Nonsupersymmetric model with unification of electroweak
and strong interactions
R. Frezzotti,1 M. Garofalo,2and G. C. Rossi1,3 1Dipartimento di Fisica, Università di Roma“Tor Vergata” INFN, Sezione di Roma 2 Via della Ricerca Scientifica—00133 Roma, Italy
2Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3JZ, Scotland, United Kingdom
3Centro Fermi—Museo Storico della Fisica, Piazza del Viminale, 1–00184 Roma, Italy (Received 9 March 2016; published 19 May 2016)
In this paper, we show that adding to the standard model particle content a set of superstrongly interacting particles with appropriately chosen hypercharges leads to unification of strong and electroweak interactions at a level comparable to that of the minimal supersymmetric standard model.
DOI:10.1103/PhysRevD.93.105030
I. INTRODUCTION
A desirable feature of a beyond-the-standard-model model (BSMM) is the unification of gauge couplings. As is well known, unification fails in the standard model (SM) [1–3], but it can be achieved, for instance, if the model is extended to incorporate supersymmetry [4]. In Fig. 1, the two-loop running of electroweak and strong couplings in the SM (black dotted lines) is compared to the running in the minimal supersymmetric standard model (MSSM) with blue and red continuous lines referring to different supersymmetry thresholds (namely 0.5 TeV blue curve and 1.5 TeV red curve) [5]. It is undeniable that inclusion of supersymmetric partners substantially improves unification.
In this short paper, we want to provide a couple of examples of nonsupersymmetric extensions of the SM where unification occurs to an accuracy level similar to that of Fig. 1. The key feature of these models is that, besides the elementary particles of the SM, a new set of superstrongly interacting particles (SIPs) with suitably chosen hypercharge quantum numbers[6]living at a scale ΛT∼ Oðfew TeVÞ, i.e. much larger than ΛQCD, is present.
The two models we want to discuss differ for the magnitude of the mass of scalar particles that we take to be either “finite” as in the SM, or as large as the GUT scale. Thus the Lagrangian (which for brevity we will generically give the label BSMM) we want to consider will have the form
LBSMM¼ L gaugeþ LSMF þ Lscalarþ LSS; ð1Þ Lgauge ¼ 14ðFBFBþ FWFWþ FAFAþ FGFGÞ; ð2Þ LSM F ¼ Xng f¼1 ½¯qf LDBWAq f Lþ ¯q f u R DBAq f u R þ¯q f d R DBAq f d R þ ¯lf LDBWl f Lþ ¯l f u R ∂l f u R þ ¯l f d R DBl f d R ; ð3Þ Lscalar¼ 12Tr½Φ†D BW μ DBWμ Φ þμ 2 0 2 Tr½Φ†Φ þ λ0 4ðTr½Φ†ΦÞ2; ð4Þ LSS¼X νQ s¼1 ½ ¯Qs
LDBWAGQsLþ ¯Qs uR DBAGQs uR þ ¯Qs dR DBAGQs dR
þX νL t¼1 ½ ¯Lt LDBWGLtLþ ¯Lt uRDBGLt uR þ ¯Lt dRDBGLt dR; ð5Þ where the new set of SIPs, including Q, L, and superstrong gauge bosons, G, is gauge invariantly coupled to SM gauge
FIG. 1. The running of electroweak and strong couplings in the SM (black dotted lines) and in the MSSM. The visible displace-ment of the blue and red curves at small scales is associated with the opening of the supersymmetry threshold taken to be either 0.5 TeV (blue curve) or 1.5 TeV (red curve) with initial conditions αsðmZÞ ¼ 0.117 and αsðmZÞ ¼ 0.121, respectively.
bosons (B, W, A) and fermions (q;l). We have indicated with DXμ the covariant derivative with respect to the group transformations of which fXg are the associated gauge bosons. The most general expression of the covariant derivative is DBW AG μ ¼ ∂μ− iYgYBμ− igwτrWrμ− igs λa 2Aaμ− igT λα T 2Gαμ; ð6Þ where Y;τrðr ¼ 1; 2; 3Þ; λaða ¼ 1; 2; …; N2 c− 1Þ and λα
Tðα ¼ 1; 2; …; N2T− 1Þ are, respectively, the UYð1Þ
hypercharge and the generators of the SULð2Þ, SUðNc ¼ 3Þ, SUðNT ¼ 3Þ group with gY, gw, gs, gT
denoting the corresponding gauge couplings.1 We notice that Q’s are subjected to electroweak, strong and super-strong interactions, while L’s are subjected to electroweak and superstrong interactions only.
For the SULð2Þ SM matter doublets, we use the
notation qL¼ ðuL; dLÞT and lL¼ ðνL; eLÞT.
Right-handed components are SULð2Þ singlets and are denoted
by qu
R, qdR and luR, ldR. A similar notation is used for Q
and L fermions. Some of the formulas below will be given for the general case of ng SM families and νQ, νL
generations of SIPs.
The scalar field, Φ, is a 2 × 2 matrix with Φ ¼ ðϕ; −iτ2ϕÞ and ϕ an iso-doublet of complex scalar fields,
that feels UYð1Þ and SULð2Þ, but not SUðNc¼ 3Þ and SUðNT ¼ 3Þ, gauge interactions.
In order to follow the running of gauge couplings from the low to very large scale (≫ ΛT), one has to
specify the order of magnitude of the masses of the elementary degrees of freedom appearing in the above Lagrangian. SM particles are taken to have their phenomenological value, while for SIPs we only need to say that they have masses OðΛT ≫ ΛQCDÞ. As we
said, the two models we want to illustrate in this work differ by the value we decide to assign to the mass of the scalars. We develop the discussion by starting with the case in which the bare parameters μ20 and λ0 in (4)
are chosen so that the scalar mass lies at some very large value, of the order of the GUT scale or larger (Secs. II and IV). This state of affairs effectively corresponds to drop the scalar field contribution in the evaluation of the β functions. The motivation for considering this somewhat unusual situation is outlined in Sec. V where we recall the scenario for nonpertur-bative elementary particle mass generation advocated in Ref.[7] (see also Ref.[8]). In Sec.VI, we deal with the case in which the scalar field is taken to have some finite value like in the SM (hence below the TeV scale).
This situation corresponds to straightforwardly adding to the SM Lagrangian the contribution of the superstrongly interacting sector. As we shall see, a quite remarkable level of unification (especially in the first case) is obtained.
Naturally the models discussed in detail in this paper are not the only possible nonsupersymmetric ones that lead to unification of gauge couplings. Interesting exam-ples where this phenomenon occurs can be found in the recent works of Refs. [9]and [10,11]. In Ref.[9], a left-right symmetric model of weak interactions with the mass of WR of the order of 2 TeV is considered in which a
consistent choice of the right handed gauge coupling satisfying gR < gL at the TeV scale is made. Heavy
neutrinos are taken as Majorana particles undergoing an inverse seesaw mechanism, entailing a non-negligible lepton number violation. The authors of Refs. [10,11]
discuss a model where electro-weak symmetry breaking is driven by a technicolor dynamics with the minimal particle content required for walking coupling and satu-ration of global anomalies. The model features three additional Weyl fermions which are singlets under tech-nicolor interactions thereby providing the extra matter necessary for the (one-loop) unification of the SM gauge couplings and yielding also a possible candidate for weakly interacting dark matter.
II. ONE-LOOPβ FUNCTIONS WITH NO SCALARS
We start by giving the relevant formulas for the evalu-ation of the one-loop running of the four gauge couplings, gY, gw, gs, gTassociated to the UYð1Þ, SULð2Þ, SUðNc ¼ 3Þ
and SUðNT ¼ 3Þ gauge groups, respectively, ignoring the
contribution of scalars.
As we shall see, a special role in achieving uni-fication is played by the hypercharge assignment of SIPs. Indeed, as worked out in the Weinberg’s book[6], there exist two possible solutions to the anomaly cancellation equations as far as hypercharge assignment is concerned. Besides the standard assignment (that we recall in Table I) in which U(1) anomalies are cancelled between quarks and leptons, there is another solution in which anomalies are cancelled within quark and lepton sectors separately. They are reported in Table II where
TABLE I. Hypercharges of SM fermions.
q l yuL¼ 2 3−12¼16 yνL¼ 0 − 1 2¼ −12 yuR¼ 2 3− 0 ¼23 yνR¼ 0 ydL¼ − 1 3þ12¼16 yelL¼ −1 þ 1 2¼ −12 ydR¼ − 1 3− 0 ¼ −13 yelR¼ −1 − 0 ¼ −1 P y2q¼2236 P y2l¼32 1We use the notation g
wfor the SULð2Þ gauge coupling. This can be a little confusing as the latter is usually denoted by g in standard textbooks[6].
we display the only choice consistent with the assumption that right-handed particles are SULð2Þ
singlets and Q ¼ T3þ Y. In Table II we hence have jQQj ¼ jQLj ¼ 1=2.
A. One-loopβ functions of the BSMM in the absence of scalars With the standard definitions,
βxðgxÞ ¼ μ
dgx
dμ ; x¼ T; s; w; Y; ð7Þ and taking the assignment of TableII for the SIP hyper-charges, one gets
βBSMM T ¼ − 11 3 NT− 4 3ðNcνQþ νLÞ g3T ð4πÞ2; ð8Þ βBSMM s ¼ − 11 3 Nc− 4 3ðNTνQþ ngÞ g3s ð4πÞ2; ð9Þ βBSMM w ¼ − 2113 −13ngðNcþ 1Þ − 1 3NTðNcνQþ νLÞ g3w ð4πÞ2; ð10Þ βBSMM Y ¼ 2 3 22 36Ncþ 3 2 ngþ 1 2NTðNcνQþ νLÞ g3Y ð4πÞ2; ð11Þ
where for generality we have left unspecified the rank of the strong and superstrong gauge groups (Nc and NT), the number of SM families (ng) and the number of SIPs generations (νQ and νL).
If, instead, also for SIPs the standard hypercharge assignment is taken, only βY is modified and becomes
βBSMM Yst ¼ 2 3 22 36Ncþ 32 ng þ 1 2NT 22 18NcνQþ 3νL g3Y ð4πÞ2: ð12Þ B. One-loop SMβ function
For comparison we report the one-loopβ functions of the SM[12] that read2 βSM s ¼ − 11 3 Nc− 4 3ng g3s ð4πÞ2; ð13Þ βSM w ¼ − 2113 −13ngðNcþ 1Þ − 1 6 g3w ð4πÞ2; ð14Þ βSM Y ¼ 2 3 22 36Ncþ 32 ngþ 1 6 g3Y ð4πÞ2: ð15Þ
III. GUT NORMALIZATION
In order to check whether or not, on the basis of the running implied by the above equations, there is (an even approxi-mate) unification, one has to determine the normalization of the couplings that are supposed to unify by requiring that the generators of the UYð1Þ, SULð2Þ, SUðNc ¼ 3Þ and
SUðNT ¼ 3Þ groups are among the generators of the
alleg-edly existing simple compact unification group, GGUT.
A. BSMM
For the BSMM of Eq. (1) the GUT normalization condition reads Tr½ðgYYÞ2 ¼ Tr 1 2gwτ3 2 ¼ Tr 1 2gsλ3 2 ¼ Tr 1 2gTλ3T 2 ; ð16Þ
where the sum in the trace is extended over all the fermions building up the putative irreducible representation of the GUT group. In our convention, each Weyl component contributes with weight one. With the alternative hyper-charge assignment of TableII, one gets
TABLE II. Nonstandard hypercharge assignments.
Q L yUL¼ 1 2−12¼ 0 yNL ¼ 1 2−12¼ 0 yUR¼ 1 2− 0 ¼12 yNR ¼ 1 2− 0 ¼12 yDL¼ − 1 2þ12¼ 0 yLL¼ − 1 2þ12¼ 0 yDR¼ − 1 2− 0 ¼ −12 yLR¼ − 1 2− 0 ¼ −12 P y2Q¼12 Py2L¼12 2
To avoid any confusion, we want to explicitly stress that in the SM scalars do contribute with the factor 1=6 in the square parenthesis of Eqs.(14) and(15).
Tr½ðgYYÞ2 ¼ 2ng 1 2 2 þ ngð−1Þ2þ 2ngNc −1 6 2 þ ngNc 2 3 2 þ ngNc −1 3 2 þ NT 1 4þ 14 ðνLþ νQNcÞ g2Y ¼ ng 3 2þ Nc 22 36 þNT 2 ðνLþ νQNcÞ g2Y; Tr 1 2gwτ3 2 ¼ ng 1 2ðNcþ 1Þ þ NT 1 2ðνQNcþ νLÞ g2w; Tr 1 2gsλ3 2 ¼ 2ðngþ NTνQÞg2s; Tr 1 2gTλ3T 2 ¼ 2ðνLþ νQNcÞg2T: ð17Þ
Setting Nc¼ NT ¼ ng¼ 3 and νL¼ νQ¼ 1 in Eqs.(17), one concludes that, up to an (irrelevant) overall multiplicative
constant, the couplings that we need to consider in order to study unification are
g21≔4
3g2Y; g22≔ g2w; g23≔ g2s; g24≔23g2T: ð18Þ
For the BSMM with the standard hypercharge assignment of Table I, one finds instead
Tr½ðgYYstÞ2 ¼ 2ng 1 2 2 þ ngð−1Þ2þ 2Ncng −16 2 þ Ncng 2 3 2 þ Ncng −13 2 þ 2NTνL 1 2 2 þ νLNTð−1Þ2þ 2NcνQNT −1 6 2 þ NcνQNT 2 3 2 þ NcνQNT −1 3 2 g2Y; ð19Þ
so that with Nc ¼ NT ¼ ng ¼ 3 and νL¼ νQ ¼ 1 the set of couplings specified in(18) should be replaced by
g21≔5
3g2Y; g22≔ g2w; g23≔ g2s; g24≔
2
3g2T: ð20Þ
B. SM
The analogous normalization formulas for the SM gauge couplings unification (for Nc ¼ 3) read
Tr½ðgYYÞ2 ¼ Tr 1 2gwτ3 2 ¼ Tr 1 2gsλ3 2 ; ð21Þ with Tr½ðgYYÞ2 ¼ 2ng 1 2 2 þ ngð−1Þ2þ 6ng −1 6 2 þ 3ng 2 3 2 þ 3ng −1 3 2 g2Y ¼ 10 3 ngg2Y; Tr 1 2gwτ3 2 ¼ ð3ngþ ngÞ 1 2 2 þ 1 2 2 g2w ¼ 2ngg2w; Tr 1 2gsλ3 2 ¼ 4ng 1 2 2 þ 4ng −1 2 2 g2s ¼ 2ngg2s; ð22Þ
from which one gets
g21≔5
C. Results for Nc= 3, ng= 3, NT= 3, νQ= 1, νL= 1
At this point we need to put together the formulas for the one-loop beta functions [Eqs.(8)–(11),(12)and(13)–(15)] with the corresponding normalization (Eqs.(18),(20)and
(23), respectively) to get the RG equations
dgi
d logμ¼ βgi; i¼ 1; 2; 3; 4; ð24Þ
where theβgi appropriate for the various cases we want to
consider are given in the next subsections.
1. BSMM one-loopβ functions with GUT normalization
With the hypercharge assignments of Table IIone gets
βg1 ¼ 8 g31 ð4πÞ2; ð25Þ βg2 ¼ 46 g32 ð4πÞ2; ð26Þ βg3 ¼ −3 g33 ð4πÞ2; ð27Þ βg4 ¼ − 17 2 g34 ð4πÞ2: ð28Þ
It turns out that (in the case of not active scalars) the nonstandard and the standard SIPs hypercharge assign-ments yield the same one-loop value for β1.
2. SM one-loopβ functions with GUT normalization βg1 ¼ 41 10 g31 ð4πÞ2; ð29Þ βg2 ¼ − 19 6 g32 ð4πÞ2; ð30Þ βg3 ¼ −7 g33 ð4πÞ2: ð31Þ
One notices the change of sign of theβg2 coefficient in the
BSMM (1) with respect to the case of the SM [compare Eqs.(26) and(30)].
IV. UNIFICATION OF COUPLINGS
In Fig. 2, we plot the one-loop running of the UYð1Þ, SULð2Þ and SUðNc¼ 3Þ inverse square gauge coupling,
α−1
i ¼ 4π=g2i, in the BSMM of Eq. (1), ignoring scalars
and with the choice of the hypercharges reported in Table II (red curves), compared to the running in the SM (black curves). The input values of the inverse gauge couplings at low energy (i.e. at mZ∼ 91 GeV) for the SM have been fixed by taking the recent PDG data [13] α−1 1 ðmZÞ ¼ 59.01 0.02; ð32Þ α−1 2 ðmZÞ ¼ 29.57 0.02; ð33Þ α−1 3 ðmZÞ ¼ 8.45 0.05: ð34Þ
Because of the different normalization of the UYð1Þ
coupling [compare Eqs.(18)and(20)], theα−11 input values
0 10 20 30 40 50 60 70 80
1e+02 1e+04 1e+06 1e+08 1e+10 1e+12 1e+14 1e+16 1e+18 1e+20
α
-1
μ(GeV)
FIG. 2. The one-loop running of electroweak and strong couplings in the BSMM (red curves) and in the SM (black curves). The visible change of slope of red curves at“low” scales is associated with the opening of the superstrong threshold that we take to be 5 TeV. NONSUPERSYMMETRIC MODEL WITH UNIFICATION OF… PHYSICAL REVIEW D 93, 105030 (2016)
of the red (BSMM) and black (SM) curve are different. For consistency the BSMM input value of α−11 must be taken to be
α−1
1 ðmZÞ ¼ 73.76 0.02: ð35Þ
The key result of the present investigation is the observation that in our favourite BSMM model(1) where scalar are very massive and do not contribute to the RG evolution, unification occurs to a much better level than in the SM. It is worth noticing the non-negligible effect due to the opening of the superstrongly interacting degrees of freedom threshold, that for definiteness we have set at 5 TeV in all figures. We have checked, however, that the quality of unification is essentially insensitive to the precise value of the superstrong threshold if the latter is varied between 1 and 10 TeV.
To appreciate the quality of unification one may compare the BSMM running of Fig. 2 with that of the MSSM controlled by the one-loopβ functions[5].
βMSSM g1 ¼ 335 g31 ð4πÞ2; ð36Þ βMSSM g2 ¼ g32 ð4πÞ2; ð37Þ βMSSM g3 ¼ −3 g33 ð4πÞ2; ð38Þ
where now g21¼ 5=3g2Y. The comparison in shown in Fig.3. The blue curves refer to the one-loop running in the MSSM,
employing the input values specified in Eqs.(32)–(34). The supersymmetry and superstrong thresholds have been set at unequal values (namely ΛMSSM¼ 1 and ΛT ¼ 5 TeV,
respectively). Notice, in fact, that the curves representing α−1
3 only differ because different values for the opening of
the supersymmetry and superstrong thresholds have been taken, while in the case of α−12 also the one-loop coef-ficients of theβ function are different (Eqs.(37)and(26), respectively). The evolution ofα−11 in the two models differ because of the input values resulting from the different GUT normalization of the UYð1Þ generator and of the
unequal one-loop coefficients ofβg1.
In our opinion, the level of unification provided by the particle content of the Lagrangian (1) without active scalars compares extremely well with what one gets in the MSSM.
A. Observations
We conclude this section with a few observations.
1. Hypercharge assignments
The hypercharge assignment in Table II is the one for which Weinberg [6] writes that it “resembles nothing observed in nature.” The reason for taking it for the hypercharges of SIPs is that with the more standard choice of TableIone would not get unification of the UYð1Þ gauge
coupling as shown in Fig. 4 where we compare the evolutions entailed in our BSMM by the two types of hypercharge assignments. Clearly no unification can be achieved if the green curve, corresponding to the TableI
assignment, describes the one-loop running ofα−11 .
0 10 20 30 40 50 60 70 80
1e+02 1e+04 1e+06 1e+08 1e+10 1e+12 1e+14 1e+16 1e+18 1e+20
α
-1
μ(GeV)
FIG. 3. The one-loop running of electroweak and strong couplings in the BSMM (red curves) and in the MSSM (blue curves). The superstrong and supersymmetry thresholds have been set atΛT¼ 5 and ΛMSSM¼ 1 TeV, respectively.
It must be noticed that the hypercharge assignment of TableII yields Q and L elementary particles with electric charges e=2, hence superstrongly confined “hadrons” have electrical charge quantized in units of e=2.
2. Two-loopβ functions and threshold effects We have extended the calculations of all the β functions up to two loops [14]. We do not report the corresponding (rather cumbersome) formulas here because two-loop terms do not modify in any essential way the previous plots, hence the quality of unification of gauge couplings visible in Fig. 2. We recall that at two loops the RG equations become much more involved as the RG evolution of each coupling depends on all the others.
Since we do not know what the full UV completion of the fundamental theory(1)could be, and consistently with our decision of (momentarily) neglecting two-loop terms, we refrain from giving estimates of possible effects due to threshold opening of GUT degrees of freedom around the GUT scale.
In any case, two-loop corrections and threshold effects tend to be of comparable numerical magnitude and may affect the values of the inverse coupling at (around) the unification scale at the level of about one unit.
3. Unification with superstrong interactions A very indirect clue on the UV structure of the GUT theory can come from the interesting observation that unification of all the four gauge couplings [UYð1Þ,
SULð2Þ, SUðNc ¼ 3Þ and SUðNT ¼ 3Þ] can be achieved
if a certain number, NS, of purely SIPs are included in
the model (1) with a Lagrangian of the form PNS
h¼1ð¯ψhDGψhþ mh¯ψhψhÞ, where mh is an OðΛGUTÞ
mass scale.
The presence of NS extra particles with purely
super-strong vector interactions modifies the last formulas in Eqs.(17) and(18)that become
Tr 1 2gTλ3T 2 ¼ ½2ðνLþ νQNcÞ þ NSg2T; ð39Þ
and (setting Nc¼ NT ¼ ng¼ 3 as well as νQ ¼ νL¼ 1)
g24¼ 8 þ NS 12 g2T: ð40Þ
This implies a modification of βg4 in Eq. (28) that now reads βg4¼ − 17 3 12 8 þ NS g34 ð4πÞ2: ð41Þ
It is remarkable that (approximate) unification of all the four couplings can be achieved with reasonable values of NS(in the range between 4 and 6) and the natural choice for
the initial condition ofα4 α−1
4 ðμ ¼ 5 TeVÞ ¼ 1: ð42Þ
The situation is illustrated in Fig.5where the cases NS¼ 4
(blue line), NS¼ 5 (black line) and NS¼ 6 (green line) are
reported. 0 10 20 30 40 50 60 70 80
1e+02 1e+04 1e+06 1e+08 1e+10 1e+12 1e+14 1e+16 1e+18 1e+20
α
-1
μ(GeV)
FIG. 4. The one-loop running of electroweak and strong couplings in the BSMM with the hyperchage assignment of TableII(red curves) and with the hyperchage assignment of TableI(green curve).
V. ON THE MODEL OF EQ. (1)
The inspiration for the model (1) and the reason for ignoring scalars came from the work of Ref.[7](see also Ref.[8]for an earlier version of the investigation) where a nonperturbative origin for elementary particle masses, that does not rely on the Higgs mechanism, was proposed. The complete Lagrangian of the model of Ref. [7] (and its extension including electroweak interactions[15]) involves a scalar SU(2) field coupled to fermions via chiral breaking Yukawa and Wilson-like terms, the latter being“irrelevant” operators of dimension d¼ 6 appearing in the Lagrangian multiplied by two powers of the inverse UV cutoff. The structure of these terms is such that the whole Lagrangian (that is formally power-counting renormalizable) enjoys an SULð2Þ × UYð1Þ symmetry (under which all particles
transform), crucial to forbid power divergent mass con-tributions in perturbation theory.
The complete model Lagrangian is not invariant, how-ever, under chiral SULð2Þ × UYð1Þ transformations of
fermions and electroweak bosons only, but one can tune some parameters (the Yukawa coupling and the coefficients of the Wilson-like terms) to critical values at which the symmetry of the Lagrangian under these chiral trans-formations is enforced up to UV cutoff effects, thus providing a solution of the naturalness problem in the way advocated by’t Hooft[16]. In the Nambu-Goldstone phase of the model, while the scalar is pushed to a very high scale and decouples, elementary particle masses result from the nonperturbative spontaneous breaking of the restored chiral symmetry triggered by the (UV cutoff remnant of the) chiral symmetry breaking terms in the critical Lagrangian.
The masses of all the elementary particle (including electroweak bosons) turn out to be proportional to the renormalization group invariant (RGI) scale of the theory times powers of the coupling constant of the strongest interaction which the particle is subjected to. This means in particular that in order to get the right order of magnitude for the top mass the RGI scale of the whole theory must be much larger than ΛQCD. This is the reason why a new superstrongly interacting sector of particles (Q, L and gauge bosons), gauge invariantly coupled to SM matter (q and l) as in equations from (2) to (5), is postulated to exist at a few TeV scale.3 0 10 20 30 40 50 60 70 80
1e+02 1e+04 1e+06 1e+08 1e+10 1e+12 1e+14 1e+16 1e+18 1e+20
α
-1
μ(GeV)
FIG. 5. The one-loop running of electroweak strong and superstrong couplings in the BSMM with the hyperchage assignment of TableIIand NS¼ 4 (blue line) NS ¼ 5 (black line), NS¼ 6 (green line).
3We might suggestively call these new degrees of freedom “techniparticles” with an eye to the well-known technicolor models of Refs. [17,18]. We refrain from doing so as the framework underlying Eq.(1) is very different from standard technicolor. One difference is the absence of tree-level FCNC
[15]. Another is that, unlike what happens in standard technicolor where techniquarks are massless particles, the SIPs we have introduced have nonperturbatively generated masses of OðΛTÞ times factors of the superstrong coupling constant. In the model described by the Lagrangian(1) (with νQ¼ νL¼ 1), mesonlike confined states will have masses that we can estimate, on the basis of what happens in QCD, to be of the order of 2 or 3 times ΛT. This remark is important in the light of the existing bounds on the parameter S[19,20] that tend to rule out standard technicolor with more than one doublet of technifermions. This is not so for the particle content of our model because, as we have argued above, SIP confined states have masses definitely “larger” than in standard technicolor, a fact that substantially reduces their contributions to S.
VI. ONE-LOOP β FUNCTIONS WITH SCALARS
We now consider the situation where all the degrees of freedom displayed in the Lagrangian(1), including scalars, are active from the electroweak scale upward. The presence of scalars (with mass smaller than ΛT) induces some little modifications in the Eqs. (10),(11),(12) and(25),(26) that become
βBSMM w → − 2113 −13ngðNcþ 1Þ − 1 3NTðNcνQþ νLÞ − 1 6 g3w ð4πÞ2; ð43Þ βBSMM Y → 2 3 22 36Ncþ 32 ngþ 1 2NTðNcνQþ νLÞ þ 1 6 g3Y ð4πÞ2; ð44Þ βBSMM Yst → 2 3 22 36Ncþ 3 2 ngþ 22 36NcνQþ 3 2νL NT þ 1 6 g3Y ð4πÞ2 ð45Þ and βg1 → 65 8 g31 ð4πÞ2; ð46Þ βg2 → 5 6 g32 ð4πÞ2; ð47Þ respectively4.
In Fig. 6, we compare the level of unification of the gauge couplings of the model(1)with and without active scalars (recall we are using the nonstandard hypercharge assignment of Table II for the superstrongly interacting matter). Unification is very satisfactory in both cases, though numerically the case with no scalars looks impres-sively good.
VII. CONCLUSIONS
In this short paper, we have shown that it is possible to build nonsupersymmetric models [see, as an example, Eq. (1)] where the unification of couplings is realized to a level comparable to the one that is achieved in the MSSM and in any case much better that in the SM.
20 22 24 26 28 30 32 34 36
1e+16 1e+17 1e+18 1e+19 1e+20
α
-1
μ(GeV)
without scalar with scalar
FIG. 6. A blow-up of the crossing region of the one-loop running of electroweak strong couplings in the BSMM model(1)with the hypercharge assignment of TableII without scalars (red lines) and with scalars (green lines). For better visibility we have slightly displaced the two coinciding evolution lines ofα−13 .
4Notice that in the present situation, where scalar fields are active, the value of the one-loopβg1 coefficient, at variance with
what happens if scalars are inactive (see Sec.IIIC1), depends on whether one takes for the superstrongly interacting matter the hypercharge assignment of TableIIor TableI. In the latter case, one would getβY st
g1 →8110
g31
ð4πÞ2, instead of Eq.(46).
The salient feature of the BSMM described by the Lagrangian(1)is the presence of a sector of superstrongly interacting (gluon-, quark-, and lepton-like) particles with a ΛT ≫ ΛQCD RGI scale set in the few TeV region. SIPs are
endowed with a somewhat unusual hypercharge assign-ment implying that the confined states have the electric charge quantized in units of e=2. Neutral bound states of SIPs with nonzero fermion number may provide candidates for cold dark matter along the lines discussed for techni-color (see e.g. [21]).
The motivation for studying the model(1)stems from the work of Ref. [7], where it is conjectured that masses of elementary particles are generated by a nonperturbative
mechanism in turn triggered by the presence of an irrelevant d¼ 6 chiral symmetry breaking operator in the fundamental Lagrangian.
The structure of the complete basic model of Refs. [7,15], the reason why it is preferable to have in the physically interesting Nambu-Goldstone phase the scalar field endowed with a very large mass, and the existence of superstrongly interacting fermions at the TeV scale are dictated by two key conceptual and phenomenological requirements, namely a neat solution of the “naturalness” problem and the correct order of magnitude of the dynamically generated top quark mass.
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