aa r X i v : . [ h e p - ph ] A p r Preprint typeset in JHEP style - PAPER VERSION hep-th08scipp-2008/14
Gauge Coupling Unification in MSSM + 5Flavors
Jeff L. Jones
Department of Physics and SCIPPUniversity of California, Santa Cruz, CA 95064E-mail: [email protected]
Abstract:
We investigate gauge coupling unification at 2-loops for theories with 5extra vectorlike SU(5) fundamentals added to the MSSM. This is a borderline casewhere unification is only predicted in certain regions of parameter space. We establisha lower bound on the scale for the masses of the extra flavors, as a function of thesparticle masses. Models far outside of the bound do not predict unification at all (butmay be compatible with unification), and models outside but near the boundary cannotreliably claim to predict it with an accuracy comparable to the MSSM prediction.Models inside the boundary can work just as well as the MSSM.
Keywords:
Gauge Coupling Unification, RG Equations, Supersymmetry. ontents
1. Introduction 12. RG Equations With Extra Flavors 33. Threshold Effects 64. Methodology 85. Bounds 126. Conclusions 207. Acknowledgements 22A. Appendix: Miscellaneous Plots 22
1. Introduction
Low energy supersymmetry is a leading candidate for solving the hierarchy problem.But its enduring popularity among model builders is in part due to another appealingfeature which the simplest models of low energy supersymmetry possess: in particular,the Minimal Supersymmetric extension of the Standard Model (MSSM) predicts gaugecoupling unification. Or phrased in another way, the combination of the MSSM andthe assumption of grand unification together successfully predicted (roughly) the rightexperimentally measured value of α at the weak scale.Despite this encouraging success, the MSSM by itself leaves many questions unan-swered. It contains many arbitrary parameters which must be set to certain values forpurely phenomenological reasons. However, it has proven difficult to construct more– 1 –atural models of low energy SUSY breaking which still maintain coupling constantunification. It is therefore of interest to identify which of the models that extend theMSSM can claim to maintain its prediction for α at a comparable level of accuracy.This may help provide direction in the search for a fully plausible theory of supersym-metry breaking.There are two primary ways in which a prediction for coupling constant unificationcan fail. One problem is perturbativity. If the coupling constants become too largebefore the would-be unification scale is reached, the perturbative expansion breaksdown and the ability to reliably make predictions based on perturbation theory is lost.The other problem is accuracy–given all coupling constants are still perturbative, dothey meet at the same energy scale, or is there a mismatch? Equivalently, how welldo they predict the measured value of α , given the assumption of unification? Is theaccuracy better or worse than the MSSM prediction?We find that while the accuracy question depends sensitively on the details ofthe model, the answer to the perturbativity question is relatively robust and dependsmostly on a few general features of the model. The most important factor for pertur-bativity is the number of matter fields (with Standard Model gauge quantum numbers)added. The more matter which appears in the loops, the less asymptotically free thetheory becomes, causing the perturbative expansion to break down earlier as the gaugecoupling constants are run up into the UV.With 5 or more vectorlike SU(5) fundamentals added to the MSSM at the elec-troweak scale, the theory becomes strongly coupled before unification. However, if themasses of the extra flavors are heavy enough, this statement does not apply. For thecase of exactly 5 extra flavors, the mass scale of the extra flavors becomes importantfor determining perturbativity. This is the primary issue explored in this paper: in theborderline case of 5 extra flavors, what are the perturbativity bounds on the massesof those flavors? These bounds on fundamentals can also be translated into boundson matter transforming in higher representations. For example, each additional mattermultiplet transforming in a 10 representation contributes the same to the RG equationsas 3 additional 5 representations.While some authors may choose to ignore models with 5 extra flavors simply be-cause the perturbativity of the gauge couplings is questionable, there have been nocareful studies to our knowledge of when this attitude is justified and why. Our goal The reason only complete SU(5) multiplets are considered is because the accuracy of unificationis usually destroyed, even at 1-loop, if incomplete multiplets are present. There are some exceptionsto this ([5],[11]). – 2 –s to analyze the situation carefully and provide more justification for rejecting suchmodels, when appropriate, and to identify the regime of parameter space where per-turbativity is not a concern.The question of which terms to keep in an asymptotic series when its expansionparameter is becoming strong is a somewhat ambiguous one. We suggest a methodfor resolving the ambiguity and estimating the theoretical uncertainty involved in theprediction for α at the weak scale. Using this method, we derive lower bounds on themasses of the extra flavors.In Section 2, we review the relevant 1- and 2-loop renormalization group equationsfor the MSSM, and for the MSSM with extra vectorlike SU(5) fundamentals added. InSection 3, we discuss the issue of 1-loop threshold corrections, both at the electroweakscale and at the GUT scale, which compete with 2-loop effects of the RG equationsin determining the value of α at the electroweak scale. In Section 4, we describe ourmethodology for analyzing gauge coupling unification. In Section 5, we discuss theresulting perturbativity bounds on the scale of the extra flavors for various slices ofparameter space. The main result is a plot of a suggested lower bound on the extraflavors as a function of the scale of the superpartners assuming degenerate superpartnermasses and a non-extremal value for tan β . Several modifications to this curve are thendiscussed, including heirarchies in the masses, large and small tan β , and the additionof a coupling to an extra singlet field. Some of these plots are included in this section,while others are saved for the appendix. In Section 6, we draw general conclusions andmake some closing remarks.
2. RG Equations With Extra Flavors
At 2-loops, the RG equations for the gauge couplings are ([1],[2]):(4 π ) ddt g i = A i g i + g i (4 π ) h X j =1 B ij g j − X x = u,d,e C i,x Tr( Y † x Y x ) i (2.1)where g i are the Standard Model gauge couplings ( i = 1 , ,
3) with normalization basedon SU(5) generators, Y x are the Yukawa coupling constants Y u , Y d , and Y e (each a 3 × t is the log of the renormalization scale.– 3 –or a supersymmetric theory with N chiral matter superfields Φ a ( a = 1 . . . N ) inrepresentations R a,i of gauge group G i , the coefficients A i and B ij are given by [2]: A i = X a S ( R a,i ) − C ( G i ) (2.2) B ij = − C ( G i )] + 2 C ( G i ) X a S ( R a,i ) + 4 X a S ( R a,i ) C ( R a,j )] (2.3)where C ( R a,i ) is the quadratic Casimir operator for the representation of the superfieldΦ a in the gauge group G i , and S ( R a,i ) is the corresponding Dynkin index. For Yukawacouplings between the matter superfields Φ a of the general form Y abc Φ a Φ b Φ c , theterm in the 2-loop RG equations from the Yukawas is: − X a,b,c Y abc Y ∗ abc C ( R c,i ) /d ( G i ) (2.4)The coefficients C i,x of the MSSM Yukawa couplings Tr( Y † x Y x ) can then be extractedfrom this.Plugging in MSSM values for these Casimirs and summing everything up leads tothe following coefficients ([1],[2],[7],[12]): A i = − B ij =
25 24 (2.5) C i,x =
265 145 185 If n additional pairs of SU(5) 5 and 5 fundamentals are added, they become: A i = + n n − n B ij = + n + n + n + n
25 + 7 n + n n (2.6)– 4 – i,x =
265 145 185 These agree with the formulas in [7] for n= n .The Yukawa couplings are also renormalized. At 1-loop, the RG equations forgeneral supersymmetric Yukawa couplings are[2]:(4 π ) ddt Y abc = 12 Y abd Y ∗ dfg Y cfg − X i C ( R c,i ) Y abc g i + ( c ↔ a ) + ( c ↔ b ) (2.7)For the Yukawa couplings of the MSSM, this becomes[2]:(4 π ) ddt Y u = Y u h Y u Y † u ) + 3 Y † u Y u + Y † d Y d − g − g − g i (2.8)(4 π ) ddt Y d = Y d h Tr(3 Y d Y † d + Y e Y † e ) + 3 Y † d Y d + Y † u Y u − g − g − g i (2.9)(4 π ) ddt Y e = Y e h Tr(3 Y d Y † d + Y e Y † e ) + 3 Y † e Y e − g − g i (2.10)Because of the way in which the Yukawa couplings enter into the gauge couplingRG equations (they are always multiplied by at least a factor of g ), the 1-loop YukawaRG’s given above are sufficient for a 2-loop calculation of the evolution of the gaugecouplings. For the purposes of this paper, we will be treating most of the entries in theYukawa matrices as zero, only keeping the top and bottom Yukawa couplings. Y u ≈ y t Y d ≈ y b Y e ≈ Another possible term which can arise in supersymmetric theories with extra fun-damentals is a coupling λSM M where S is a singlet field (which can, for example, be– 5 –sed to help generate the µ term for the Higgs) and M i and M i are the n extra flavors( i = 1 . . . n ). In a gauge mediation context, M and M are the messengers which trans-mit SUSY breaking to the visible sector. If this term is present in the SU(5)-symmetrictheory at the GUT scale, then there will be two such terms at energies where SU(5)is broken to SU(3) × SU(2) × U(1), λ SM (2) a M (2) a and λ SM (3) b M (3) b ( a = 1 ,
2, and b = 1 . . . λ and λ are equal at the GUT scale, but evolveseparately in the broken phase below the GUT scale.The 1-loop RG equations for the evolution of λ and λ can be derived from equation2.7. The result is: (4 π ) ddt λ = 4 λ + 3 λ λ − g λ − g λ (2.11)(4 π ) ddt λ = 5 λ + 2 λ λ − g λ − g λ (2.12)The contribution of λ and λ to the RG equations for the gauge couplings can bederived from equation 2.4: (4 π ) β g = . . . − g λ − g λ (2.13)(4 π ) β g = . . . − g λ (2.14)(4 π ) β g = . . . − g λ (2.15)
3. Threshold Effects
Given the assumption of unification, the predicted value of α ( m Z ) can be written asa function of other weak-scale observables, by computing the values for α and α atthe weak scale, evolving them up to the scale where they cross, and then evolving α back down from there. To leading order in the coupling constants, this function canbe computed simply by integrating the 1-loop RG equations. However, at next order,the terms coming from integrating the 2-loop RG equations compete with terms arisingfrom 1-loop threshold effects, some from the weak scale and others from the GUT scale.Both of these effects are model dependent[10].There are two types of 1-loop weak scale thresholds. The first type are the leadinglogarithms, which arise from the RG equations themselves and the fact that the contri-bution from each mass comes in at a different scale. Including the leading logarithms– 6 –s an alternative to integrating a separate set of RG equations between every massscale in the theory. They allow one to include all MSSM particles in the RG equationsstarting at m Z , even though many of the MSSM particles are heavier than m Z . Thesetwo approaches differ at 2-loop, but because all of the MSSM particles are close on alogarithmic scale to m Z compared to the full integration range (from m Z to m GUT ),the resulting difference in the GUT couplings as a function of the weak-scale parame-ters is comparable to 3-loop effects. In addition to these logarithmic terms, there arealso “finite” 1-loop radiative corrections to sin ˆ θ W , which affect the determination ofˆ α ( m Z ) and ˆ α ( m Z ) from electroweak observables.The M S renormalization scheme does not preserve supersymmetry, so it is appro-priate for us to use the DR scheme instead[3]. All of the variables with hats in thissection represent DR parameters. The parameter sin ˆ θ W can be defined by the relation([4],[8]): cos ˆ θ W sin ˆ θ W = π ˆ α ( m Z ) √ m Z G F (1 − ∆ r )where G F = 1 . × − GeV − [9] is the Fermi constant measured from muondecay, ∆ r is a sum of a long list of radiative corrections, and ˆ α ( m Z ) is the value of theelectromagnetic coupling at the weak scale which is computed from:ˆ α ( m Z ) = α EM − ∆ ˆ αα EM = 1 / . α represents the radiative corrections to ˆ α . Because the value of ˆ α is used to compute sin ˆ θ W , ∆ ˆ α also affects the value of sin ˆ θ W , indirectly.At 1-loop, the radiative corrections to ˆ α and ˆ α in DR are given by[4]:∆ ˆ α = 0 . ± . − α EM π n − (cid:16) m W m Z (cid:17) + 169 ln (cid:16) m t m Z (cid:17) + 13 ln (cid:16) m H + m Z (cid:17) + 49 X i =1 ln (cid:16) m ˜ u i m Z (cid:17) + 19 X i =1 ln (cid:16) m ˜ d i m Z (cid:17) + 13 X i =1 ln (cid:16) m ˜ e i m Z (cid:17) + 43 X i =1 ln (cid:16) m ˜ χ + i m Z (cid:17)o ∆ ˆ α = α ( m Z )2 π n −
23 ln (cid:16) m t m Z (cid:17) − (cid:16) m ˜ g m Z (cid:17) − X i =1 ln (cid:16) m ˜ q i m Z (cid:17)o – 7 – α ( m Z ) = α ( m Z )1 − ∆ ˆ α (3.1)where α ( m Z ) on the righthand side of the equations represents the accepted M S valueof the strong coupling constant at the weak scale.The supersymmetric corrections to ˆ α ( m Z ) and ˆ α ( m Z ) depend in a simple way ononly two masslike parameters, each a particular weighted geometric mean of differentsuperpartner masses: M α = h m / H + Y i =1 m / u i Y i =1 m / d i Y i =1 m / e i Y i =1 m / χ + i i / (3.2) M α = h m g Y i =1 m / q i i / (3.3)The calculation of the sin ˆ θ W corrections, by comparison, is far more complicated, in-volving pages of finite corrections rather than just logarithmic corrections, and dependson a large number of parameters. (These expressions are hidden in the parameter ∆ r above, and can be found in reference [4].) It is therefore useful to consider sin ˆ θ W ,tan β , M α , and M α as the inputs, rather than the entire MSSM parameter space.In addition, to obtain even approximate gauge coupling unification, one must requirea correlation between the two SUSY mass parameters, the scale of the extra flavors,and sin ˆ θ W . This can be used to eliminate sin ˆ θ W as an input, making it possible torepresent the perturbativity bounds visually with a small number of plots.
4. Methodology
The 2-loop gauge coupling RG equations, coupled to the 1-loop Yukawa RG equations,were integrated numerically from the weak scale (defined as m Z ) up to the unificationscale. The unification scale, for our purposes, is defined as the scale M GUT at which α ( M GUT ) = α ( M GUT ). The boundary conditions imposed at the weak scale arethe weak-scale DR parameters, determined in terms of physical observables as well as M α , M α , tan β , and sin ˆ θ W which represent the MSSM parameters. A fifth input, theenergy scale of the extra 5 flavors, is not involved in the weak-scale boundary conditionsbut is important in determining the evolution of the coupling constants.One measure of how well the couplings unify is how close α ( M GUT ) is to the value ofthe other two gauge couplings at the unification scale. On some plots in this paper, this– 8 –iscrepancy is reported as a percentage difference. This way of measuring the accuracyof unification is most useful when considering what unknown GUT thresholds would benecessary in order to obtain perfect unification. If the required GUT thresholds are toolarge (more than a few percent), then the theory looks implausible from the standpointof unification.Another measure for how well the couplings unify is to use the unification scaledetermined by α and α to predict the value of α at the weak scale, comparing itto the accepted α ( m Z ). This way of measuring the discrepancy is more useful forassessing whether particular models can claim to have made a successful prediction (orat least a retrodiction).Our method was to evolve the couplings up in energy from m Z using the MSSMRG equations (2.1,2.5), until reaching the energy scale of the 5 extra flavors. At thatscale, the RG equations including 5 additional flavors (2.1,2.6 for n = 5) were used tocontinue evolving the couplings into the UV. Both 1-loop and 2-loop expressions forthe running coupling constants were evolved simultaneously in this manor. Dependingon the input parameters, several different outcomes are possible during the numericalevolution.One possibility is that even in the 1-loop approximation, one or more of the cou-plings becomes strong before unification ( α = α ) is reached. Obviously, in this case,there is no prediction for unification. Another possibility is that the 1-loop couplingsremain weak and unify, but one or more of the 2-loop couplings becomes strong (mean-ing α i > α i = 1, the 2-loop contribution to the beta function begins to dominate the1-loop contribution. (As mentioned by [7] and confirmed in our results, a typical placefor this to happen is around α i = 0 . α about α = 0. Only a finite number of terms n in the expansion convergeon the true non-perturbative value of the function–while the rest of the terms beginningwith n + 1 diverge (as more terms are kept, they get further and further from the rightanswer, rather than closer to it). As α becomes larger, the number of convergent terms n decreases. Although it depends on the function being expanded and there are known– 9 –athematical exceptions, a general rule of thumb for determining how many terms n to keep is to look for the place where the ( n + 1) th term is larger than the n th term.When the full non-perturbative function is unknown, this rule of thumb is the onlyguide we have for knowing how many terms in the series to keep. Hence if the secondterm in the asymptotic expansion for the beta function is larger than the first term,it is unreasonable to expect that the second term adds any additional information tothe first. It is possible in some cases that it makes the approximation better, butgenerally it would make the approximation worse. For example, in [13] a case of thecoupling constant becoming strong is presented where the beta function is argued tobe Pade-Borel summable–under such an assumption the full beta function is computedand found to be closer to the 1-loop approximation than the 2-loop approximation.Keeping this rule of thumb for asymptotic series in mind, more and more of theterms in the series become useless as the coupling constants become stronger. The fewerthe terms in the series we keep, the more theoretical uncertainty in the beta functionand any predictions extracted from it. After all of the terms have become unreliable,the coupling constant finally becomes “strong,” and the entire perturbation series hasbroken down. Looking for a formal Landau pole in the RG equations overestimatesthe scale at which the series breaks down, so instead we have looked at which terms inthe beta function are dominant and used that to estimate the theoretical uncertaintyinvolved.As will be discussed in Section 5, the MSSM 2-loop prediction for α ( m Z ) (whichdepends on the masses of the superpartners) is generally worse than the 1-loop predic-tion. So it would appear that only 1-loop perturbativity is required in order to com-pete with the “success” of the MSSM prediction. However, we argue that a strongerrequirement–that the expansion at 2-loops is still reliable in the sense discussed above–isa more appropriate (though admittedly still somewhat arbitrary) condition for deter-mining whether a model can compete with the MSSM in claiming to predict α ( m Z ).This was not an assumption going in to this project, but a conclusion that emergedafter running numerical simulations of the evolution of the coupling constants for manydifferent input parameters and comparing their fates.A key issue here is estimating how reliable a 1-loop calculation is in the case wherethe 2-loop expansion breaks down before unification. Already above, we have dividedthe parameter space into 3 regions: one where the 2-loop expansion holds all the wayto unification, another where the 1-loop holds but the 2-loop breaks down, and a thirdwhere the coupling becomes fully non-perturbative before unification and even the 1-loop calculation is meaningless. Now we divide the second of these 3 regions into two– 10 –ubregions, by identifying a window where the 2-loop calculation fails near enough theunification scale that there is a way to estimate roughly the theoretical uncertaintyinvolved in the 1-loop calculation.Our method for estimating the theoretical uncertainty in the 1-loop computationgoes as follows. First, run the couplings up until the 2-loop term in the beta function forone of them (typically α ) becomes larger than the corresponding 1-loop term. Abovethis point, we presume that the 1-loop approximation to that beta function is a betterapproximation than the 2-loop approximation (following the rule of thumb describedabove). After that, continue to run the couplings up into the UV, but using the 1-loopbeta function for the coupling which has become too strong for the 2-loop to makesense. In the event that the same thing happens to another coupling, switch over tothe 1-loop approximation for that beta function as well. Assuming that unification isreached before any of the couplings become fully non-perturbative, use the value of α ( M GUT ) = α ( M GUT ) computed in this way to run all of the couplings back down tothe weak scale. On the way down, the same procedure is used where either the 1-loopor the 2-loop beta function for each coupling constant is used, depending on which ismore appropriate. Then compare this hybrid method for computing α ( m Z ), whichcould perhaps be described as a “1.5-loop calculation”, to the purely 1-loop calculationof α ( m Z ) using the same input parameters. The theoretical uncertainty in the 1-loopresult is then estimated to be the difference between these two estimates for α ( m Z ).Another way to view the method of 1.5-loop calculation described above is as thecontinuum limit of a function which depends on the sum of a large discrete number ofterms, each involving a different coupling, where some of the couplings are too strongto include their 2-loop contribution and others are weak enough to include their 2-loopcontribution. However, this “sum” is a bit exotic in that each term depends on thelast, and the renormalization scale is different for each term. Since the running iscomputed numerically by discretizing the integral, this is not only an analogy, but alsoa description of how the calculation was performed.Our method has some arbitrariness, but it is only employed to get a rough ideaof how trustworthy the 1-loop calculation is, for the purpose of deciding whether 1-loop perturbativity is enough to say that a given model predicts unification. Based onthis method, it appears that 1-loop perturbativity is not enough because of the largetheoretical uncertainty when the coupling constant is becoming strong.– 11 – . Bounds The value of α ( m Z ) has been measured by a number of different experiments. Thecurrent global average is reported as 0 . ± .
002 by the particle data group [9]. Aglobal fit of all Standard Model parameters yields α ( m Z ) = 0 . ± . . σ , from the average taking into account all ex-periments (including lattice QCD). The disagreement could be statistical in nature, orit could be that adding physics beyond the Standard Model will reconcile these twovalues. Non-supersymmetric GUT theories predict α ( m Z ) = 0 . ± . α ( m Z ) = 0 . ± . . . ± . α to within ∼ α ( m Z ) is 0 . .
2% larger than the experimental value.However, since the 2-loop RG equations and the 1-loop thresholds make the predictionworse, the seemingly impressive closeness of the 1-loop prediction to the experimentalvalue can only be regarded as a coincidence.Figures 1 and 2 show two examples of the procedure described in Section 4: thecoupling constants are evolved using both 1- and 2-loop expansions simultaneously,and the 2-loop evolution switches over to 1-loop once the first and second terms inthe corresponding beta function become of equal magnitude. The black lines representthe 1-loop evolution, while the blue lines represent the “1.5-loop” evolution (2-loopor 1-loop, whichever is more appropriate as the energy scale changes). In both plots,there is a kink in the evolution at the scale where the 5 extra flavors appear. Thereare also kinks where the 2-loop evolution switches to 1-loop for α and α . In Figure 1,the extra matter is at 2800 TeV, and unification at 2-loops happens just before the2-loop expansion is lost. In Figure 2, the extra matter is at a lower scale of 500TeV, and the 2-loop becomes bad before unification happens. Figure 2 is an exampleof the window in parameter space nearby where 2-loop perturbativity fails where thetheoretical uncertainty in the 1-loop calculation can be roughly estimated. As is typical,the accuracy of the 1.5-loop unification is poor and when the couplings are run backdown to the weak scale, there is a fairly large uncertainty in the prediction for α ( m Z ).For this case, the 1-loop prediction for α is 0.1136 while the 1.5-loop prediction for α is 0.1394, for a theoretical uncertainty of about 23% (one is too large and the other istoo small). This uncertainty typically becomes worse as the energy scale for the extra– 12 –avors is lowered further, becoming incalculable once the 1.5-loop calculation becomesnon-perturbative before unification. As is evident from the plot, the uncertainty inthe strength of the unified coupling at the GUT scale is even worse, since the 1- and1.5-loop crossing points are far apart from each other. Μ H GeV L Α i - Figure 1: ˆ θ W = 0 . The value of sin ˆ θ W ( m Z ) used here was chosen for these plots simply because itprovides good unification when the threshold for the extra matter is high enough. Theactual value depends on all of the SUSY masses and could take on many different valuesfor fixed values of M α and M α (which are set equal in these plots). While the StandardModel M S value of this parameter is known[9] to be 0 . ± . DR valuein the MSSM can range all the way from 0.230 to 0.238 depending on the details of theSUSY spectrum[6].After experimenting with different input parameters, it appears that the most im-portant boundary in parameter space is between the region where 2-loop perturbativity– 13 –
000 10 Μ H GeV L Α i - Figure 2: ˆ θ W = 0 . holds all the way to unification, and where the 2-loop expansion in at least one of thecouplings breaks down before unification. Figure 3 shows a plot of this boundaryfor a fixed value of sin ˆ θ W = 0 . M SUSY = M α = M α . This was computed by scanning over the corresponding slice ofparameter space and running the couplings from weak to GUT scale many times foreach point on the plot. However, this plot is not the most useful because the accuracyof unification is sensitive to sin ˆ θ W , and different values of sin ˆ θ W are required fordifferent values of M SUSY . Figure 4 shows a plot of the percentage difference betweenthe coupling constants at the GUT scale (cid:16) α ( M GUT ) − α ( M GUT ) α ( M GUT ) × (cid:17) . Equivalently,this is an estimate of the magnitude of the threshold corrections required in the GUTtheory to attain perfect unification. For this value of sin ˆ θ W , the scale of the SUSYpartners must be somewhere around 220 GeV in order to avoid requiring large GUTthreshold corrections. – 14 –
00 200 300 400 500 600 SUSY mass scale @ GeV D @ TeV D Figure 3:
A plot of the boundary below which 2-loop perturbativity is lost before unificationoccurs. sin θ = .
100 200 300 400 500 600 SUSY mass scale @ GeV D - - @ % D Figure 4:
A plot of the magnitude of the GUT threshold correction required in order tomake the gauge couplings unify, for sin ˆ θ W = 0.234 where the 2-loop expansion is barelyperturbative at unification. To compensate for the problem of a fixed value of sin ˆ θ W only allowing a narrowrange of realistic values of M SUSY by realistic accuracy constraints, we have fed thiscurve back into another algorithm which finds the most appropriate value of sin ˆ θ W – 15 –o use for each point on the boundary. This is done by scanning over different valuesof sin ˆ θ W and finding the minimum unification error, rerunning the boundary-findingalgorithm with the new array of sin ˆ θ W values, and iterating this procedure until thenew boundary no longer moves appreciably. The result is the far more useful plotshown in Figure 5.Figure 5 is the central result of this paper. It is a plot of the boundary where2-loop perturbativity is lost, but where sin ˆ θ W is tuned at each point to a value whichmaximizes the ability of the couplings to unify well. With a few exceptions to be soondiscussed, any model with 5 extra fundamentals whose masses fall below this boundarymust either sacrifice accuracy or perturbativity (predictivity), regardless of what thevalue of sin ˆ θ W is. The exceptions to this are if the two SUSY scales set equal hereare split, or if there is a Yukawa coupling which is very near its own perturbativitylimit. These exceptions can shift the bound somewhat, but not by much. Notice thatthe lower bound in Figure 5 is raised somewhat relative to the plot in Figure 3 for alow SUSY scale, and relaxed somewhat for a higher SUSY scale. None of the boundarypoints differ by more than a factor of 2. The range of sin ˆ θ W values required to generatethis plot was from 0.2323-0.2356.
100 200 300 400 500 600 SUSY mass scale @ GeV D @ TeV D Figure 5:
A plot of the boundary below which 2-loop perturbativity is lost before unificationoccurs, with sin ˆ θ W tuned at each point to minimize unification error. Linear-linear scale.Error bars indicate vertical scanning resolution in numerical algorithm (2%). On a log-log scale, the same plotted points can be closely approximated by astraight line (Figure 6). The empirical formula for the mass relationship at the bound-– 16 – @ GeV D @ TeV D Figure 6:
A plot of the boundary below which 2-loop perturbativity is lost before unificationoccurs, with sin ˆ θ W tuned at each point to minimize unification error. Log-log scale. ary extracted from this is M . M − . SUSY ≥ . M α and M α , were taken tobe equal. However, there is no reason to assume there could not be a separation betweenthem. M α is a weighted average of SUSY partners with electromagnetic charge, while M α is a weighted average of SUSY partners with color charge. If the squarks andgluinos are much heavier than the sleptons and charginos, then M α and M α would beseparated. In gauge mediation, there is a natural heirarchy between the color chargedand color neutral sparticles. However, because the squarks are included in both of theaverages, the separation between M α and M α will not be as much as the heirarchybetween the color charged and color neutral partner masses themselves. For example,the typical heirarchy of between 3:1 and 6:1 in gauge mediation would only lead to aratio of M α /M α ≈ M α . By comparison, the boundis relatively insensitive to changes in M α , but it has a weak effect which goes in the opposite direction as M α . The result is that the bound is relaxed somewhat comparedto M α = M α if the ratio M α /M α is large. The case which allows the lowest scale for– 17 –he extra flavors then, is when M α is as high as possible and M α is as low as possible.Fig 7 shows a plot of the perturbativity bound as a function of M α /M α for fixed M α = 800 GeV. The required value of sin ˆ θ W for this range of parameter space is0 . − . M α /M α = 2 point on the plot ( M ≥
820 TeV). H M Α (cid:144) M Α L @ TeV D Figure 7:
A plot of the perturbativity boundary for M α = 800 GeV, M α /M α = 1 −
8, andsin ˆ θ W tuned to minimize unification error. In order to achieve M α = 800 GeV and M α = 100 GeV (the value 8 on theindependent axis of Fig 7), one would need the gluinos to be at 6.4 TeV and for therest of the sparticles including the squarks to all be at 100 GeV. Raising the squarksup closer to the gluinos would raise both M α and M α . The ratio between the scaleswould then be less than 8, however the overall bound would be slightly more relaxed.In principle, it is possible to lower the lower bound on the extra 5 flavors all the waydown to about 300 TeV, but only at the expense of requiring a large heirarchy andgluinos several hundreds of times heavier than the lightest supersymmetric partners.This is much larger than could be attained naturally in gauge mediation (or any otherSUSY breaking scenario we are aware of).One may wonder why we have assumed the 5 extra flavors are mass degenerate,and if anything changes if they are non-degenerate. If they are non-degenerate thenthe mass scale used here can be taken to be the geometric mean of all of the masses.Because all of the extra flavors contribute the same type of term to the RG equations(each with a different mass), the terms can be combined in the same way we have– 18 –ombined the masses of the SUSY particles into two composite mass parameters. Thisshould hold at 2-loops as long as the masses are roughly near the same scale (withinan order of magntiude or so). The reason for this is the same reason that only 1-loopthreshold corrections are required for a 2-loop unification analysis: the distance on alogarithmic scale over which the RG equations differ is small compared to the entirerange over which the couplings are run, and higher order corrections would matter onlyat higher loops. So the answer here is that our analysis is valid as long as all 5 flavorsare roughly on the same order of magntiude; but the situation may be different if thereis a huge hierarchy in their masses. We have nothing more to say about such models–itwould be simple to extend our analysis to any single one, but it does not seem worthcategorizing or testing all of the many possibilites at present.In all of the above, we have set tan β = 10. However, the curves have been recom-puted for other values of tan β , and it was found that the resulting plots do not changenoticeably at all unless tan β is close to its own perturbativity limit. So the above plotscan be taken to represent a wide range of values for tan β , leaving out only a tiny bitof parameter space near the small tan β limit.For large tan β there is a minimum value for the SUSY scale, in order for thebottom Yukawa coupling to stay perturbative until the unification scale. If the SUSYscale is too low, then regardless of the masses of the extra flavors, the 2-loop calculationis lost before unification. If the SUSY scale is high enough, then the lower bound onthe mass scale for the extra flavors roughly matches the one for tan β = 10. The curveis very slightly lowered, but the difference is hardly noticeable on the plot and onlyslightly greater than the uncertainty inherent in the algorithm used.A similar thing happens for small tan β (tan β ∼ β result. (The difference is because the top Yukawa has a larger effecton the gauge couplings through its larger coefficient in the RG equations.) The boundis still within a factor of 2 of the non-extremal tan β bound, probably not significantenough of a difference for model builders to worry about.Some plots of the extreme tan( β ) bounds are included in the Appendix (Fig 9,Fig 10).The quoted values for tan β in the captions should not be taken too seriously, becausethe 2-loop Yukawa RG equations were not used, nor were the 1-loop Yukawa thresh-olds considered. This leaves the value of tan( β ) that corresponds to particular valuesof the DR Yukawa couplings approximate, however this uncertainty does not extendto their effect on the gauge couplings. For our purposes (analyzing the perturbativityof the gauge couplings when the Yukawa couplings take extreme values), this should– 19 –ot matter.The final modification we considered was the addition of a coupling between theextra fundamentals and an extra singlet field. The single coupling λ at the GUT scaleruns as two separate couplings λ (2) and λ (3) below the GUT scale. If they are unifiedat the GUT scale, the typical result is that λ (3) is greater than λ (2) after they are rundown to the weak scale. For most values λ could take at the GUT scale, there is nodiscernable effect on the runnings of the gauge couplings. But as with the top andbottom Yukawa couplings, the exception is when λ is right near its own perturbativitylimit ( λ = √ π ). Fig 11 in the appendix shows an example of this case and whathappens when it is run down to the weak scale.The result is that if λ is near its perturbativity limit, then it can relax the boundin Fig 5 by just barely more than a factor of 2. Its effect can be larger than that ofthe top or bottom Yukawa couplings, because the coefficient in front of its term in thegauge coupling RG equations is larger. In the appendix, Fig 12 and Fig 13 show twoexamples of how the 2-loop perturbativity boundary for the gauge couplings is shiftedfor fixed large values of λ (2) and λ (3) at the weak scale. If they are too large, thenthere is an upper limit on the SUSY scale otherwise one or both of them will becomenon-perturbative before unification. Ideally, it would be better to have plots based ona fixed value of λ at the GUT scale rather than the weak scale. The running dependson the SUSY scale as well as the scale for the extra flavors, and for much of the rangeplotted the two couplings do not actually unify (although they were chosen to comeclose for the middle of the range). To do the analysis for fixed GUT scale couplingwould be more complicated and could be a direction for future improvements, howeverour opinion is that it is unlikely to change the results significantly.
6. Conclusions
Dealing with asymptotic expansions near the limit where they are becoming non-perturbative is a murky business, and it is difficult to make precise or absolute claims.Nevertheless, we hope we have convincingly demonstrated in this paper using a fairlyreasonable methodology that models including 5 extra flavors below a certain scale inthe neighborhood of a thousand TeV, depending on the masses of the superpartners,cannot reliably claim to make predictions of gauge coupling unification on par with theMSSM. We have provided a fairly robust lower bound as a guideline for model buildersto stay above, if they wish to do as well as the MSSM on this front.– 20 –odels above the bound can do just as well as the MSSM in predicting unifica-tion. However, whether they actually predict unification (that is, whether the couplingconstants meet each other or miss each other) depends on the details of the model–for instance, the wrong distribution of SUSY partners may lead to a value of sin ( θ W )which predicts that the couplings do not unify. From the point of view of gauge couplingunification, models with 5 extra flavors which predict unification should be consideredjust as seriously as candidates with fewer flavors. But because the scale must be sepa-rated by at least a few orders of magnitude from the electroweak scale, the fine-tuningproblems in these models will likely be more severe, and the prospects for detectablesignatures at LHC worse. Models below the bound can still be considered as candidatesto solve other problems in high energy physics, but if they are to be believed then thesuccessful MSSM prediction for α must be seen as a coincidence.One example of a valid UV-completion of the Standard Model involving 5 or moreextra flavors below the bound would be a duality cascade, where the couplings becomestrong at some energy scale and above that scale there is a dual description of the theorywhich is weakly coupled. However, it is hard to know how to relate the couplings ofthe high energy theory to the couplings of the low energy theory. Unless this challengeis overcome, these models cannot claim to predict unification either.If the masses of the gluinos (and optionally, the squarks) were a couple orders ofmagnitude above the masses of the other sparticles, then the separation of scales couldallow the 5 extra flavors to be lowered to a few hundred GeV, however the requiredheirarchy would be much larger than gauge mediation or other natural models of SUSYbreaking would predict, and some of the motivation for low energy supersymmetry (ie,solving the heirarchy problem) would be undone.Small tan β (and to a lesser extent large tan β ) can relax the perturbativity boundsslightly, but do not significantly change the other conclusions. To say something moreprecise about particular values of tan β near 1, a more detailed analysis including 2-loopYukawa coupling RGE’s is needed. Adding a coupling between the extra fundamentalsand an extra singlet field has no effect unless the coupling is near its own perturbativitybound, in which case the effect is similar to the effect of the top Yukawa (and slightlylarger). If both of these effects were combined in just the right way, along with aheirarchy between color charged and color neutral objects, it is reasonable to believethat the bound in this paper could be relaxed even more. However, to get such a modelwhere all of these things line up favorably might be challenging, and even if it wereaccomplished it does not appear that the bound could be relaxed by more than anorder of magnitude (still at least 3 orders of magnitude above the Z mass).– 21 – . Acknowledgements The author would like to acknowledge useful conversations on this topic with MichaelDine, Thomas Banks, Howard Haber, Guido Festuccia, and David E. Kaplan. Theauthor would especially like to thank John Mason, who was involved in the early stagesof this project, for many useful discussions, suggestions, and help.
A. Appendix: Miscellaneous Plots
This appendix contains some plots involving large and small tan β and a large λ couplingin front of a SM M term. Log Μ Y t , b Y b Y t Figure 8:
A sample plot of the running of the top and bottom Yukawa couplings for smalltan β , where the top Yukawa reaches its perturbativity limit of Y t = √ π just before unifica-tion. This tends to happen for small tan β if the SUSY scale is too low or if the extra flavorsare too high. The value of tan β at the electroweak scale used here is 1.35, however the precisevalue of tan β corresponding to this plot would be modified because the 1-loop correctionsto the top and bottom masses and Yukawa couplings at the weak scale were not included inour analysis. This subtlety should not affect our conclusions about the perturbativity of thegauge couplings themselves. – 22 –
00 200 300 400 500 600 SUSY mass scale @ GeV D @ TeV D Figure 9:
A plot of the boundary below which 2-loop perturbativity is lost before unificationoccurs, with sin ˆ θ W tuned at each point to minimize unification error, and tan β = 48. If theSUSY scale is below about 150 GeV, then the Yukawa couplings become non-perturbative,regardless of where the extra flavors are. Above 150 GeV, the curve is not appreciably differentfrom the tan β = 10 plot in Figure 5.
100 200 300 400 500 600 SUSY mass scale @ GeV D @ TeV D Figure 10:
A plot of the boundary below which 2-loop perturbativity is lost before unificationoccurs, with sin ˆ θ W tuned at each point to minimize unification error, and tan β = 1 .
4. If theSUSY scale is below about 130 GeV, then the top Yukawa coupling becomes non-perturbative,regardless of where the extra flavors are. Above 130 GeV, the shape of the curve is modifiednoticeably from the tan β = 10 plot in Figure 5, and overall the boundary is a bit lower. – 23 – Μ H GeV L Λ H L Λ H L Λ H L Figure 11:
A sample plot of the running of λ (2) and λ (3) , where they unify at the GUTscale right at their perturbativity limit ( λ = λ (2) = λ (3) = √ π ). Since these are couplingsbetween the extra fundamentals and an extra singlet, they run only above the scale wherethe extra flavors enter. Their values at the weak scale are λ (2) = 0 .
638 and λ (3) = 1 . – 24 –
100 200 300 400 500 600 SUSY mass scale @ GeV D @ TeV D Figure 12:
A plot of the boundary below which 2-loop perturbativity is lost before unifi-cation occurs, with sin ˆ θ W tuned at each point to minimize unification error, and boundaryconditions at the weak scale of λ (2) = 0 . λ (3) = 1 . λ couplings are about as strongas they can get without becoming non-perturbative for any of the plot range. The shape ofthe curve is similar to when the λ coupling is turned off, but shifted downward from the plotin Figure 5 by roughly a factor of 2. The rightmost point represents the lowest possible scalefor the extra flavors of any of the scenarios run. – 25 –
100 200 300 400 500 600 SUSY mass scale @ GeV D @ TeV D Figure 13:
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