The generation of the superpartners of the particles of the standard model is implicitly taken care of by promoting the field of the latter particles to superfields.
For example, we may define the left-chiral superfield associated with the electron, as in (2.6.23), by
E.x; /DA.x/Ci .Le/.x/C1
2LF.x/C i
45 @A.x/
1
45 @ .Le/.x/ 1
32.5/2A.x/; (2.13.1) where .Le/ is the left-hand part of the electron field, and A, here, is the field associated with its superpartner of spin0, referred to as the selectron, andF is an auxiliary field. Similarly, we may introduce the superfield of a gauge boson, say,
of aW vector boson, in the Wess-Zumino gauge, as in (2.6.41), by V.W/.x; /D 1
45W.x/C i 2p
25 .x/ 1
16.5/2D.x/;
(2.13.2) whereWrepresent theWvector boson field, in question, suppressing other indices, and, here, denotes its superpartner, a Majorana spinor, referred to as the wino, and Ddenotes an auxiliary field. In general, the superpartner of a gauge field is called a gaugino.
By continuing, the above process, we may generate the superfields associated, in particular, with the quarks and leptons of the standard model, conveniently, denoted in the following manner:
Ni
Ei
Li; .E/ciI Ui
Di
Qi; .U/ci; .D/ci; (2.13.3) where the subscriptispecifies the various generations, with theNidenoting the left- chiral superfields associated with the left handed neutrino fields, theUi denoting the left-chiral superfields associated with the left-handedu;c;tquarks, and theDi
denoting the left-chiral superfields associated with the left-handedd;s;b quarks, respectively, fori D 1; 2; 3..E/ciI .U/ci; .D/ci are associated with the left-handed positron, and the left-handed anti-quarks, respectively. The superpartner of a lepton is referred to as a slepton, and of the quark as a squark. The superparticles of the Gluons, eight of them, are referred as gluinos.
To consider the interaction of the above fermion fields, with the Higgs bosons, recall how theF term of a superpotentialWˇˇ
F, in Sects.2.11,2.12, was defined.
To this end the following elementary properties, involving left-chiral superfields˚L
and their adjoints should be noted:
ı.2/.R/ .˚jLAj/D0; (2.13.4) ı.2/.R/ .˚iLAi/.˚jLAj/D0; (2.13.5) ı.2/.R/ .˚i1LAi1/.˚i2LAi2/.˚jLAj/D0; (2.13.6) from which we may infer that for describing Yukawa interactions of the quark and leptons, in a supersymmetric setting, the superpotential must be a functions of the (left-) chiral fields but not of their adjoints. By attempting to define their interactions with only one Higgs left-chiral superfield doublet, say with weak hyperchargeY D C1, we will encounter a difficulty as will be evident below. To this end, one defines
two Higgs doublets of left-chiral superfields H1D
H10
H1 ; H2D H2C
H20 ; (2.13.7)
respectively, of weak hypercharges Y D 1. We note that using the identity Y D2.QT3/, the weak hypercharges associated with the superfields in (2.13.3) are respectively, Y D 1; 2I1=3;4=3; 2=3. Accordingly, SU.3/ SU.2/ U.1/singlets may be introduced to define the Lagrangian density of the Yukawa interactions of the fermions with the Higgs bosons and the superpotential will involve linear combinations ofF parts of the following anti-symmetric terms:
p1
2.EiH10NiH1/.Ej/c; p1
2.DiH10UiH1/.Dj/c; p1
2.DiH2CUiH20/.Uj/c; (2.13.8) with coefficients depending on i and j, where we note that the anti-symmetric combinations of the two doublets, in each term, give rise to SU.2/singlets. Clearly, if we have only one Higgs doublet, sayH1, then the last term will involve the adjoints of the superfield components of H1. Similarly, if we have the Higgs doubletH2 the first two terms will involve the adjoints of the superfield components of H2. In particular, the two Higgs fields now give masses tou-type andd-type quarks consistent with supersymmetry. The presence of two Higgs doublets turns out to be also important in the elimination of higgsino related anomalies. In addition to the Yukawa couplings in (2.13.8), we may also consider a mass-like term involving the two Higgs doublets in the form/.H20H10H2CH1/ˇˇ
F.44Interactions with gauge fields occur through the kinetic energy terms.
The theory under consideration, with just two Higgs doublets, with minimal number of couplings and minimal number of fields is referred to as the Minimal Supersymmetric Model (MSSM).
An important contribution of this model is to the problem of the unification of gauge couplings at high energies, that is, beyond the MSSM model. This is considered next.
The beta functions of the MSSM are given in Table 2.1 together with the corresponding ones for the SM for comparison,45where
2 d d2
1
˛#.2/ Dˇ#; ˛# D g2#
4 ; (2.13.9)
44Here it is worth noting that selecting renormalizable supersymmetric interactions terms which conserve baryon and lepton numbers, may be equivalently achieved by imposing a symmetry known as the conservation ofR-parity, but we will not go into it here. See, e.g., [11,14].
45See Appendix B of this chapter for a discussion of the SM in this context.
Table 2.1 Table depicting the expressions for12 beta functions, with the beta functions, as introduced in (2.13.1), for the SM, and MSSM models as functions of the number of generations ng, and the number of complex Higgs doubletsnH
12 ˇ# SM MSSM
12 ˇs .334ng/ .276ng/
12 ˇ .224ng12nH/ .186ng32nH/
12 ˇ0 .203 ngC12nH/ .10ngC32nH/
with B:C::
˛s.2/ˇˇˇ
2DM2 D˛.2/ˇˇˇ
2DM2 D 5
3˛0.2/ˇˇˇ
2DM2; (2.13.10) at a unifying energy scale to be determined below. The number of complex Higgs bosons doublets are taken to benH D 1for the SM, andnH D 2for the MSSM models.46
The following two equations immediately follow:
2 d d2
1
˛s.2/ 1
˛.2/ D 1
; (2.13.11)
2 d d2
1
˛s.2/3 5
1
˛0.2/ D 12
5 : (2.13.12)
Upon subtracting (2.13.12) from 12/5 times (2.13.11), and using the unifying boundary conditions (2.13.10), at a unifying energy specified by a mass parameter M, as well as the defining equations47
1
˛0 D cos2W
˛em ; 1
˛ D sin2W
˛em ; (2.13.13)
we obtain at an energy scale specified by the massMZof theZvector boson, sin2Wˇˇˇ
M2ZD 1 5C 7
15
˛em.M2Z/
˛s.M2Z/ : (2.13.14)
46For details concerning these beta functions see [13,22,23]. Their computations parallel very closely to those computed in Sect. 6.6, and already used in Sect. 6.18 in Vol. I.
47See Appendix B of this chapter for a review and for the relevant analysis in the standard model.
On the other hand, the second defining equation in (2.13.11), (2.13.13) and the equality in (2.13.14) lead upon integration to
ln M2
M2Z D
5˛em.M2Z/
18 3
˛em.M2Z/
˛s.M2Z/
: (2.13.15)
Using the experimentally input,48 ˛s.MZ2/ D 0:1184˙0:0007,1=˛em.M2Z/ D 127:916˙0:015, give, sin2Wˇˇ
M2Z'0:231, which compares remarkably well with the experimental result.49 (2.13.15) gives rise to a scale M ' 2:2 1016GeV.
Moreover,
2 d d2
1
˛s.2/D .276ng/
12 ; (2.13.16)
gives, upon integration forngD3, the expression 1
˛s.M2/D 1
˛s.M2Z/C 3 4 ln
M2 M2Z D 3
5 ˛s.M2Z/C 3
20 ˛em.M2Z/: (2.13.17) The latter gives1=˛s.M2/ 24:3,1=˛em.M2/ 65.
As a measure of the accuracy of the approach of the coupling parameters to eventual unification in (2.13.10), one may introduce the following critical parameter50
D ˛1.M2Z/˛s1.M2Z/
.3=5/˛01.M2Z/˛1.M2Z/; (2.13.18) which experimentally takes the value'0:74. On the other hand, by integrating the renormalization group equations (2.13.9)/Table2.1from2 DMZ2to2DM2and using the boundary conditions in (2.13.10), we obtain for the theoretical expression for (2.13.18)
theorD ˇˇs
.3=5/ˇ0ˇ D 39
.3=5/33C3 D0:714; (2.13.19) which compares well with the experimental value. This is unlike the value of0:5 obtained for the non-supersymmetric version as discussed in Appendix B of the
48Beringer [5].
49See, e.g., [5].
50Peskin [29].
Fig. 2.2 Quadratic divergent contributions to a scalar particle self-mass squared from the two graphs, with thesolidand thedashed linesdenoting a spin 1/2 and a scalar particle, respectively
chapter. Also with the lifetime of the proton/ M4, one obtains the very welcome additional power of104for the lifetime to that of the non-supersymmetric theory.
The recent experimental large lower bound51for the lifetime of the proton justifies, however, the need of further investigations of unifications schemes.
In Sect.2.10, we have seen, by a particular example, how supersymmetry eliminates quadratic divergences in the radiative corrections to the mass squared, through the study of the propagator, of a scalar particle. Due to opposite signs of the statistics of the fermion and the scalar particle, and with the particular relationship existing between the respective dimensionless couplings, as imposed by supersymmetry, such quadratic divergences cancel out. As a matter of fact, in a supersymmetric theory, some radiative corrections may not only be finite but may be completely absent in perturbation theory.52
A fundamental energy scale arises in the SM from the vacuum expectation value of the Higgs boson field ' 246GeV,53 which sets up the scale for the masses of the particles in the theory, including its own mass. This energy scale is much smaller in comparison to the Planck energy scale 1019GeV or lower, at which gravitation is expected to be significant. It is even much smaller than a grand unified energy scale, say 1016GeV. The question then arises as to what amounts for the enormous energy scale difference between a grand unified energy scale and the energy scale characteristic of the SM? This is known as the hierarchy problem.
What kind of new physics arises in this huge range of energy? As ascalarparticle, the self-mass squared •M2H of the Higgs boson, in the non-supersymmetric SM model, is quadratically divergent,54 as inferred, by simple power counting, from such diagrams as shown in Fig.2.2.
51See [27].
52See [20,34]. The underlying theorems are referred to as non-renormalization theorems.
53See, e.g., (6.14.34) in Chap. 6 of Vol. I [26].
54See also [39].
With an ultraviolet cut-off taken of the order, say,M 1016 1019GeV, an unnatural cancelation,55 referred to as fine-tuning, has to occur between the bare mass squared of the Higgs boson and radiative corrections of the order M2 in order to give a net finite mass for the Higgs boson not much different from the energy scale characterizing the SM.56 This unnatural cancelation of enormously large numbers has been termed a facet of the hierarchy problem. We have seen in Sect. 6.14 of Vol. I, that at the tree level approximation, the electroweak theory provides very good agreement with experiments for the masses of the gauge bosons.
A supersymmetric removal of an unwanted quadratic divergence is a positive contribution to the hierarchy problem. Supersymmetry is of significance in dealing with the hierarchy problem, as in supersymmetric field theories cancelations of such large quadratic corrections, a priori, generally, occur between loops involving particles and loops involving their supersymmetric counterparts in a supersymmetric version of a non-supersymmetric field theory, similar to the two loops shown in Fig.2.2. Moreover, this cancelation is, possibly, up to divergences of logarithmic type which are tolerable, thus protecting a scalar particle from acquiring a large bare mass.
Since no superparticles are expected to have the same masses as their particle counterparts, otherwise some of them would have been discovered so far, super- symmetry is to be broken. For supersymmetry to provide a solution to the hierarchy problem, the relationship between the dimensionless coupling constants, in the unbroken supersymmetric theory, must be maintained without spoiling renormal- izability, and without re-introducing quadratic divergences that supersymmetry was here to eliminate. This may be done by introducing supersymmetry breaking terms in the Lagrangian density with mass terms and coupling terms with positive mass dimensionalities, referred to as “soft” supersymmetry breaking terms.57 This, in turn, introduces new physics beyond the SM, specified by an energy scale which may be denoted byMs. With the quadratic divergence now removed, the self mass squared•MH2, will, by dimensional reasoning, be proportional toMs2, up to the well known logarithmic corrective factors of perturbation theory. Clearly, such a mass scaleMscannot be too large, say of the order of a TeV or so, otherwise the hierarchy problem would re-emerge all over, or perhaps give rise to a little hierarchy problem.
55In QED the self-mass of the electron is only logarithmically divergent and the shift between the bare and the physical mass is not huge in comparison with a cut-off mass scale, say, of the order of the Planck energy.
56Aad, et al. [1], Chatrchyan, et al. [6].
57“Soft” supersymmetry breaking terms have been classified systematically by Girardello and Grisaru [18]. Such terms are spelled out in detail in [38, p. 68].