{'fraction_non_alphanumeric': 0.04049726878884912, 'fraction_numerical': 0.021535756890814342, 'mean_word_length': 4.174788823911631, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Minimal string-scale unification of gauge couplings

We look for the minimal particle content which is necessary to add to the standard model in order to have a complete unification of gauge couplings and gravity at the weakly coupled heterotic string scale. Using the current precision electroweak data, we find that the presence of a vector-like fermion at an intermediate scale and a non-standard hypercharge normalization are in general sufficient to achieve this goal at two-loop level. If one requires the extra matter scale to be below the TeV scale, then it is found that the addition of three vector-like fermion doublets with a mass around 700 GeV yields a perfect string-scale unification, provided that the affine levels are kY=13/3, k2=1 and k3=2, as in the SU(5) X SU(5) string-GUT. Furthermore, if supersymmetry is broken at the unification scale, the Higgs mass is predicted in the range 125 GeV - 170 GeV, depending on the precise values of the top quark mass and tan(beta) parameter.


Introduction
Superstring theory has emerged as the most promising candidate for a quantum theory of all known interactions. The phenomenology of E 8 ×E 8 heterotic string theory [1] exhibits many of the attractive features of the low-energy physics that we see today. In particular, the four-dimensional standard model (SM) gauge group G SM = SU(3) C ⊗ SU(2) L ⊗ U(1) Y and its generations can be easily incorporated. String theory also offers an elegant explanation for the doublet-triplet splitting problem [2]. Moreover, the unification of gauge couplings and gravity is an intrinsic property of heterotic string theory. Remarkably, unification of couplings is a prediction of string theories even without any grand unified theory (GUT) below the Planck scale. Indeed, gauge and gravitational couplings unify at tree level as [3] where α string = g 2 string /4π is the string-scale unification coupling constant, G N is the Newton constant, α ′ is the Regge slope, α i = g 2 i /4π (i = Y, 2, 3) are the gauge couplings and k i are the so-called affine or Kač-Moody levels at which the group factors U(1) Y , SU(2) L and SU(3) C are realized in the fourdimensional string. The appearance of non-standard affine levels k i plays an important role in string theories. While the non-Abelian factors k 2 and k 3 should be positive integers, the Abelian factor k Y can take a priori any arbitrary value, only constrained to be k Y > 1 for the right-handed electron to have a consistent hypercharge assignment. Furthermore, these factors determine the value of the mixing angle sin θ W at the string scale.
Since string theory relates a dimensionless gauge coupling to a dimensionful gravitational coupling, Eq. (1) itself predicts the unification scale Λ = g string M P , where M P = 1.22 × 10 19 GeV is the Planck mass. This scale is lowered by the inclusion of one-loop string effects and in the weak coupling limit one finds [4] Λ = g string Λ S , where Λ S is given by γ ≈ 0.577 is the Euler constant. It has also been noted [5,6] that the unification scale in the strong coupling limit can be much lower than the perturbative result given by Eq. (2). Yet, it is not clear whether unification is a robust prediction in this case.
Gauge coupling unification is one of the few solid pieces of evidence in favor of supersymmetry. It is well known that the extrapolation of low-energy data within the framework of the MSSM yields an almost perfect unification of gauge couplings at the scale Λ M SSM ≈ 2 × 10 16 GeV, which is about a factor of 25 lower than the string scale of Eq. (3). The resolution of this discrepancy has been the subject of many studies and several paths to unification have been proposed [7,8,9]. On the other hand, it is remarkable that, in the nonsupersymmetric SM, the one-loop g 2 and g 3 gauge couplings already unify at a scale Λ SM ≈ 10 17 GeV, which is close to the unification scale predicted by the string theory. In this case, gauge coupling unification could be achieved for a hypercharge normalization k Y ≈ 13/10 [8]. However, if two-loop effects are taken into account, the above scale should be at most Λ SM ≈ 4×10 16 GeV [10], which is one order of magnitude smaller than the expected string scale. For high-scale supersymmetry breaking, it has been recently shown that gauge coupling unification can be achieved at about 2 × 10 16 GeV in axion models with SM vector-like fermions [11], or at 10 16−17 GeV in the SM with suitable normalizations of the U(1) Y , which can be realized in specific orbifold GUTs [12]. Nevertheless, the unification scale in all of these cases is somehow below the expected string scale.
In view of the above considerations and in the light of the current precision electroweak data, in this letter we study the problem of gauge coupling unification within string theory, with the aim to look for the minimal particle content which is necessary to add to the SM in order to achieve unification at the weakly coupled heterotic string scale. We show that the addition of vector-like fermions with an intermediate mass scale leads to such a unification, provided that the hypercharge affine level is non-canonical, i.e. k Y = 5/3. The existence of such matter states is in general expected in realistic string theories. We also study the possibility that the extra matter content, which leads to unification, has a mass scale below the TeV scale. Among the possible minimal solutions, we find that unification can be achieved with the introduction of three vector-like fermion doublets with a mass around 700 GeV, if the affine levels are k Y ≃ 13/3, k 2 = 1 and k 3 = 2. This normalization is consistent with an SU(5) ⊗ SU(5) or SO(10) ⊗ SO(10) string-GUT compactification [13,14].

One-loop analysis
The evolution of the gauge coupling constants at one loop is governed by the renormalization group equations (RGE) where α i Z ≡ α i (M Z ) and the β-function coefficients b i are given by for non-supersymmetric theories. The function s(R) is 1 for complex scalars, 2 for chiral fermions and 4 for vector-like fermions. The Casimir group invariant for the adjoint representation, C 2 (G n ), is n for SU(n) groups and null for a U(1) group. The functions N i (R) encode the group structure contributions as where d i (R) is the dimension of the representation concerning the invariant subgroup G i and T i (R) is the Dynkin index which, in our convention, is 1/2 for the fundamental representations of SU(n) groups and y 2 for the U(1) Y group. We use the convention that the hypercharge Y = Q − T 3L . In particular, for the SM with N generations and n H complex Higgs doublets one finds Let us now examine the one-loop running of the gauge couplings. The unified coupling constant α string at the scale Λ is expressed in terms of the SU(3) C , SU(2) L and U(1) Y gauge couplings and the corresponding affine levels k i through Eqs. (1)- (3). Thus, at the unification scale Λ, Eq. (4) implies with the additional constraint which reflects the stringy nature of the unification.
These equations can be analytically solved to determine the scale Λ. We obtain where W −1 (x) is the k = −1 real branch of the Lambert W k function [15].
In our numerical calculations we shall use the following electroweak input data at the Z boson mass scale M Z ≃ 91.2 GeV [16,17]: for the fine structure constant α, the weak mixing angle θ W and the strong coupling constant α s , respectively. The top quark pole mass M pole t is taken as [18] M pole and the Higgs vacuum expectation value v = 174.1 GeV.
Using the SM coefficients b i given in Eq. (7) and assuming k 2 = k 3 = 1, we obtain from Eq. (10), Λ ≈ 2.7×10 17 GeV, which in turn implies α string = 0.021. Substituting these values into Eq. (8) we find α s (M Z ) ≃ 0.1239, a value which is clearly outside the experimental range given in Eq. (11). The above result already indicates that the string-scale unification of gauge couplings requires either non-perturbative (or higher-order perturbative) string effects to lower the unification scale or extra matter particles to modify the RGE evolution of the gauge couplings. It is precisely the second possibility that we consider in this work.
Anticipating a possible string-GUT compactification scenario, we shall restrict our analysis to the inclusion of fermions in real irreducible representations. The addition of chiral fermions leads in general to anomalies and their masses are associated to the electroweak symmetry breaking, which imposes further constraints. Also, the introduction of new light scalars requires additional finetunings. Thus, we shall consider the following fermionic states [19]: These can naturally appear in extensions of the SM as a part of some incomplete GUT multiplets. They are present, for instance, in the 5 +5, 10 + 10 and 24 irreducible representations of SU (5). The addition of such matter states gives corrections to the b i coefficients in the gauge coupling running. Denoting by ∆ i these corrections, one has ∆ Y = 2 9 n Q + 2 3 n L + 16 9 n U + 4 9 n D + 4 3 n E + 50 9 n X , where n r denotes the number of multiplets belonging to the irreducible representations r given in Eq. (13). The string unification conditions (8) also get modified, where M is the new-physics threshold. Notice that we assume a common mass scale for the extra matter content, once we are interested in minimal scenarios which could lead to a successful unification. The solution of the above equations is now given by for the unification scale and for the threshold, where Finally, having obtained Λ and M, it remains to determine the hypercharge normalization k Y from Eq. (17): Using Eqs. (18)- (21), it is straightforward to obtain all the possible solutions that lead to the string-scale unification of couplings at one-loop order. Here we present only those which are minimal, i.e. those which require the addition of a single extra particle with a mass scale M. The results are given in Table 1. There exist 3 minimal solutions, namely, n U = 1 , n D = 1 and n G = 1, which correspond to the addition of an up-type or down-type vector-like fermion or one gluino-type fermion, respectively, with quantum numbers as given in Eqs. (13). In all three cases the presence of a non-canonical hypercharge normalization, k Y = 5/3, is required. We have taken the non-Abelian affine levels k 2 and k 3 to be equal to 1 or 2, which are the preferred values from the stringmodel building viewpoint [8]. We also notice that no minimal solution was found with k 2 = k 3 .

Two-loop gauge coupling unification
To perform a more precise analysis of string unification, a two-loop RGE study becomes necessary. We make use of the two-loop RGEs of gauge couplings [20], Table 1 Minimal extra matter content which leads to string-scale unification at one loop. The results for the new-physics threshold M , the unification scale Λ and the hypercharge affine level k Y are presented for the central values given in Eq. (11).  Table 2 Minimal solutions which lead to string-scale unification at two-loop order. We use the central values for the electroweak input data given in Eqs. (11) and (12). which include the one-loop Yukawa coupling running and take properly into account the new physics contributions and threshold. In Table 2 we present the two-loop results for the minimal one-loop solutions given in Table 1. As in the one-loop case, no solution was found with k 2 = k 3 .
It turns out that the unification scale Λ and the hypercharge normalization are not very sensitive to higher order corrections. This can be readily seen by comparing the one-loop results of Eqs. (18) and (21) with the two-loop values numerically obtained (see Table 2). On the other hand, the new-physics threshold M can be significantly altered by such corrections. In particular, we notice that while at one loop the solutions n U = 1 and n D = 1 require an intermediate scale of the order of 10 15 −10 16 GeV, this scale is lowered to 10 12 − 10 13 GeV at two-loop order. One may ask whether such an intermediate mass scale could be naturally generated. In principle, it might be due to the possible presence of nonrenormalizable higher-order operators or could be associated with an approximate global symmetry, such as a chiral symmetry of Peccei-Quinn type.
We have also searched for minimal solutions where the new matter states have Table 3 Minimal extra particle content with a mass below the TeV scale, which leads to unification at two-loop order. We use the electroweak input data given in Eqs. (11) and (12). The non-Abelian affine levels are k 2 = 1 and k 3 = 2 in all cases. The quantities in brackets reflect the effects of the α s (M Z ) uncertainty. a mass scale below the TeV scale. Seven solutions were found, which are listed in Table 3. All of them require the non-Abelian affine levels to be k 2 = 1 and k 3 = 2. Of particular interest is the first solution with three vector-like fermion doublets, i.e. n Q = 3. Not only it yields a perfect string-scale unification at g string ≈ 1, but also, for α s (M Z ) = 0.119 and M = 710 GeV, it implies the hypercharge normalization k Y = 13/3 , thus suggesting an SU(5) ⊗ SU(5) or SO(10) ⊗ SO(10) string-GUT compactification [13,14].

Higgs boson mass
In the string landscape [21], the supersymmetry breaking scale can be high and the SM (with, eventually, some residual matter content) is the simplest effective theory all the way down to low energies. In this scenario, the mass of the yet undiscovered Higgs boson appears to be the most relevant parameter. In general, supersymmetric models contain one pair of Higgs doublets H u and H d . The combination φ ≡ sin β H u − cos β iσ 2 H * d is typically chosen as the fine-tuned SM Higgs doublet φ with a small mass term. If supersymmetry is broken at the string scale, the Higgs boson quartic coupling λ at the unification scale is then given by  Table 2 (n U = 1 and n G = 1) is similar to the one depicted in the figure.

Conclusion
String theory offers us a consistent framework for the unification of all the fundamental interactions including gravity. For a weakly coupled heterotic string, the unification scale is expected around 5 ×10 17 GeV, which is too high to be achieved in the SM or MSSM, even with a non-canonical normalization of the hypercharge. A possible way to reconcile the GUT and string scales is the addition of new matter states to the particle spectrum. In this letter we have presented some minimal solutions based on the introduction of vectorlike fermions. Working at two-loop order, three minimal solutions were found, which correspond to the presence at an intermediate scale of an up-type, down-type or gluino-type fermion with affine levels k 2 = k 3 = 1 and k Y ≈ 6/5 , 5/4 , 63/50 , respectively.
Another interesting issue is the existence of new particles with masses relatively close to the electroweak scale. Imposing the new-physics threshold to be below the TeV scale, we have found several minimal solutions for stringscale unification. All of them require at least three new matter states. It is remarkable that the addition of three vector-like fermion doublets (n Q = 3) yields unification at the string scale Λ S for (k Y , k 2 , k 3 ) = (13/3 , 1 , 2) . These values are consistent with the affine levels of an SU(5) ⊗ SU(5) string-GUT. In this case, the strong coupling constant at the M Z scale is α s (M Z ) = 0.119, with all the other electroweak input data given at their central values.
The string landscape allows for a high-scale supersymmetry breaking. If supersymmetry is broken at the string scale, most of its problems, such as fast dimension-five proton decay, excessive flavor and CP violation and stringent constraints on the Higgs mass, are avoided. In this scenario, the Higgs boson mass is predicted in the range 125 GeV m H 170 GeV, for the minimal string unification solutions presented here.