Gauge Coupling Unification in the Standard Model

The string landscape suggests that the supersymmetry breaking scale can be high, and then the simplest low energy effective theory is the Standard Model (SM). We show that gauge coupling unification can be achieved at about 10^{16-17} GeV in the SM with suitable normalizations of the U(1)_Y. Assuming grand unification scale supersymmetry breaking, we predict that the Higgs mass range is 127 GeV to 165 GeV, with the precise value strongly correlated with the top quark mass and SU(3)_C gauge coupling. We also present 7-dimensional orbifold grand unified theories in which such normalizations for the U(1)_Y and charge quantization can be realized.

Introduction -There exists an enormous "landscape" for long-lived metastable string/M theory vacua [1]. Applying the "weak anthropic principle" [2], the landscape proposal may be the first concrete explanation of the very tiny value of the cosmological constant, which can take only discrete values, and it may address the gauge hierarchy problem. Notably, the supersymmetry breaking scale can be high if there exist many supersymmetry breaking parameters or many hidden sectors [3,4]. Although there is no definite conclusion that the string landscape predicts high-scale or TeV-scale supersymmetry breaking [3], it is interesting to consider models with high-scale supersymmetry breaking [4,5].
If the supersymmetry breaking scale is around the grand unification scale or the string scale, the minimal model at low energy is the Standard Model (SM). The SM explains the existing experimental data very well, including electroweak precision tests, and it is easy to incorporate aspects of physics beyond the SM through small variations [4,5,6]. However, even if the gauge hierarchy problem can be solved by the string landscape proposal, there are still some limitations of the SM, for example, the lack of explanation of gauge coupling unification and charge quantization.
Charge quantization can be easily explained by embedding the SM into a grand unified theory (GUT). Should the Higgs particle be the only new physics observed at the Large Hadron Collider (LHC) and the SM is thus confirmed as a low energy effective theory, an important question will be: can we achieve gauge coupling unification in the SM without introducing any extra multiplets between the weak and GUT scales [7] or having large threshold corrections [8]? As is well known, gauge coupling unification cannot be achieved in the SM if we choose the canonical normalization for the U (1) Y hypercharge interaction, i.e., the Georgi-Glashow SU (5) normalization [9]. Also, to avoid proton decay induced by dimension-6 operators via heavy gauge boson exchanges, the gauge coupling unification scale is constrained to be higher than about 5 × 10 15 GeV.
In this Letter we reconsider gauge coupling unification in the SM. The gauge couplings for SU (3) C and SU (2) L are unified at about 10 16−17 GeV, and the gauge coupling for the U (1) Y at that scale depends on its normalization. If we choose a suitable normalization of the U (1) Y , the gauge couplings for the SU (3) C , SU (2) L and U (1) Y can in fact be unified at about 10 16−17 GeV, and there is no proton decay problem via dimension-6 operators. Therefore, the real question is: is the canonical normalization for U (1) Y unique?
For a 4-dimensional (4D) GUT with a simple group, the canonical normalization is the only possibility, assuming that the SM fermions form complete multiplets under the GUT group. However, the U (1) Y normalization need not be canonical in string model building [10,11], orbifold GUTs [12,13] and their deconstruction [14], and in 4D GUTs with product gauge groups: (1) In weakly coupled heterotic string theory, the gauge and gravitational couplings unify at tree level to form one dimensionless string coupling constant g string [10] where g Y , g 2 , and g 3 are the gauge couplings for the U (1) Y , SU (2) L , and SU (3) C , respectively, G N is the gravitational coupling and α ′ is the string tension. Here, k Y , k 2 and k 3 are the levels of the corresponding Kac-Moody algebras; k 2 and k 3 are positive integers while k Y is a rational number in general [10].
(2) In intersecting D-brane model building on Type II orientifolds, the normalization for the U (1) Y (and other gauge factors) is not canonical in general [11].
(3) In orbifold GUTs [12], the SM fermions need not form complete multiplets under the GUT group, so the U (1) Y normalization need not be canonical [13]. This statement is also valid for the deconstruction of the orbifold GUTs [14] and for 4D GUTs with product gauge groups.
We shall assume that at the GUT or string scale, the gauge couplings in the SM satisfy where g 2 1 ≡ k Y g 2 Y , and k Y = 5/3 for canonical normalization. We show that gauge coupling unification in the SM can be achieved at about 10 16−17 GeV for k Y = 4/3, 5/4, 32/25. Especially for k Y = 4/3, gauge coupling unification in the SM is well satisfied at two loop order. Assuming GUT scale supersymmetry breaking, we predict that the Higgs mass is in the range 127 GeV to 165 GeV. In addition, we construct 7-dimensional (7D) orbifold GUTs in which such normalizations for the U (1) Y and charge quantization can be realized. A more detailed discussion will be presented elsewhere [15].
Gauge Coupling Unification -We define α i = g 2 i /(4π) and denote the Z boson mass by M Z . In the following, we choose a top quark pole mass m t = 178.0±4.3 GeV [16], α 3 (M Z ) = 0.1182 ± 0.0027 [17], and the other gauge couplings, Yukawa couplings and the Higgs vacuum expectation value at M Z from Ref. [18].
We first examine the one-loop running of the gauge couplings. The one-loop renormalization group equations (RGEs) in the SM are where t = ln µ, µ is the renormalization scale, and We consider the SM with k Y = 4/3, 5/4, 32/25 and 5/3. In addition, we consider the extension of the SM with two Higgs doublets (2HD) with b = (7/k Y , −3, −7) and k Y = 4/3, and the Minimal Supersymmetric Standard Model (MSSM) with b = (11/k Y , 1, −3) and k Y = 5/3. For the MSSM, we assume the supersymmetry breaking scale 300 GeV for scenario I (MSSM I), and the effective supersymmetry breaking scale 50 GeV to include the threshold corrections due to the mass differences between the squarks and sleptons for scenario II (MSSM II) [19]. We use M U to denote the unification scale where α 2 and α 3 intersect in the RGE evolutions. There is a sizable error associated with the α 3 (M Z ) measurement. To take into account this uncertainty, we also consider α 3 − δα 3 and α 3 + δα 3 as the initial values for the RGE evolutions, whose corresponding unification scales are called In Table I we compare the convergences of the gauge couplings in above scenarios. We confirm that the SM with canonical normalization k Y = 5/3 is far from a good unification. Introducing supersymmetry significantly improves the convergence. Meanwhile, the same level of convergences can be achieved in all the nonsupersymmetric models. In particular, the SM with k Y = 32/25 and the 2HD SM with k Y = 4/3 have very good gauge coupling unification. The two-loop running of the gauge couplings produces slightly different results. We perform the two-loop running for the SM with k Y = 4/3, as it has an excellent unification. We use the two-loop RGE running for the gauge couplings and one-loop for the Yukawa couplings [20]. With the central value of α 3 , we show the gauge coupling unification in Fig. 1. At the unification scale of 4.3 × 10 16 GeV, the value of α 1 precisely agrees with those of α 2 and α 3 . If the Higgs particle is the only new physics discovered at the LHC and the SM is thus confirmed as a low energy effective theory, the most interesting parameter is the Higgs mass. To be consistent with string theory or quantum gravity, it is natural to have supersymmetry in the fundamental theory. In the supersymmet-ric models, there generically exist one pair of the Higgs doublets H u and H d . We define the SM Higgs doublet H, which is fine-tuned to have a small mass term, as H ≡ − cos βiσ 2 H * d + sin βH u , where σ 2 is the second Pauli matrix and tan β is a mixing parameter [4,5]. For simplicity, we assume that supersymmetry is broken at the GUT scale, i.e., the gauginos, squarks, sleptons, Higgsinos, and the other combination of the scalar Higgs doublets (sin βiσ 2 H * d + cos βH u ) have a universal supersymmetry breaking soft mass around the GUT scale. We can calculate the Higgs boson quartic coupling λ at the GUT scale [4,5] and then evolve it down to the weak scale. Using the one-loop effective Higgs potential with top quark radiative corrections, we calculate the Higgs boson mass by minimizing the effective potential [5]. For the SM with k Y = 4/3, the Higgs boson mass as a function of tan β for different m t and α 3 is shown in Fig. 2. We see if we vary α 3 within its 1σ range, m t within its 1σ and 2σ ranges and tan β from 1.5 to 50, the predicted mass of the Higgs boson ranges from 127 GeV to 165 GeV. A large part of this uncertainty is due to the present uncertainty in the top quark mass. The top quark mass can be measured to about 1 GeV accuracy at the LHC [21]. Assuming this accuracy and the central value of 178 GeV, the Higgs boson mass is predicted to be between 141 GeV and 154 GeV.

FIG. 2:
The predicted Higgs mass for the SM with kY = 4/3. The red (lower) curves are for α3 + δα3, the blue (upper) α3 − δα3, and the black α3. The dotted curves are for mt ± δmt, the dash ones for mt ± 2δmt, and the solid ones for mt.
Furthermore, for the SM with k Y = 5/4 and 32/25, the gauge coupling unifications at two loop are quite similar to that of the SM with k Y = 4/3, and the predicted Higgs mass ranges are almost the same [15].
Orbifold GUTs -In string model building, the orbifold GUTs and their deconstruction, and 4D GUTs with product gauge groups, the normalization for the U (1) Y need not be canonical. As an explicit example, we show that k Y = 4/3 can be obtained in the 7D orbifold SU (6) model on the space-time M 4 × T 2 /Z 6 × S 1 /Z 2 where charge quantization can be realized simultaneously. Here, M 4 is the 4D Minkowski space-time. Similarly, k Y = 5/4 and k Y = 32/25 can be obtained in the 7D orbifold SU (7) models with charge quantization [15].
We consider the 7D space-time M 4 × T 2 × S 1 with coordinates x µ , z and y where z is the complex coordinate for the torus T 2 and y is the coordinate for the circle S 1 . The radii for T 2 and S 1 are R and R ′ . The T 2 /Z 6 × S 1 /Z 2 orbifold is obtained from T 2 × S 1 by moduloing the equivalent classes where ω = e iπ/3 . (z, y) = (0, 0) and (0, πR ′ ) are the fixed points under the Z 6 × Z 2 symmetry. N = 1 supersymmetry in 7 dimensions has 16 supercharges and corresponds to N = 4 supersymmetry in 4 dimensions; thus, only the gauge multiplet can be introduced in the bulk. This multiplet can be decomposed under the 4D N = 1 supersymmetry into a vector multiplet V and three chiral multiplets Σ 1 , Σ 2 , and Σ 3 in the adjoint representation, where the fifth and sixth components of the gauge field (A 5 and A 6 ) are contained in the lowest component of Σ 1 , and the seventh component of the gauge field (A 7 ) is contained in the lowest component of Σ 2 . The SM quarks, leptons and Higgs fields can be localized on 3-branes at the Z 6 × Z 2 fixed points, or on 4-branes at the Z 6 fixed points.
We obtain that, for the zero modes, the 7D N = 1 supersymmetric SU (6) gauge symmetry is broken down to the [15]. Also, we have only one zero mode from Σ i with quantum number (3, 1, −2/3) under the SM gauge symmetry, which can be considered as the right-handed top quark [15]. On the 3-brane at the Z 6 ×Z 2 fixed point (z, y) = (0, 0), the preserved gauge symmetry is SU [13]. Thus, on the observable 3-brane at (z, y) = (0, 0), we can introduce one pair of Higgs doublets and three families of SM quarks and leptons except the right-handed top quark [13]. Because the U (1) Y charge for the right-handed top quark is determined from the construction, charge quantization can be achieved from the anomaly free conditions and the gauge invariance of the Yukawa couplings on the observable 3brane. Moreover, the U (1) ′ anomalies can be cancelled by assigning suitable U (1) ′ charges to the SM quarks and leptons, and the U (1) ′ gauge symmetry can be broken at the GUT scale by introducing one pair of the SM singlets with U (1) ′ charge ±1 on the observable 3-brane. Interestingly, this U (1) ′ gauge symmetry may be considered as a flavour symmetry, and then the fermion masses and mixings may be explained naturally. Furthermore, supersymmetry can be broken around the compactification scale, which can be considered as the GUT scale, for example, by Scherk-Schwarz mechanism [22].
Conclusions -The string landscape suggests that the supersymmetry breaking scale can be high and then the simplest low energy effective theory is just the SM. We showed that gauge coupling unification in the SM with k Y =4/3, 5/4, and 32/25 can be achieved at about 10 16−17 GeV. Assuming GUT scale supersymmetry breaking, we predicted that the Higgs mass is in the range 127 GeV to 165 GeV. We also presented the 7D orbifold GUTs where such normalizations for the U (1) Y and charge quantization can be realized.