Oblique corrections from triplet quarks

We present general formulas for the oblique-correction parameters $S$, $T$, $U$, $V$, $W$, and $X$ in an extension of the Standard Model having arbitrary numbers of singlet, doublet, and triplet quarks with electric charges $-4/3$, $-1/3$, $2/3$, and $5/3$ that mix with the standard quarks of the same charge.

The total number of h-type quarks is n h . The specific h-type quarks h and h ′ have masses m h and m h ′ , respectively. Similar notations are utilized for the u-type, d-type, and l-type quarks.
The mass of the gauge bosons W ± is m W . The mass of the gauge boson Z is m Z . We define c w ≡ m W /m Z and s w ≡ 1 − c 2 w . In our model there are arbitrary numbers of the following gauge-SU(2) multiplets of quarks [1]: SU(2) singlets with weak hypercharge 1 In Eqs. (1)- (7), the letter σ denotes singlets of gauge SU (2), the letter δ stands for doublets, and the letter τ means triplets; the first number in the subscript is two times the third component of weak isospin; the second number in the subscript is six times the weak hypercharge; the letter ℵ stands for either L, in the case of left-handed quarks, or R, in the case of right-handed quarks.
The purpose of this paper is to compute the oblique parameters in this generic model. The oblique parameters are defined as [2] 2,3 , (9a) where g is the SU(2) gauge coupling constant. The A V V ′ (q 2 ) are the coefficients of the metric tensor g µν in the vacuum-polarization tensor between gauge bosons V µ and V ′ ν carrying four-momentum q. In A V V ′ (q 2 ) one only takes into account the dispersive part-one discards the absorptive part; one subtracts the Standard-Model contribution from the New-Physics-model one.
The outline of this paper is as follows. In Section 2 we present our notation for the gauge interactions. In Section 3 we present our notation for the Passarino-Veltman (PV) functions. In Section 4 we display the results for the oblique parameters. Thereafter, three appendices deal with technical issues: Appendix A gives technical details of the computations, Appendix B gives analytic formulas for the PV functions, and Appendix C demonstrates the cancellation of the ultraviolet divergences in S, T , and U. The reader does not need to read the appendices in order to fully understand the scope and results of this paper.

Notation for the gauge interactions
The interactions of the quarks with the photon field A µ are given by The interactions of the quarks with the gauge bosons W ± are given by where γ L = (1 − γ 5 )/ 2 and γ R = (1 + γ 5 )/ 2 are the projectors of chirality. Note the presence of the n h × n u mixing matrices N ℵ , and n d × n l mixing matrices Q ℵ .
The matrix V L is the generalized Cabibbo-Kobayashi-Maskawa matrix. The interactions of the quarks with the gauge boson Z are given by with Hermitian mixing matricesH ℵ ,Ū ℵ ,D ℵ , andL ℵ . Note the minus signs in the third and fourth lines of Eq. (13). Since the Z couples to a current proportional to (g /c w ) (T 3 − Qs 2 w ), those matrices are of the formH where ½ always is the unit matrix of the appropriate dimension.
Because of the SU(2) algebra relation T 3 = [T + , T − ], where T + and T − are the SU(2) raising and lowering operators, respectively, there are relations between the mixing matrices appearing in Eq. (13) and the ones in Eq. (12), viz.
Thus, the matrices N ℵ , V ℵ , and Q ℵ are the fundamental ones, while the matrices H ℵ , U ℵ , D ℵ , and L ℵ are derived ones.

Notation for the Passarino-Veltman functions
Our notation for the relevant PV functions [12] is the one of LoopTools [13]: where Q ≡ q 2 and I and J have mass-squared dimensions. The quantities Q, I, and J are assumed to be non-negative. In Eqs. (16), µ is an arbitrary quantity with mass dimension and d = 4 − ǫ (where eventually ǫ → 0 + ) is the dimension of space-time. We also define B ′ 00 (Q, I, J) ≡ ∂B 00 (Q, I, J) ∂Q . (17c) All the functions in this section may be computed through softwares like LoopTools [13] or COLLIER [14]. They may as well be computed analytically; the results of that computation are presented in Appendix B.
4 Results for the oblique paramaters where N c = 3 is the number of quark colors, and the last line of Eq. (18) means that, in the end, one should not forget to subtract from T the same quantity computed in the context of the Standard Model.

Simplified notation
In order to present the expressions for the oblique parameters in a compact way, we introduce a new notation wherein all the quarks are denoted by letters a and/or b. The symbol a means a sum over all the quarks. The symbol " a,a ′ " means firstly a sum over the h-type quarks h and h ′ , then a sum over the u-type quarks u and u ′ , ..., and finally a sum over the l-type quarks l and l ′ . The matrices A ℵ andĀ ℵ correspond to the quarks a just as the matrices H ℵ andH ℵ correspond to the quarks h, ..., and the matrices L ℵ andL ℵ correspond to the quarks l. We also use the symbol " a b " when we sum both over the quarks a and over the quarks b such that the electric charge Q a of the quarks a is equal to the electric charge Q b of the quarks b plus one unit: Q a = Q b + 1; in this case, we have to deal with charged-current mixing matrices M ℵ that are • N ℵ when a = h and b = u; • V ℵ when a = u and b = d; • Q ℵ when a = d and b = l.
In this way, the expression for T in Eq. (18) gets shortened to

S and U
We have

V and W
We have

A Technical details
Suppose the fermions f 1 and f 2 with masses m 1 and m 2 , respectively, interact with the gauge bosons V θ and V ′ ψ through the Lagrangian Then, the vacuum polarization between a V θ and a V ′ ψ with four-momenta q caused by a loop of where It follows from the definitions (A3) that for a loop with two identical quarks a with electric charge Q a . (We use the notation of Section 4.2.) • In the computation of A γZ (q 2 ), for a loop with two identical a-type quarks. (We use once again the notation of Section 4.2.) • In the computation of A ZZ (q 2 ), in a loop with quarks a and a ′ carrying identical electric charges.
• In the computation of A W W (q 2 ), in a loop with quarks a and b carrying electric charges Q a and Q a − 1, respectively. (We use once more the notation of Section 4. with the functions g andĝ defined in Eqs. (22b) and (22c), respectively. The function h defined in Eq. (22d) appears in The functions relevant for the computation of the oblique parameters V and W are defined in Eqs. (24). They appear in The function l that appears in the expression for the oblique parameter X is given by Eq. (26) and originates in If in Eq. (A12) one sets G A = 0 and m 1 = m 2 , as happens if V = γ is a photon, then one obtains because of Eq. (A9). Hence, the contributions to A γγ (0) and to A γZ (0) from fermion loops both vanish. Notice, though, that A γγ (0) is necessarily zero because of gauge invariance, while A γZ (0) does not need to vanish in general.

B Formulas for the PV functions
In the limit ǫ → 0 + , we define the divergent quantity div where γ is the Euler-Mascheroni constant. We furthermore define ∆ ≡ Q 2 + I 2 + J 2 − 2 (QI + QJ + IJ) .
The quantity ∆ is positive if and only if it is not possible to draw a triangle with sides of lengths √ Q, We define the function The function f (Q, I, J) is continuous and well-behaved everywhere except at the point

The analytic formulas for the relevant PV functions are
+absorptive part.
When Q = 0, the PV functions are and their derivatives are When both Q = 0 and I = J one has All the formulas in this appendix were numerically checked by using LoopTools.

C Cancellation of the divergences
In this appendix we demonstrate that the ultraviolet divergences cancel out in the oblique parameters S, T , and U. In the other three parameters such divergences are a priori absent.

C.1 The quark mass terms
The quarks in Eqs. The matrices M 1 , . . . , M 7 are assumed to have adequate dimensions that we do not, however, specify.

C.2 The gauge interactions
The interactions of the quarks with the gauge bosons W ± are given by We rewrite these interactions using the general notation of Eq. (12). We obtain The interactions of the quarks with the gauge boson Z are given by Rewriting these interactions by using the general notation of Eq. (13), we obtain where we have used the notation of Section 4.2. According to Eqs. (14), Therefore, the quantity inside square brackets in Eq. (C17) is Thus, the a-type quarks produce in S a divergence proportional to Therefore, the oblique parameter S is finite if the equation holds for both ℵ = L and ℵ = R. Now, according to Eqs. (C15), Therefore, the right-hand side of Eq. (C20) is equal to (C26) The terms inside the square brackets in Eq. (C26) clearly cancel out.
Let us demonstrate each of the two identities (C34) in turn.