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I. SUPERSYMMETRIC GRAND UNIFICATION

A. Supersymmetrizing the : content

The content of the standard model (SM) consists of three families of chiral and . 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 that transform in the ( 2 , 0) representation of the Lorentz (they have an undotted index α). The subscripts R, L do not specify the repre- sentation of the but instead are used to indicate the different transformation properties under the SU(2)L gauge group. In addition, there is a single representation, that is the Higgs field:

h(1, 2)−1/2. (2)

As we learned in the previous section, requires the presence of additional states which form supermultiplets with the known . Since all states of a super- multiplet carry the same gauge numbers, we need at least a doubling of states: For every field of the SM, one has to postulate a with the exact same gauge quantum numbers and a 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 1 ˜ ˜ ˜ are promoted to vector multiplets by adding the corresponding spin- 2 (G, W , B) to the spectrum. Finally, the Higgs is also promoted to a chiral multiplet with a 1 spin- 2 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 of the Higgsino. Another way to see the necessity of a second Higgs doublet is the fact that the Higgsino is a chiral which carries U(1) hypercharge and hence it upsets the cancellation condition. A second Higgsino of opposite U(1)Y 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 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) ! a ! s5 1 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 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 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 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 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 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 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 colliders, most of the squarks and must be heavier than

300 GeV. A more precise of αs(mZ ) gives a slightly higher value, 0.13 instead of 0.12. However, these corrections can easily be compensated by similar corrections to the upper limit of the integration, following the details of the particle spectrum at the grand unification scale. It remains a remarkable fact that the minimal supersymmetric extension of the standard model is approximately compatible with grand unification ‘out of the box’, with no need for further model building.

5 C. Additional aspects of supersymmetric GUT

1. Yukawa unification

GUTs unify quarks and leptons in the sense that they are embedded in the same GUT representations. In particular, the 5¯ has the down singlets and doublets, while the 10 has the doublets and lepton singlets. Consequently, the minimal SU(5) model with its Yukawa structure, u d ¯ ¯ W ∼ Yab 5H2 10a10b + Yab 5H1 10a5b, (25) gives T YD = YE . (26)

Thus, the eigenvalues or, equivalently, the of the down quarks and the charged leptons of the same generation should be equal at the scale of unification. This relation can then be renormalized down to the weak scale [2]:

d yd 1 yd 2 2 2 16 2 4 2 = 2 yu + 3(yd − yℓ ) − g3 − g1 . (27) d ln Q yℓ ! 16π yℓ !  3 3  This should be tested against the measured fermion masses. Note that the fermion masses are the running masses evaluated at, say, mZ where mb ≃ 3 GeV and mτ ≃ 1.7 GeV. Indeed, it is found to be correct for the third generation couplings (b − τ unification) for either tan β ≃ 1 − 2 (large top Yukawa coupling, yt(mZ ) ≃ 0.95/ sin β) or tan β >∼ 50

(large bottom Yukawa coupling, yb(mZ ) ≃ 0.017 tan β), and for a much larger parameter once finite corections to the quark masses from sparticle loops (not included here) are considered.

Clearly, the successful renormalization of the unification relation yd/yℓ requires large

Yukawa couplings (which renormalize yd). The large Yukawa couplings are needed to coun- terbalance the QCD corrections. It is not surprising then that these relations fail for the lighter generations. Yukawa unification (in its straightforward form) applies to and dis- tinguishes the third generation. The first and second generation fermion masses may be assumed to vanish at leading order and could be further understood as setting the mag- nitude of the perturbations for any such relations, either from higher dimensional Higgs representations or from mpl-suppressed operators, or both.

6 2. decay

The p → K+ν¯ channel is predicted to be the dominant for supersymmetric SU(5) theories. We concentrate on this channel here. It is enough to exclude minimal SU(5). The p → K+ν¯ decay results from dimension 5 operator, and the associated dressing diagram. The dimension five operators,

QQQ˜L˜ + ucdcu˜ce˜c, (28) come from colored Higgs triplet exchange, and arise from the following terms: 1 W = λuQ ucH + V ∗λdQ dcH¯ + λdecL H¯ + λueiφi Q Q H Y i i i f ij j i j f i i i f 2 i i i C ∗ d ¯ u c c −iφi ∗ d c c ¯ +Vijλj QiLjHC + λi Vijui ejHC + e Vijλj ui djHC. (29)

Here, the H and H¯ represent two different Higgs multiplets that give the up and down type quarks their masses. The Hf is the doublet, while the HC is the colored Higgs triplet. All fields are superfields. Requiring that the proton lifetime obeys the experimental limit [3], τ(p → K+ν¯) > 6.7 × 1032 years, puts a lower bound on the colored Hihhs triplet mass:

16 MHC ≥ 7.6 × 10 GeV. (30)

Taking into account that the GUT-scale fields – the adjoing Higgs field Σ, the colored

Higgs triplet HC , and the new vector bosons V – may have different masses, the RGEs at one loop are [4]

−1 −1 1 2 msusy Λ α3 (mZ ) = α5 (Λ) + (−2 − ng)log +(−9+2ng)log 2π  3 mZ mZ Λ Λ Λ −4log + 3log + log , (31) MV MΣ MHC #

−1 −1 1 13 2 msusy Λ Λ Λ α2 (mZ ) = α5 (Λ) + (− − ng)log +(−5+2ng)log − 6log + 2log , 2π  6 3 mZ mZ MV MΣ  −1 −1 1 1 2 msusy 3 Λ Λ 2 Λ α1 (mZ ) = α5 (Λ) + (− − ng)log +( +2ng)log − 10 log + log . 2π " 2 3 mZ 5 mZ MV 5 MHC #

By an appropriate choice of combination of couplings, we can eliminate MΣ and MV :

−1 −1 −1 1 12 MHC msusy 3α2 (mZ ) − 2α3 (mZ ) − α1 (mZ)= log − 2log . (32) 2π  5 mZ mZ  One can invert this equation to determine the colored Higgs mass independently of the other masses at the GUT scale. For msusy =1 T eV , one gets

14 15 3.5 × 10 ≤ MHC ≤ 3.6 × 10 GeV. (33)

7 Note that this limit will not be drastically affected in the case where we take the scalaras of the first and secomd generations to have masses of order 10 TeV, as done in some models to alleviate flavor problems. This is because changing an energy scale of an entire SU(5) multiplet does not change the unification condition, and hence the RGE bound, at one loop. The upper bound in (33) is in conflict with the lower bound of (30), thus excluding the minimal SU(5) model. There are several ways of extending the minimal model to make it viable. One strategy it to keep the colored Higgs triplet heavy enough to obey the proton lifetime constraint, and add gauge multiplets to modify the RGE constraint. The simplest way to do this is to include a second pair of Higgs bosons in the 5 + 5¯ representation without any Yukawa coupling to matter multiplets. However, in this pair one makes the triplet lighter than the doublet. The second strategy is to suppress the dimension five operators in some way. That has been done in an extra-dimensional framework, and in extended Higgs sectors.

3. Doublet-triplet splitting

Supersymmetry guarantees that once the Higgs SU(2)-doublets and SU(3)-triplets are split so that the former are and the latter are heavy, this is preserved to all orders in perturbation theory. Nevertheless, it does not specify how such a split may occur. This is the doublet-triplet splitting problem which is conceptually, though not technically, a manifestation of the .

More generally, it is an aspect of the problem of fixing the µ-parameter at µ = O(mZ ). Extensive model building efforts and many innovative solutions exist. They typically involve extending the model representations, symmetries and/or dimensions.

[1] M. E. Peskin, arXiv:0801.1928 [hep-ph]. [2] N. Polonsky, Lect. Notes Phys. M68, 1 (2001) [arXiv:hep-ph/0108236]. [3] Y. Hayato et al. [Super-Kamiokande Collaboration], Phys. Rev. Lett. 83, 1529 (1999) [arXiv:hep-ex/9904020]. [4] H. Murayama and A. Pierce, Phys. Rev. D 65, 055009 (2002) [arXiv:hep-ph/0108104].

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