MUON g − 2 AND ELECTRIC DIPOLE MOMENTS IN SUGRA MODELS

arXiv:hep-ph/0108082v1 8 Aug 2001 MUON g − 2 AND ELECTRIC DIPOLE MOMENTS IN SUGRA MODELS R. ARNOWITT, B. DUTTA, B. HU AND Y. SANTOSO Center For Theoretical Physics, Department of Physics, Texas A&M University, College Station TX 77843-4242, USA E-mail: [email protected] The SUSY contribution to the muon magnetic moment anomaly, aSUGRA , and the µ electron electric dipole moment, de , is discussed within the framework of a modified mSUGRA model where the magnitudes of the soft breaking masses are universal, but arbitrary phases are allowed. It is shown analytically how the cancellation < mechanism can allow for large phases (i.e. θB ∼ 0.4) and still suppress the value of de below its current experimental bound. The dependence of aSUGRA on the CP µ violating phases are analytically examined, and seen to decrease it but by at most a factor of about two. This reduction would then decrease the upper bound on m1/2 due to the lower bound of Brookhaven data, and hence lower the SUSY mass spectrum, making it more accessible to accelerators. At the electroweak scale, the phases have to be specified to within a few percent to satisfy the experimental bound on de , but at the GUT scale, fine tuning below 1% is required for lower values of m1/2 . This fine tuning problem will become more serious if the bound on de is decreased. 1 Introduction In supersymmetry, the interaction of charginos (χ̃± i , i = 1, 2), neutralinos (χ̃0j , j = 1, 2, 3, 4) and sleptons (ẽk , µ̃k , k = 1, 2; ν̃e , ν̃µ ) with the leptons (lm = e, µ, τ ) gives rise to electromagnetic vertices lm − ln − γ. Thus the basic diagrams for the diagonal muon interaction shown in Fig. 1 gives rise to anomalous contributions to the muon magnetic moment aµ = (g − 2)/2, and could possibly produce an electric dipole moment, dµ . Similar diagrams with µ → e and ν̃µ , µ̃ → ν̃e , ẽ can give rise to ae and de , while the off diagonal diagram could allow for the decay µ → e + γ. The different possibilities, however, involve different physics. Corrections to the anomalous magnetic moments of the leptons are always present in supersymmetry, and the recent results of the Brookhaven E821 experiment 1 showing a 2.6 σ deviation from the Standard Model prediction, SM −10 aexp µ − aµ = 43(16) × 10 (1) shows the possibility of a SUSY contribution. Indeed, in the initial calculations 2,3 based on supergravity grand unification models (mSUGRA 4 ), it was predicted 3 that a deviation should show up when the experimental sensitivity arnowitt-susy01: submitted to World Scientific on November 7, 2018 1 reached that of the Brookhaven experiment. However, for electric dipole moments to be non-zero requires in addition the presence of CP violating phases. The current experimental bounds on de is 5 de < 4.3 × 10−27 (2) and this bound is expected to be reduced by a factor of 2-3 in the near future 6 . χ̃0j ν̃µ q q µ q µ χ̃± i q µ µ µ̃ γ γ Figure 1. Diagrams contributing to aµ and dµ involving intermediate chargino-sneutrino states and intermediate neutralino-smuon states. In this paper, we examine the magnetic moment and electric dipole moment phenomena within the framework of gravity mediated SUGRA models, and assume that the Brookhaven anomaly is real 7 and due to SUSY. We will assume a 2 σ range for aSUGRA µ 11 × 10−10 < aSUGRA < 75 × 10−10 µ (3) If we assume universal soft breaking in the first two generation, then mµ̃k = mẽk , and mν̃µ = mν̃e . Then both aµ and de can be obtained from the same complex amplitude A: α m2µ Re[A] (4) aSUGRA =− µ 4π sin2 θW de = − α me Im[A] 8π sin2 θW (5) There has been a great deal of recent analysis of aµ within the framework of mSUGRA models 8,9 showing that indeed for current theory constrained by all accelerator and non-accelerator bounds, aSUGRA can be expected to lie within µ the range of Eq. (3). It is thus at first sight puzzling that de is so small, since it arises from the same amplitude A that gives rise to a large aSUGRA . Two µ possible explanations for this are the following: (1) The CP violating phases appearing in A are anomalously small, and in fact it is possible to build reasonably natural models where this can happen 10 . (2) The CP violating phases are indeed O(1) (as the CKM phase is), but there are cancelations arnowitt-susy01: submitted to World Scientific on November 7, 2018 2 between the two diagrams of Fig. 1 suppressing the value of de 11 . This possibility appears more preferable, and there has been considerable analysis within that framework 12 and we will consider it here. However, this leads immediately to the following question: if cancellations occur in de , why do corresponding cancellations not occur also in aSUGRA ? In the following we µ will answer this question and show analytically how one may have large phases such that de is suppressed to nearly zero, but aSUGRA is reduced by less than µ a factor of 2 (so that agreement with the Brookhaven data is maintained). The remaining question then is whether fine tuning of the phases is needed to suppress de to the level of Eq. (2) when the phases are large. We will see that significant fine tuning has started to occur at the GUT scale, and this problem will become more serious if the upper bound on de is reduced further. 2 SUGRA Models We consider here a generalization of the usual mSUGRA model allowing for CP violating phases. At the GUT scale MG , the theory depends upon the following parameters: m0 , the universal scalar soft breaking mass; mi = |m1/2 |eiφi , i = 1, 2, 3 the three gaugino masses; A0 = |A0 |eiα0 , the cubic soft breaking mass; B0 = |B0 |eiθB0 , the quadratic soft breaking mass; and µ0 = |µ0 |eiθµ , the Higgs mixing mass. One is always free to set one of the gaugino phases to zero and we chose φ2 = 0. Radiative breaking of SU (2) × U (1) at the electroweak scale determines θµ according to (with the convenient choice that the Higgs VEVs be real) θµ = −θB + f1 (−θB + αl , −θB + αq ) where f1 is a loop correction, α(l,q) are the (lepton, quark) phases of A at the electroweak scale, and θB is the B phase at the electroweak scale. In addition |µ|2 and |B| are determined by the usual formulae. Thus the theory is defined by four real parameters, m0 , |m1/2 |, |A0 |, and tan β = hH2 i/hH1 i, and four new CP violating phases: θB0 , φ1 , φ3 , α0 . However, since we are here examining only the electron EDM, our results depend only weakly on φ3 and α0 . Thus the two important phases are θB0 and φ1 . In carrying out the calculations discussed below, it is important to impose all the known accelerator and non-accelerator bounds on the SUSY parameter space (including coannihilation effects in Ωχ̃01 h2 ), and these bounds need to be calculated accurately. For a discussion of these see Ref. 13. Finally we mention that we scan the parameter space over the range m0 , |m1/2 | < 1 TeV, |A0 /m1/2 | < 4, and 2 < tan β < 50. arnowitt-susy01: submitted to World Scientific on November 7, 2018 3 3 Suppression of de It was realized from the beginning that aµ is an increasing function of tan β 2,3 . The Brookhaven data favors tan β > 5 − 7 8 , and larger tan β is consistent with the data. In order to see analytically the effect of the CP violating phases then, we consider the leading part of the complex amplitude for large tan β calculated in Ref. 13. Using Eq. (4), one can write aSUGRA in the form µ aSUGRA = A cos θµ + B cos(θµ + φ1 ) + b cos θµ + c µ (6) A comes from the chargino diagram, while B, b and c arise from the neutralino diagram. In general, the chargino contributions are largest and the parameter a = B/(A+ b) is ∼ = 0.10 − 0.45. (b and c are generally small). Similarly, Eq.(5) for de has the general form de me = − 2 (A + b) [sin θµ + a sin(θµ + φ1 )] e 2mµ (7) One sees that a priori, the amplitudes for aSUGRA and de are of large size, µ being scaled by the largest amplitudes A + b. However, one can suppress de , and in fact even obtain de = 0 if the phases obey sin θµ + a sin(θµ + φ1 ) = 0, or alternately, using Eq. (6) (neglecting the small loop corrections), if 13 tan β(θB ) = a sin φ1 (1 + a cos φ1 ) (8) Since a is not small, one sees that θB need not be small to accommodate even de = 0. This is essentially the origin of the cancellation effect 11 for large SUSY CP violating phases. One may now insert Eq. (8) into Eq. (6) to see the effect the phases have on aSUGRA : µ aSUGRA (θB , φ1 ) = aSUGRA (0, 0) µ µ cos θB Q(φ1 ) | cos θB | (9) where Q(φ1 ) = [1 + 2a cos φ1 + a2 ]1/2 ; (1 + a) < 0.5 ∼ Q ≤ 1 (10) Thus the effect of the non-zero phases which reduce de to zero is to reduce the magnitude of aSUGRA by at most a factor of two. Further, since experimentally µ aSUGRA > 0, one requires cos θB > 0 and so µ θB > 0 for 0 < φ1 < π; θB < 0 for π < φ1 < 2π arnowitt-susy01: submitted to World Scientific on November 7, 2018 (11) 4 < and so as φ1 varies over the entire range, one has |θB | ∼ 0.4. In summary then, even if de were zero, the cancellation mechanism 11 can < accommodate large phases (|θB | ∼ 0.4), and give acceptable predictions for the muon magnetic moment anomaly 13 . The question, however, is: given the current bounds Eq.(2) on de , can one have large CP violating phases without unreasonable fine tuning of θB or φ1 ? We turn to this question next. 4 Large Phases and Fine Tuning Since the experimental upper bound on de is so small, one might expect that when the SUSY CP violating phases are large, fine tuning might be required to obtain the necessary amount of suppression. In order to quantify this, we define for any phase φ the quantity R(φ) = (φ1 − φ2 ) (φ1 + φ2 )/2 (12) where φ1 and φ2 are the largest and smallest values of φ that satisfy the bound of Eq. (2). Thus R(φ) is a measure of how tightly constrained a phase must be to satisfy the experimental bounds. How much fine tuning one can tolerate is, of course, a matter of individual taste. However, we will take here as a benchmark the requirement that for any phase, R(φ) > 0.01. For a grand unified model, presumably parameters at the GUT scale are the more fundamental ones, and they are to be determined by some higher theory. Thus fine tuning at MG can represent a serious problem. The sensitive parameter in this case is θB0 (the B phase at MG ). To understand analytically what may be occurring, we consider the RGE for the low and intermediate tan β region where an analytic expression exists. One finds 10 1 B = B0 − (1 − D0 )A0 − ΣΦi |m1/2 | eiφi 2 (13) < where D0 = 1 − m2t /(200 sin β)2 ∼ 0.25 and Φi = O(1). Taking the imaginary part one finds 1 |B| sin θB = |B0 | sin θB0 − (1 − D0 )|A0 | sin α0 − ΣΦi m1/2 sin φi 2 (14) For fixed α0 and φi , one can relate the range of θB , ∆θB , allowed by radiative electroweak breaking in terms of the range of θB0 , ∆θB0 : |B| ∆(θB0 ) ∼ ∆(θB ) = |B0 | arnowitt-susy01: submitted to World Scientific on November 7, 2018 (15) 5 However, radiative breaking at the electroweak scale shows that |B| gets small as tan β grows i. e. |B| = (1/2) sin 2β(m23 /|µ|). Hence we expect that ∆θB0 ≪ ∆θB (16) and fine tuning may occur at the GUT scale. An example of what happens is shown in Fig.3 where R(θB0 ) is plotted as a function of m1/2 for tan β = 40, A0 = 0 for φ1 = 1.6, 1.2, 0.9, 2.3, and 2.6. One sees that for a wide range of φ1 , R(θB0 ) falls below 0.01, and if one were to impose the condition that R > 0.01, it would eliminate a significant portion of the low m1/2 parameter space. In the near future, one may expect the bounds on de to be reduced by a factor of 2 to 3 6 . This would exacerbate the fine tuning problem at the GUT scale. 0.025 0.02 R (θΒ ) 0 0.9 φ 1= .6 2 φ= 1 1.2 φ 1= 2.3 φ 1= 0.015 0.01 φ =1.6 1 400 500 600 700 m1_ (GeV) 2 Figure 2. R(θB0 ) as a function of m1/2 for tan β = 40, A0 = 0 for (from bottom to top) φ1 = 1.6, 1.2, 0.9, 2.3 and 2.6. 13 5 Conclusion If the SUSY CP violating phases are small, i.e. O(10−2 ), then the bounds on de can be satisfied, and the effects of de disconnect from gµ − 2. The Brookhaven E821 experiment, plus other experimental constraints, then leads to the following results for mSUGRA models 8 : The light Higgs, h, mass bound combined with the b → s + γ constraint gives rise to a lower bound > on m1/2 of m1/2 ∼ 300 − 400 GeV. The lower bound on the aµ anomaly then produces an upper bound on m1/2 of m1/2 < 585(845) GeV at the 90% (95%) arnowitt-susy01: submitted to World Scientific on November 7, 2018 6 C.L. for tan β ≤ 50. For mSUGRA, aSUGRA is bounded from above with µ < aSUGRA ∼ 50 × 10−10 . One can then predict the mSUGRA discovery reaches µ for accelerators. Thus at the 90% C.L. one finds that the Tevatron should see the light Higgs only, while a 500 GeV NLC would be able to see only h, the light stop squark and perhaps the light selectron. (One would need a higher energy LC to see more of the SUSY spectrum.) The LHC, of course, could see the entire SUSY spectrum. In addition, the lower bounds of dark matter detection rates would be raised, since the upper bound on m1/2 has been lowered, and µ > 0. If the CP violating phases are large, they effect both de and aµ . The experimental bound on de can then still be satisfied if the cancelation mechanism 11 occurs. One finds then the following 13 : The cancelations can be understood analytically [Eq. (11)] and are seen to lead to θB as large as ∼ 0.4 with large gaugino phase φ1 . The value of aSUGRA is reduced by at most a µ factor of about two, and so agreement with the Brookhaven aµ data can still be satisfied. (The reduction of aSUGRA will, however, lower the upper bound µ on m1/2 and thus lower the SUSY mass spectrum, increasing the reach of accelerators.) The experimental bounds on de can generally be satisfied with a fine tuning of phases of a few percent at the electroweak scale. However at MG , fine tuning < 1% is needed for θB0 in the lower m1/2 region. This fine tuning will become more serious if the experimental bound on de is lowered further. Acknowledgments This work was supported in part by a National Science Foundation grant number PHY-0070964. References 1. H.N. Brown et al., Muon (g-2) Collaboration, Phys. Rev. Lett. 86, 2227 (2001). 2. D.A. Kosower, L.M. Krauss and N. 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