arXiv:hep-ph/0304238v2 3 Jun 2003 eoeoeo h onrtn o eivn h uesmercS supersymmetric the believing for cornerstone T the well[5]. of very one measurement experimental th become the of with prediction agrees angle the equivalently supersymmet berg co or the the elegantly[4], in produces constants while particles, coupling point additional to single of energies presence a low the at at intercept extension, measured not values do the scale from cation ex constants the coupling supersymmetry, Model. 3 the Standard of without the Model of Standard str the extension a because lends supersymmetric This the angle[1],[3]. to Weinberg the support of prediction const the coupling and strong energies and weak electromagnetic, the of vergence eateto hsc,Crei elnUiest,Pittsb University, Mellon Carnegie Physics, of Department n fteatatv etrso h rn nfidTer stec the is Theory Unified Grand the of features attractive the of One opigCntn nfiainin Unification Constant Coupling osblte hc a logv aeresult. same th give discuss also We can experiments. which with possibilities well W very of value agrees a which yields angle unification model this standard add of that with remarkable extension model persymmetric is It standard of scalars. extension and fermions the in studied is stants xesoso tnadModel Standard of Extensions nfiaino lcrmgei,wa,adsrn opigc coupling strong and weak, electromagnetic, of Unification igFn Li Ling-Fong a 8 2018 28, May Abstract , 1 n egWu Feng and rh A15213 PA urgh, ntesu- the in einberg vrec of nvergence other e nsa high at ants itional trapolation on- tandard Wein- e hsis This i has his unifi- ong on- ric Model and the experimental search for the supersymmetry will be one of the main focus in the next round of new accelerators. In this paper we will explore the general possibilities of getting coupling constants unification by adding extra particles to the Standard Model[2] to see how unique is the Supersymmetric Standard Model in this respect[?]. We will compute the contribution of various types of new particles to the evolution of the coupling constants and study their effects on the unification. These results will also be useful for checking models with extra particles to see whether they have a satisfactory unification of coupling constants.
1 Evolution of Coupling Constants
In this section we will set up the framework to discuss the unification in the SU (5) type of model. Recall that the evolution of the coupling constants g3,g2,g1 of the subgroups SU (3)C , SU (2)W , U (1)Y at one loop are given by 1 1 µ = +2b ln , i =1, 2, 3 (1) g2 (µ) g2 (µ ) i µ i i 0 0 where bi are the coefficients in β−function of the renomalization group equations[6],
dg i = −b g3, (2) d (ln µ) i i
Note that the coupling constants g2,g1 are related to the usual coupling constants, g,g′ in SU (2) × U (1) by
′ 3 g = g2, g = g1 (3) r5 and Weinberg angle is given by
g′2 sin2 θ = (4) W g2 + g′2
2 At the unification scale MX , we have
g1 (MX )= g2 (MX )= g3 (MX )= g5 (5)
where g5 is the gauge coupling of the SU (5) group. Using the relation, e = g sin θW , we can write
1 1 µ = +8πb ln (6) α (µ) α 3 M 3 5 X
1 1 µ sin2 θ = +8πb ln (7) α (µ) W α 2 M 5 X 3 1 1 µ cos2 θ = +8πb ln (8) 5 α (µ) W α 1 M 5 X e2 g2 g2 where α = , α = 3 , α = 5 . Eliminating the unification scale M 4π 3 4π 5 4π X from Eqs(6,8) we get
1 3 α sin2 θ = b + (9) W 8 5 α 1+ b 3 5 where b − b b = 3 2 (10) b2 − b1 and all coupling constants αi are evaluated at µ = MZ . For convenience, we 2 can solve b in terms of sin θW ,
α − sin2 θ α W b = 3 (11) 8 3 sin2 θ − 5 5
3 α 2 Using the experimental values, = .0674 and sin θW =0.231, [7] we get α3
b =0.71 (12)
2 which is value needed in the unification to get the right value for sin θW . Note that in the unification of the Standard Model without supersymmetry, 1 we have b = which gives sin2 θ = 0.204, and is significantly different 2 W from the experimental value. The inclusion of Higgs scalars will have a very small effect and will not change this result significantly. Or equivalently, 2 if we use the experimental value for sin θW , coupling constants, when the extrapolated to the unification scales, will not converge to a single coupling constant. Comparing the Standard Model value with Eq (12) we see that we need to increase b in the Standard Model to satisfy the experimental value. Recall that various contributions to the β−function for the gauge coupling constants are of the form,
−g3 11 2 1 β = i t(i) (V ) − t(i) (F ) − t(i) (S) (13) i 16π2 3 2 3 2 3 2
(i) (i) (i) Here t2 (V ) , t2 (F ) and t2 (S) are the contributions to βi from gauge bosons, Weyl fermions, and complex scalars respectively. For fermions in the representation with representation matrices T a, they are given by
ab a b t2 (F ) δ = Tr T (F ) T (F ) (14) where the trace means summing over all members of the multiplets. Similarly, for the gauge bosons and scalar contributions, t2 (V ) and t2 (S) . For example, for the gauge bosons, T a′s are the matrices for the adjoint representations
4 (n) and for the gauge group SU (n) , t2 (V ) is given by,
(n) t2 (V ) = n, for n> 2, (15) (n) t2 (V ) = 0, for n =1
(i) Note that t2 (R) , R = V, S, F are pure group theory factors and are inde- pendent of the spin of the particles. For the low rank representations in the SU (n) group they are given by,
(n) t2 1 fundamental rep 2 adjoint rep n n − 2 2nd rank antisymmetric tensor 2 n +2 2nd rank antisymmetric tensor 2 1 Since is common to all β−functions and will be cancelled out in the 16π2 ratio in b in Eq (10) we will neglect it in writing out the coefficient bn,
11 2 1 b = t(n) (V ) − t(n) (F ) − t(n) (S) n 3 2 3 2 3 2
For convenience, we can use the parameter b given in Eq (12) to discuss the contribution to the Weinberg angle θW from models with new particles.
b b = 32 (16) b21 where
b32 ≡ b3 − b2, b21 ≡ b2 − b1 (17)
5 We will separate the contributions into gauge bosons, fermions, and scalars,
b32 = b32 (V )+ b32 (F )+ b32 (S) (18)
b21 = b21 (V )+ b21 (F )+ b21 (S) (19)
Note that for a complete multiplet of SU (5) , their contributions to b32 and b21 cancel out in the combination in Eq (16). This can be seen as follows. Let T (a) (R) be the matrices for some representation R of SU (5) group. Then the coefficient t2 (R) of the second Casmir operator given by
ab (a) (b) t2 (R) δ = tr T (R) T (R) (20) will be the same when we take T (a) (R) ∈ SU (3) , SU (2) or U (1),
(3) (2) (1) t2 (R)= t2 (R)= t2 (R) (21)
Thus their contributions will cancel out in the combination b3−b2 or b2−b1. In other words, a complete multiplet of SU (5) , e.g. 5 or 10, will not effect the prediction of the Weinberg angle, even though they will effect the evolution of each individual coupling constant. Here we assume that all members of the SU (5) multiplets survive to low energies. This is the case for the fermion contributions in the Standard Model. Thus for the studies of the Weinberg angle, we need to consider only the incomplete SU (5) multiplets, i.e. those SU (3) × SU (2) × U (1) multiplets which do not combine into a complete SU (5) multiplets. The reason that we can have the incomplete multiplets in the unification theory has to do with the decoupling of superheavy particles. For example, in the standard SU (5) unification, only gauge bosons in sub- groups SU (3)×SU (2)×U (1) survive to low energies, while the other gauge bosons, e.g. leptoquarks, are superheavy and decouple. Similarly, only Higgs bosons in SU (2) doublet will contribute to the β−function.
6 2 Contributionto θW from Various Multiplets
The contributions of SU (3)×SU (2)×U (1) gauge bosons are given by, using Eq (15) 22 b (V )=11, b (V )= , b (V )=0 (22) 3 2 3 1 and they give 11 22 b (V )= , b (V )= (23) 32 3 21 3 The contribution from the Higgs scalars are
1 1 b (S)= , b (S)= − (24) 32 6 21 15
Since these contributions are very small, they are not included in the usual analysis. We now consider the effects of various low rank multiplets to the Weinberg angle, or the parameter b in Eq (16). The results for the scalars and fermions are listed separately in the following tables: (a) Scalars:
7 SU (3) SU (2) b1 (s) b2 (s) b3 (s) b32 (S) b21 (S) φ0 1 1 1 1 color singlet doublet − , − − 0 − φ ! 10 6 6 15 φ+ 2 2 2 ′′ triplet φ0 , 0 − 0 − 3 3 3 φ− φ++ 3 2 2 1 ′′ triplet φ+ , − − 0 − 5 3 3 15 φ0 ′′ 1 1 Singlet φ+, − 0 0 0 5 5 4 4 ′′ Singlet φ++, − 0 0 0 5 5 ′′ Singlet φ0, 0 0 0 0 0 u/ 1 1 1 1 7 color triplet doublet − − − − d/ 30 2 3 6 15 !L 4 1 1 4 ′′ singletu / − 0 − − 15 6 6 15 1 1 1 1 ′′ Singletd / − 0 − − 15 6 6 15 11 1 2 1 2 ′′ 1 generation of squarks − − − − − 30 2 3 6 15
8 (b) Fermions:
SU (3) SU (2) b1 (f) b2 (f) b3 (f) b32 (f) b21 (f) ν 1 1 1 2 − − − color singlet doublet − 0 l ! 5 3 3 15 l+ 4 4 4 ′′ triplet l0 0 − 0 − 3 3 3 l− l++ 6 4 4 2 ′′ triplet l+ − − 0 − 5 3 3 15 l0 ′′ 2 2 singlet l+ − 0 0 0 5 5 ′′ singlet l0 0 0 0 0 0 u 1 2 1 14 color triplet doublet − −1 − − d ! 15 3 3 15 8 1 1 8 ′′ singlet u − 0 − − R 15 3 3 15 2 1 1 2 ′′ singlet d − 0 − − R 15 3 3 15 11 4 1 4 ′′ 1 generation of quarks − −1 − − − 15 3 3 15 color octet glunios 0 0 −2 −2 0
Remarks :As it is evident in Eq (13) with same SU (3) × SU (2) × U (1) 1 quantum numbers the contribution from the scalars are of the fermions 2 contributions.
9 3 Supersymmetric extension of standard model and generalization
As we have discussed before, we need to consider only the incomplete multi- plets. The particles needed to be included are given below,
1. Scalars:
H+ 1 2 2 Higgs doublets H = b32 (H)= , b21 (H)= − H0 ! 3 15
2. Fermions
a a a gluniosg ˜ , a =1,... 8 b32 (˜g )= −2 b21 (˜g )=0 W˜ + 4 4 Winos W˜ = W˜ 0 b W˜ = b W˜ = − 32 3 21 3 W˜ − binos ˜b b32 ˜b =0 b21 ˜b =0 h˜0 2 4 ˜ ˜ ˜ − 2 Higginos h = − b32 h = b21 h = h˜ ! 3 15
Thus the total contributions are
1 26 b = , b = − (25) 32 3 21 15 and using Eq (16), we get for b,
11 1 + b = 3 3 = .715 (26) 22 26 − 3 15
This agrees with the experimental measurement very well and gives a strong support for the supersymmetric extension of Standard Model. Note that the
10 1 result b = comes from a cancellation between large glunios contribution 32 3 and that of winos and Higginos. Thus the mechanism to produce the right value for the Weinberg angle in the supersymmetric extension of Standard Model is quite complicate. This makes the SSM somewhat unique in the sense the extra particles which give all these cancellations are not arbitrary but are required by the principle of supersymmetry. We will now discuss the other examples which can give satisfac- tory answer for the Weinberg angle. Since the supersymmetric extension of Standard Model has been so successful in the prediction of Weinberg an- gle, we will now give the possibilities of other multiplets which give the same contribution. We will use a simpler notation for the multiplet structure, f (a, b)Y , denote a fermion multiplet transforms as representations a, b under × S SU (3) SU (2) with hypercharge Y. Similarly, (a, b)Y for the scalar mul- tiplets. To present the result, we group together the multiplets with same contributions for the coefficients functions, b3, b2, and b1.
1. b32 = −2, b21 = 0 (glunio-like states)
f f f ⊕ f × s × (8, 1)0 (6, 1)0 3, 1 (3, 1)0 6 (8, 1)0 2 0 f f s ⊕ s s s × ⊕ s ⊕ × (8, 1)0 (6, 1)0 (3, 1)0 (6, 1)0 2 3, 1 (6, 1)0 3, 1 2 0 0 4 4 2. b = , b = − (Wino-like states) 32 3 21 3
f s × s × (1, 3)0 (1, 3)0 2 (1, 2)0 8
2 4 3. b = , b = − (Higgio-like states) 32 3 21 15
f × s × s ⊕ f (1, 2)−1 2 (1, 2)−1 4 (1, 3)0 (1, 1)2 f ⊕ s × s ⊕ s × (1, 2)−1 (1, 2)−1 2 (1, 3)0 (1, 1)2 2
11 1 2 4. b = , b = − (Higgs-like states) 32 3 21 15
s × (1, 2)−1 2
Note that this last group includes the Standard Model Higgs contribu- tion.
Clearly, if take one set of multiplet from each group, the total contribution will give the same Weinberg angle as the supersymmetry Standard Model. The list given above is constructed by mimicking those multiplets of the supersymmetry Standard Model and are by no means the only possibilities. We just want to demonstrate the existence of other examples. None of these possibilities, except for the supersymmetric one, are realized in any realist phenomenological model. For the case of new fermions multiplets, one might worry about the anomaly cancellation. However, as we have discussed before, only the incomplete multiplets are relevant for the Weinberg angle and it is possible to cancel the anomalies by fermion multiplets which decouple at very high energies.
References
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