigl( \bigl \vert Q_{L}^{l} \bigr \vert ^{2}+\bigl \vert Q_{R}^{l}\bigr \vert ^{2} \bigr), $$

(A.12)

and

$$ B_{5}=0. $$

(A.13)

For the scenario 2

$$ \bigl[ B_{L}^{u} \bigr]_{32}=Q_{L}^{u} ( x-1 ) A\lambda^{2}, $$

(A.14)

$$ \bigl[ B_{R}^{u} \bigr]_{32}=Q_{R}^{u} ( x-1 ) s_{t}s_{c}A\lambda^{2}, $$

(A.15)

$$ \bigl[ B_{L,R}^{l} \bigr]_{33}=Q_{L,R}^{l}. $$

(A.16)

Then

$$ B_{1}=2 ( x-1 )^{2}A^{2}\lambda^{4} \bigl[ \bigl( Q_{L}^{u}Q_{L}^{l} \bigr)^{2}+ \bigl( Q_{R}^{u}Q_{R}^{l} \bigr)^{2} \bigr] , $$

(A.17)

$$ B_{2}=2 ( x-1 )^{2}A^{2}\lambda^{4} \bigl[ \bigl( Q_{L}^{u}Q_{R}^{l} \bigr)^{2}+ \bigl( Q_{R}^{u}Q_{L}^{l} \bigr)^{2} \bigr], $$

(A.18)

$$ B_{3}=Q_{R}^{u}Q_{L}^{l} ( x-1 )^{3}s_{t}s_{c}A^{3} \lambda^{6} \bigl[ \bigl( Q_{L}^{u} \bigr)^{2}+ \bigl( Q_{R}^{u} \bigr)^{2} \bigr], $$

(A.19)

$$ B_{4}=Q_{R}^{l}Q_{L}^{u} ( x-1 ) A\lambda^{2} \bigl( \bigl \vert Q_{L}^{l} \bigr \vert ^{2}+\bigl \vert Q_{R}^{l}\bigr \vert ^{2} \bigr), $$

(A.20)

and

$$ B_{5}=4s_{t}s_{c}Q_{L}^{u}Q_{R}^{u}Q_{L}^{l}Q_{R}^{l} ( x-1 )^{2}A^{2}\lambda^{4}. $$

(A.21)

Finally, for scenario 3

$$ \bigl[ B_{L}^{u} \bigr]_{32}=Q_{L}^{u} \bigl( x_{1}V_{td}V_{cd}^{\ast }+x_{2}V_{ts}V_{cs}^{\ast }+x_{3}V_{tb}V_{cb}^{\ast } \bigr) , $$

(A.22)

$$ \bigl[ B_{R}^{u} \bigr]_{32}=0, $$

(A.23)

$$ \bigl[ B_{L,R}^{l} \bigr]_{33}=Q_{L,R}^{l}. $$

(A.24)

Then

(A.25)

(A.26)

$$ B_{3}=0, $$

(A.27)

(A.28)

and

$$ B_{5}=0. $$

(A.29)

1.2 A.2 Lepton and charm quark contribution

The a _i coefficients in (12) contain the charged lepton and charm quark contribution. They are given by

(A.30)

(A.31)

(A.32)

(A.33)

(A.34)

where $\mu_{1}=\frac{m_{\mathrm{lepton}}}{m_{\mathrm{top}}}$ and $\mu_{2}=\frac{m_{\mathrm{charm}}}{m_{\mathrm{top}}}$, see Table 4. The I _i are integrals in the three phase space with analytical solution

(A.35)

(A.36)

(A.37)

(A.38)

(A.39)

The numerical values are shown in Table 3.

Table 3 Numerical values of the integrals for each charged lepton

Table 4 Numerical values of the B _i , i=1,…,5