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The Matter Content of the Standard Model (SM) Consists of Three Families of Chiral Quarks and Leptons. Each Family Has Five Diff

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The Matter Content of the Standard Model (SM) Consists of Three Families of Chiral Quarks and Leptons. Each Family Has Five Diff I. SUPERSYMMETRIC GRAND UNIFICATION A. Supersymmetrizing the standard model: particle content The matter content of the standard model (SM) consists of three families of chiral quarks and leptons. Each family has five different gauge representations: qL(3, 2)+1/6, uR(3¯, 1)−2/3, dR(3¯, 1)+1/3, lL(1, 2)−1/2, eR(1, 1)+1. (1) 1 All of these fields are Weyl fermions that transform in the ( 2 , 0) representation of the Lorentz group (they have an undotted spinor index α). The subscripts R, L do not specify the repre- sentation of the Lorentz group but instead are used to indicate the different transformation properties under the SU(2)L gauge group. In addition, there is a single scalar representation, that is the Higgs field: h(1, 2)−1/2. (2) As we learned in the previous section, supersymmetry requires the presence of additional states which form supermultiplets with the known particles. Since all states of a super- multiplet carry the same gauge quantum numbers, we need at least a doubling of states: For every field of the SM, one has to postulate a superpartner with the exact same gauge quantum numbers and a spin such that it can form an appropriate supermultiplet. More specifically, the quarks and leptons are promoted to chiral supermultiplets by adding scalar I I ˜I ˜I I (spin-0) squarks (˜qL,u ˜R, dR) and sleptons (lL,e ˜R) to the spectrum. The gauge bosons 1 ˜ ˜ ˜ are promoted to vector multiplets by adding the corresponding spin- 2 gauginos (G, W , B) to the spectrum. Finally, the Higgs boson is also promoted to a chiral multiplet with a 1 spin- 2 Higgsino superpartner. However, the supersymmetric version of the SM cannot ‘live’ with only one Higgs doublet and at least a second doublet, of opposite hypercharge, has to be added. This can be seen from the fact that one cannot write down a supersymmetric version of the Yukawa interactions of the SM without introducing a second Higgs doublet. The reason is the definite chirality of the Higgsino. Another way to see the necessity of a second Higgs doublet is the fact that the Higgsino is a chiral fermion which carries U(1) hypercharge and hence it upsets the anomaly cancellation condition. A second Higgsino of opposite U(1)Y charge is necessary. 1 B. Gauge coupling unification There is an interesing conclusion that follows from the quantum number assignments of the new particles that we have introduced to make the SM supersymmetric. An attractive feature of the SM is that the quarks and leptons of each generation fill out multiplets of the simple gauge group SU(5). This suggests a very beautiful picture, called grand unification, in which SU(5), or a group such as SO(10) or E6 for which SU(5) is a subgroup, is the fundamental gauge symmetry at very short distances. This unified symmetry is spontaneously broken to the SM gauge group SU(3) × SU(2) × U(1). For definiteness, we focus on SU(5). The generators of SU(5) can be represented as 5× 5 Hermitian matrices acting on the 5-dimensional vectors in the fundamental representations. To see how the SM is embedded in SU(5), it is convenient to write these matrices as block with 3 and 2 rows and columns. Then, the SM generators can be identified as 1 ta 0 3 − 1 SU(3) : ; SU(2) : ; U(1) : 3 . (3) 1 ! a ! s5 1 ! 0 σ /2 2 a a A B 1 AB The matrices t and σ /2 are normalized to tr[T T ] = 2 δ , while the last matrix is identifed with 3/5 Y . The symmetryq breaking can be caused by the VEV of a Higgs fields in the adjoint representation of SU(5). The VEV − 1 1 hΦi = V · 3 (4) 1 1 ! 2 commutes with the generators in (3) but not with the off-diagonal generators. It thus breaks SU(5) → SU(3) × SU(2) × U(1). Matter fermions can be organized as left-handed Weyl fermions in the SU(5) representa- tions 5¯ and 10, where the latter is the antisymmetric matrix: d¯ 0u ¯ u¯ u d d¯ 0u ¯ u d 5¯ : d¯ ; 10: 0 u d . (5) e 0e ¯ ν 0 The SU(5) covariant derivative is A A Dm = ∂m − ig5AmT , (6) 2 where g5 is the SU(5) gauge coupling. There is only room for one value here. So this model predicts that the three SM gauge couplings are related by g3 = g2 = g1 = g5, (7) where 5 ′ g3 = gs, g2 = g, g1 = g . (8) s3 ′ To test this picture, we must check whether the measured values of gs,g,g evolve at very short distances into values that obey (7). 2 gi Let αi = 4π for i = 1, 2, 3. The one-loop renormalization group equations for the gauge couplings are dg b dα b i = − i g3 or i = − i α2. (9) d log Q (4π)2 i d log Q 2π i For U(1), the coefficient b1 is 2 3 1 3 b = − Y 2 − Y 2, (10) 1 3 5 f 3 5 b Xf Xb where the two sums run over multiplets of left-handed Weyl fermions (f) and complex- 3 2 valued bosons (b). The factors 5 Y are the squares of the U(1) charges defined by (3). For non-Abelian groups, the b-coefficients are 11 2 1 b = C (G) − C(r ) − C(r ), (11) 3 2 3 f 3 b Xf Xb where C2(G) and C(r) are the standard group theory coefficients. For SU(N), 1 C (G)= C(G)= N, C(N)= . (12) 2 2 The solution of the RGE (9) is − − b Q α 1(Q)= α 1(M)+ i log . (13) i i 2π M Now consider the situation where the three couplings gi become equal at the mass scale MU , the mass scale of the SU(5) symmetry breaking. Using (13), we can then determine the SM couplings at any lower mass scale. The three αi(Q) are determined by two parameters. We can thus eliminate those parameters and obtain the relation −1 −1 −1 α3 = (1+ B)α2 − Bα1 , (14) 3 where b − b B = 3 2 . (15) b2 − b1 The values of the αi are known very accurately at Q = mZ : −1 −1 −1 α3 =8.50 ± 0.14, α2 = 29.57 ± 0.02, α1 = 59.00 ± 0.02. (16) Inserting these values into (14), we find B =0.716 ± 0.005 ± 0.03. (17) The first error is that propagated from the errors in (16). The second is an estimate of the systematic error from neglecting the two-loop renormalization group coefficients and other higher-order corrections. We can compare the value of B in (17) to the values of (15) from different models. The hypothesis that the three SM gauge couplings unify is acceptable only if the gauge theory that describes physics between mZ and MU gives a value of B consistent with (17). The minimal SM fails this test: • SU(3): each fermion generation has four triplets. The Higgs fields are color singlets. • SU(2): each fermion generation has four doublets. The Higgs fields are doublets. • U(1): Each fermion generation have Y 2 = 6(1/6)2 + 3(2/3)2 + 3(1/3)2 + 2(1/2)2 + 1(1)2 = 10/3. Each Higgs doublet hasP Y 2 = 2(1/2)2 =1/2. P Thus, the SM values of the bi are 4 b = 11 − n , 3 3 g 22 4 1 b = − n − n , 2 3 3 g 6 h 4 1 b = − n − n , (18) 1 3 g 10 h where ng is the number of generations and nh is the number of Higgs doublets. Notice that ng cancels out of (15). This is to be expected. The SM fermions form complete representations of SU(5), and so their renormalization effects cannot lead to differences among the three couplings. For the SM with any number of generations and any number of Higgs doublets, we have (11/3)+(1/6)nh BMHDM(ng, nh)= . (19) (22/3) − (1/15)nh 4 For the minimal SM, with nh = 1, we have 115 B = =0.528, (20) SM 218 35 far away from (17). To get a value consistent with (17) we need nh = 6: B6HDM = 52 =0.673. We can redo the calculation in the minimal supersymmetric version of the SM (SSM). First of all, we should rewrite (11) for a supersymmetric model with one vector supermul- tiplet, containing a vector and a Weyl fermion in the adjoint representation, and a set of chiral supermultiplets indexed by k, each with a Weyl fermion and a complex boson. Then (11) becomes 11 2 2 1 b = C (G) − C (G) − + C(r ) i 3 2 3 2 3 3 k Xk = 3C2(G) − C(rk). (21) Xk The formula (22) undergoes a similar rearrangement: 3 b = − Y 2. (22) 1 5 k Xk For a supersymmetric model with ng generations and nh Higgs doublets, we have b3 = 9 − 2ng, 1 b = 6 − 2n − n , 2 g 2 h 3 b = −2n − n . (23) 1 g 10 h The SSM has nh = 2, yielding 5 B = =0.714, (24) SSM 7 in excellent agreement with (17). Actually, the results here overstate the case for supersymmetry by ignoring two-loop terms in the RGEs, and also by integrating these equations all the way down to mZ , even though, from searches at high energy colliders, most of the squarks and gluinos must be heavier than 300 GeV.
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