The Strong Coupling Constant in Grand Uni ed Theories Damien M. Pierce Stanford Linear Accelerator Center Stanford University Stanford, California 94309 ABSTRACT The prediction for the strong coupling constant in grand uni ed the- ories is reviewed, rst in the Standard Mo del, then in the sup ersym- metric version. Various corrections are considered. The predictions in b oth sup ergravity-induced and gauge-mediated sup ersymmetry break- ing mo dels are discussed. In the region of parameter space without > large ne tuning, the strong coupling is predicted to b e M 0:13. s Z Imp osing M = 0:118, we require a uni cation scale threshold s Z correction of typically 2, which is accommo dated by some GUT mo dels but in con ict with others. Work supp orted by Department of Energy Contract DE{AC03{76SF00515. 1 bf Intro duction In the Standard Mo del, given values of the three gauge couplings, the Yukawa couplings, the Higgs b oson self-coupling, and a dimensionful observable e.g., the muon lifetime, a gauge b oson mass, or a quark mass, any other observable can then b e computed. In a grand-uni ed theory GUT, one need only input two of the gauge couplings, then the third one can b e predicted, as well as other observables. Thus, GUT theories are more predictive. Because the SU2 and U1 cou- plings are measured quite precisely,we will examine the prediction of the strong coupling constant in grand-uni ed theories, b oth in the Standard Mo del and in sup ersymmetric mo dels. We start with the one-lo op results and then pro ceed to discuss the next higher-order corrections. The inclusion of these corrections leads to a precise prediction of the strong coupling constant. 2 bf Renormalization Group Equations We are interested in measuring gauge couplings at the weak-scale or b elow, and running the gauge couplings up to higher scales. The solution of the renormaliza- tion group equations RGE's accurately describ es the evolution of the couplings, even as they are evolved over many decades. At one-lo op, the renormalization group equations for the gauge couplings are b Q dg i i 3 = g ; t =ln ; 1 i 2 dt 16 Q 0 where the b are the one-lo op b eta-constants. These constants receive contribu- i tions from every particle which circulates in the one-lo op gauge-b oson self-energy diagram. For example, for a fermion doublet ; e, b =1=3. 2 The one-lo op renormalization group equation is easily solved. The inverse of the gauge coupling evolves linearly with the log of the scale, Q b i 1 1 ln : 2 Q= Q 0 i i 2 Q 0 If we assume that the couplings do unify at some scale, wehave that b b b b b b 1 2 2 3 3 1 + + =0 : 3 Q Q Q 3 1 2 This equation is valid for any scale Q between the weak scale and the uni cation MS values of the U1 and SU2 gauge couplings from scale. Next, we deduce the 1 2 1 2 the quantities ^ = 127:90 0:09 ands ^ =0:2315 0:0004 Ref. ^ ands ^ are the MS values of the electromagnetic coupling and sine-squared weak mixing angle evaluated at M Z 1 1 M =58:97 0:05 ; M =29:61 0:05 : 4 Z Z 1 2 Given the Standard Mo del values of the one-lo op b eta-constants 19 41 ; b = ; b = 7 ; 5 b = 2 3 1 10 6 we determine the prediction of the strong coupling constant in the Standard Mo del M =0:07 Standard Mo del, one-lo op 6 s Z with negligible error. Comparing with the measured value as quoted by the Par- 1 ticle Data Group, M =0:118 0:003, we see that the Standard Mo del pre- s Z diction of the strong coupling constant is ab out 16 standard deviations to o small. Including higher order corrections cannot help repair this situation by more than a few standard deviations. In order to remedy this huge discrepancy,we need to go b eyond the Standard Mo del, adding matter to change the b eta-constants b . i We also need to sp ecify the mass scale of the new matter. Rather than determining what additional matter content will lead to successful gauge coupling uni cation in an ad ho c fashion, we will examine the implications of a motivated mo del. It is widely accepted that the Standard Mo del is an e ective theory. Some new physics must b ecome manifest at scales b elow the multi-TeV 2 scale. A promising mo del of new physics is sup ersymmetry, which, among other virtues, explains how a theory with a weak-scale elementary Higgs b oson is stable with resp ect to the Planck scale. Sup ersymmetry also naturally explains the 3 breaking of electroweak symmetry, and it typically contains a natural dark matter 4 candidate. In order to make the Standard Mo del sup ersymmetric, we are required to add a sp eci c set of new particles, the sup erpartners. For every Standard Mo del gauge b oson, wemust add a spin-1/2 ma jorana particle, a gaugino. And for every Standard Mo del fermion wemust add a spin-0 partner, a squark or a slepton. In order to give b oth up- and down-typ e fermions a mass, and to ensure anomaly cancellation, wemust also add an additional Higgs doublet. We refer to the two Higgs doublets as \up-typ e" and \down-typ e." This additional particle content de nes the minimal sup ersymmetric standard mo del MSSM. In the simplest and most often considered versions of the MSSM with high-energy inputs, the entire sup ersymmetric sp ectrum scales with one or two parameters whichmust b e near the weak scale by naturalness considerations. Given this new matter content and the sup erpartner mass scale, whichwe taketobeM for the moment, we can take a rst-order lo ok at the prediction Z of the strong coupling constant in the MSSM. As b efore, we use Eq. 3 to solve for M as a function of M and M . Now the b eta-constants are s Z 1 Z 2 Z di erent. Because of the new matter content, wehave the following changes from the Standard Mo del values b =4:1! 6:6 ; b =3:2! 1 ; b =7!3 : 7 1 2 3 We use the central values and errors in Eq. 4 to nd M =0:116 0:001 MSSM, one-lo op. 8 s Z The new matter content mandated by sup ersymmetry brings the prediction for the strong coupling constant to the measured value 0:118 0:003 within one standard deviation! This could b e a coincidence, or it could b e a hint that we are on the right trackby considering minimal low-energy sup ersymmetry. Next we will consider corrections to the prediction of in the MSSM. s 3 bf Corrections to the prediction of s In this section, we consider two sources of corrections to the prediction of the strong coupling constant in the MSSM. The rst involves improving the evolution of the couplings from their initial values at the grand uni cation scale to the weak scale or, the sup ersymmetry breaking scale. We simply extend the renormal- ization group equations to one higher lo op order. This means that in addition to n resumming logarithms of the form [ =4 lnM =M ] from the one-lo op RGE, GU T Z 2 n wenow resum logarithms of the form [ =4 lnM =M ] from the two-lo op GU T Z terms. Attwo lo ops, it is necessary to run b oth the gauge and Yukawa couplings together, as they form a coupled set of di erential equations. The general form 5 for either a gauge or Yukawa coupling RGE at two-lo ops is b b dg ij k ij i 2 2 2 : 9 + g = g g g i k j j 2 2 2 dt 16 16 Since we can safely ignore the small Yukawa couplings of the rst two generations, the set g includes fg ; g ; g ; ; ; g. The equations are readily solved i 1 2 3 t b numerically. The correction due to the two-lo op RGE's is large and p ositive. The prediction of M increases by ab out 0:01. s Z The second source of corrections we will consider in this section are the sup er- symmetric threshold corrections. These are divided into twotyp es, the logarithmic corrections and the nite i.e., nonlogarithmic corrections. We will discuss these in turn. When we arrived at prediction 8, we assumed that the masses of the sup er- partners were equal to M . However, we know from collider searches that many Z of the sup erpartner masses must b e heavier than M , and in general, they could Z b e an order of magnitude heavier without straining naturalness considerations to o much. As weevolve the gauge couplings down from high scales, wemust decouple each sup erpartner contribution in turn as we cross the mass threshold. Hence, we arrive at logarithmic corrections of the form b lnM =M . From these i susy Z corrections, we determine the shift in the prediction of the strong coupling and nd, in general, for a particle of mass M >M with b contributions to the Z i b eta-constants M 1 1 : 10 b b b +b b b +b b b ln M = 2 3 1 3 1 2 1 2 3 Z s 2b b M 1 2 Z Note that the b eta-constants b in this equation include the entire sup erpartner i sp ectrum.