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Comparison Between Theory and Experiment and Future Perspectives

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The Anomalous Magnetic Moment of the Muon

Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 274))

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Abstract

The anomalous magnetic moment of the muon provides one of the most precise tests of quantum field theory as a basic framework of elementary particle theory and of QED and the electroweak SM in particular.

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Notes

  1. 1.

    With the correct sign of the hadronic LbL term the deviation would have been 1.5 \(\sigma \) based on the smallest available hadronic vacuum polarization. With larger values of the latter the difference would have been smaller.

  2. 2.

    To mention the sign error and the issue of the high energy behavior in the LbL contribution or errors in the applied radiative corrections of \(e^+e^-\)–data or missing possible real photon radiation effects by the muons.

  3. 3.

    The small spread in the central values does not reflect this fact, however.

  4. 4.

    The terminology “theory–driven” means that we are not dealing with a solid theory prediction. As in some regions only old data sets are available, some authors prefer to use pQCD in place of the data also in regions where pQCD is not supposed to work reliably. The argument is that even under these circumstances pQCD may be better than the available data. This may be true, but one has to specified what “better” means. In this approach non–perturbative effects are accounted for by referring to local quark–hadron duality in relatively narrow energy intervals. What is problematic is a reliable error estimate. Usually, only the pQCD errors are accounted for, essentially only the uncertainty in \(\alpha _s\) is taken into account. It is assumed that no other uncertainties from non–perturbative effects exist; this is why errors in this approach are systematically lower than in more conservative data oriented approaches. Note that applying pQCD in any case assumes quark–hadron duality to hold in large enough intervals, ideally from threshold to \(\infty \) (global duality). My “conservative” evaluation of \(a_\mu ^{{\mathrm {had}}}\) estimates an error of 0.8%, which for the given quality of the data is as progressive as it can be, according to my standards concerning reliability. In spite of big progress in hadronic cross section measurements the agreement between different measurements is not as satisfactory as one would wish. Also more recent measurements often do not agree within the errors quoted by the experiments. Thus, one may seriously ask the question how such small uncertainties come about. The main point is that results in different energy ranges, as listed in Table 5.2 in Sect. 5.1.7, are treated as independent and all errors including the systematic ones are added in quadrature. By choosing a finer subdivision, like in the clustering procedure of [29], for example, one may easily end up with smaller errors (down to 0.6%). The subdivision I use was chosen originally in  [30] and were more or less naturally associated with the ranges of the different experiments. The problem is that combining systematic errors is not possible on a commonly accepted basis if one goes beyond the plausible procedures advocated by the Particle Data Group.

  5. 5.

    The analysis [41] does not include exclusive data in a range from 1.43 to 2 GeV; therefore also the new BaBar data are not included in that range. It also should be noted that CMD-2 and SND are not fully independent measurements; data are taken at the same machine and with the same radiative correction program. The radiative corrections play a crucial role at the present level of accuracy, and common errors have to be added linearly. In [42, 43] pQCD is used in the extended ranges 1.8–3.7 GeV and above 5.0 GeV; furthermore [43] excludes the KLOE data.

  6. 6.

    The variety of speculations about new physics is mind–blowing and the number of articles on “physics beyond the SM” (BSM) almost uncountable. This short essay tries to reproduce a few of the main ideas for illustration, since a shift in one number can have many reasons and only in conjunction with other experiments it is possible to find out what is the true cause for an observed deviation from the SM prediction. My citations may be not very concise and I apologize for the certainly numerous omissions.

  7. 7.

    GUT extensions of the SM are not very attractive for the following reasons: the extra symmetry breaking requires an additional heavier Higgs sector which makes the models rather clumsy in general. Also, unlike in the SM, the known matter–fields are not in the fundamental representations, while an explanation is missing why the existing lower dimensional representations remain unoccupied. In addition, the three SM couplings (as determined from experiments) allow for unification only with at least one additional symmetry breaking step \(G_{\mathrm {GUT}} \rightarrow G' \rightarrow G_{\mathrm {SM}}\). In non-SUSY GUTs the only possible groups are \(G_{\mathrm {GUT}} =E_6~\mathrm {or}~ SO (10)\) and \(G'=G_{LR}= SU (3)_c \otimes SU (2)_R \otimes SU (2)_L \otimes U(1)~\mathrm {or}~G_{PS}= SU (2)_R \otimes SU (2)_L \otimes SU (4)\) [62]. \(G_{LR}\) is the left–right symmetric extension of the SM and \(G_{PS}\) is the Pati–Salam model, where \( SU (3)_c \otimes U(1)_Y\) of the SM is contained in the \( SU (4)\) factor. Coupling unification requires the extra intermediate breaking scale to lie very high \(M'\sim 10^{10}~\text {GeV}\) for \(G_{LR}\) and \(M'\sim 10^{14}~\text {GeV}\) for \(G_{PS}\). These are the scales of new physics in these extensions, completely beyond of being phenomenologically accessible. The advantage of SUSY GUTs is that they allow for unification of the couplings with the new physics scale being as low as \(M_Z\) to 1 TeV [63], and the supersymmetrized \(G_{\mathrm {GUT}} = SU (5)\) extension of the SM escapes to be excluded.

  8. 8.

    It should be noted that heavy sequential fermions are constrained severely by the \(\rho \)–parameter (NC/CC effective coupling ratio), if doublet members are not nearly mass degenerate. A doublet \((\nu _L,L)\) with \(m_{\nu _L}=45~\text {GeV}\) and \(m_L=100~\text {GeV}\) only contributes \(\varDelta \rho \simeq 0.0008\), which however is violating already the limit from LEP electroweak fits (7.5). Not yet included is a similar type contribution from the 4th family \((t',b')\) doublet mass–splitting, which also would add a large positive term

    $$\varDelta \rho = \frac{\sqrt{2}G_\mu }{16 \pi ^2}\,3\,m_{t'}^{2}\left( 1+\frac{m_{b'}^{2}}{m_{t'}^{2}}\,\ln \frac{m_{b'}^{2}}{m_{t'}^{2}}\right) + \cdots $$

    in case of a large mass splitting \(m_{t'}^{2} \gg m_{b'}^{2}\), or a small correction \(\varDelta \rho = \frac{\sqrt{2}G_\mu }{16 \pi ^2}\,\frac{2\varDelta ^2}{\Sigma }\), which vanishes for small mass splitting \(\varDelta = |m_{t'}^{2}-m_{b'}^{2}| \ll \Sigma =m_{t'}^{2}+m_{b'}^{2}\). In this context it should be mentioned that the so called custodial symmetry of the SM which predicts \(\rho _0=1\) at the tree level (independent of any parameter of the theory, which implies that it is not subject to subtractions due to parameter renormalization) is one of the severe constraints on extensions of the SM. Virtual top effect contributing to the radiative corrections of \(\rho \) allowed a determination of the top mass prior to the discovery of the top by direct production at Fermilab in 1995. The LEP precision determination of \(\varDelta \rho =\frac{\sqrt{2}G_\mu }{16\pi ^2}\,3\,m_{t}^{2}\) (up to subleading terms) from precision measurements of Z resonance parameters yields \(m_t=172.3^{+10.2}_{-7.6} ~\text {GeV}\) in excellent agreement with the direct determination \(m_t=171.4(2.1)~\text {GeV}\) at the Tevatron and with the recent determinations \(m_t=172.84(0.70)~\text {GeV}\) [76] from ATLAS and \(m_t=172.44(0.13)(0.47)~\text {GeV}\) [77] from CMS (for CDF and D0 see  [78]). In extensions of the SM in which \(\rho \) depends on physical parameters on the classical level, like in GUT models or models with Higgs triplets etc. one largely looses this prediction and thus one has a fine tuning problem [67]. But, also “extensions” which respect custodial symmetry like simply adding a 4th family of fermions should not give a substantial contribution to \(\varDelta \rho \), otherwise also this would spoil the indirect top mass prediction.

  9. 9.

    Searches for Technicolor states like color–octet techni–\(\rho \) were negative up to 260–480 GeV depending on the decay mode.

  10. 10.

    Of course, there are more non-renormalizable extensions of the SM than renormalizable ones. For the construction of the electroweak SM itself renormalizability was the key guiding principle which required the existence of neutral currents, of the weak gauge bosons, the quark-lepton family structure and last but not least the existence of the Higgs. However, considered as a low energy effective theory one expects all kinds of higher dimension transition operators coming into play at higher energies. Specific scenarios are anomalous gauge couplings, little Higgs models, models with extra space–dimensions à la Kaluza–Klein. In view of the fact that non-renormalizable interactions primarily change the high energy behavior of the theory, we expect corresponding effects to show up primarily at the high energy frontier. The example of anomalous \( W^+W^-\gamma \) couplings, considered in the following subsection, confirms such an expectation. Also in non-renormalizable scenarios, effects are of the generic form (7.10) possibly with \(M_\mathrm{NP}\) replaced by a cut-off \(\varLambda _\mathrm{NP}\). On a fundamental level we expect the Planck scale to provide the cut–off, which would imply that effective interactions of non-renormalizable character show up at the 1 ppm level at about \(10^{16}~\text {GeV}\,.\) It is conceivable that at the Planck scale a sort of cut-off theory which is modeling an “ether” is more fundamental than its long distance tail showing up as a renormalizable QFT [86]. Physics-wise such an effective theory, which we usually interpret to tell us the fundamental laws of nature, is different in character from what we know from QCD where chiral perturbation theory or the resonance Lagrangian type models are non-renormalizable low energy tails of a known renormalizable theory, as is Fermi’s non-renormalizable low energy effective current–current type tail within the SM.

  11. 11.

    In my opinion “natural” here is misleading. Imposing ad hoc \(Z_2\) selection rules have no natural explanation.

  12. 12.

    \(M_\mathrm{Pl}= (G_N/c\hbar )^{-1/2}\simeq 1.22\times 10^{19}~\text {GeV}\), \(G_N\) Newton’s gravitational constant, c speed of light, \(\hbar \) Planck constant.

  13. 13.

    One could add other gauge invariant couplings like

    $$(\tilde{U}_L^c\tilde{D}_L^c\tilde{D}_L^c)\,,\; (\tilde{Q}_L\tilde{L} \tilde{D}_L^c) \,,\;m (\tilde{L}H_2)\,,\; (\tilde{L}\tilde{L} \tilde{E}_L^c)$$

    which violate either B or L, however. In the minimal model they are absent.

  14. 14.

    We label \(U=(u,c,t)\), \(D=(d,s,b)\), \(N=(\nu _e,\nu _\mu ,\nu _\tau )\) and \(E=(e,\mu ,\tau )\).

  15. 15.

    Even with the constraints mentioned, SUSY extensions of the SM allow for about 100 free symmetry breaking parameters. Free parameters typically are masses and mixings of the neutralinos, the higgsino mass \(\mu \) (the \({+\mu H_1H_2}\) term of the 2HDM Higgs potential) and \({\tan \beta \,.}\) This changes if one merges GUT concepts with SUSY, in fact SUSY-GUTs (e.g. as based on \( SU (5)\)) are the only theories which allow for grand unification broken at a low scale (\(\sim \)1 TeV). This provides strong constraints on the SUSY breaking mechanism, specifically we distinguish the constrained CMSSM a SUSY-GUT with soft breaking masses universal at the GUT scale. The NUHM is as CMSSM with non-universal Higgs masses: \(\bullet \) the CMSSM defined to have universal couplings at the GUT scale has the free parameters: \(m_0,m_{1/2},A_0,\tan \beta \) and \(\mathrm {sign}(\mu )\). \(\bullet \) NUHM1 considers \(M_A\) as an additional free parameter at the EW scale. \(\bullet \) NUHM2 in addition assumes \(\mu \) to be independent at the EW scale. These models assume many degeneracies of masses and couplings in order to restrict the number of parameters. Typically, SM parameters are supplemented by \( m_{1/2}\) (scalar-matter mass, like \(m_{\tilde{q}}\), \(m_{\tilde{\ell }}\)), \(m_{0}\) (the \(U(1)_Y\otimes SU (2)_L\) gaugino masses, \(m_{\tilde{\gamma }}\), \(m_{\tilde{Z}}\), \(m_{\tilde{W}}\) and gluino mass \(m_{\tilde{g}}\)), \(\mathrm {sign}(\mu ), \tan \beta ,A\) (trilinear soft breaking term), and more for less constrained models.

  16. 16.

    Stating that a small parameter (like a small mass) is unnatural unless the symmetry is increased by setting it to zero. The equivalent hierarchy problem addresses the fine–tuning problem encountered in Higgs mass renormalization: the renormalized (observed) low energy effective mass square

    $$m^2_\mathrm{ren}=m^2_\mathrm{bare}-\delta m^2$$

    is \(O(v^2)\) of the order of the electroweak scale square, while in the bare theory exhibiting the Planck mass as a UV cutoff, \(m^2_\mathrm{bare}\) and the counterterm \(\delta m^2\) are of order \(\varLambda _\mathrm{Planck}^2\). So the observed Higgs mass appears as a highly fine–tuned difference of two very large numbers. Exact supersymmetry eliminates the fine–tuning by canceling positive bosonic contributions to \(\delta m^2\) exactly by negative fermionic ones, such that quadratic UV singularities are absent.

  17. 17.

    Conformal symmetry would require severe fine tuning of parameters, just what we want to avoid in this context.

  18. 18.

    In [138] the CP conserving 2HDM case is considered without imposing the \(\varPhi _2 \rightarrow -\varPhi _2\) symmetry, which allows for two more terms in the potential \(V \rightarrow V+\left[ \lambda _6\,\left( \varPhi ^+_1\varPhi _1\right) +\lambda _7\,\left( \varPhi ^+_2\varPhi _2\right) \right] \,\left( \varPhi ^+_1\varPhi _2\right) + \mathrm {h.c.}\). The CP-even mass matrix is of the form

    $$\begin{aligned}\mathcal{M}^2 = \left( \begin{array}{cc} \lambda _1 v^2 &{} \lambda _6 v^2 \\ \lambda _6 v^2 &{} M_A^2+\lambda _5 v^2 \end{array}\right) \end{aligned}$$

    and one has to distinguish the following special limits:

    • Decoupling limit: \(M_A^2 \gg \lambda _iv^2\) implying \(M_h^2\sim \lambda _1 v^2\) and \(|c_{\beta -\alpha }\ll 1|\) and the lighter scalar h is the SM like one.

    • Alignment limits: \(\lambda _6=0\) with two possibilities:

      1. (1)

        \(\lambda _1 < \lambda _5+M_A^2/v^2\) and again h is identical with the SM Higgs and \(c_{\beta -\alpha }=0\)

      2. (2)

        \(\lambda _1 > \lambda _5+M_A^2/v^2\) in which case H is identical with SM Higgs and \(c_{\beta -\alpha }=1\). This is an unexpected possibility, namely the discovered Higgs is to be identified the heavier scalar. The lighter would have masses in the range 20–90 GeV and would have escaped detection so far, because of suppressed couplings to SM states.

  19. 19.

    Denoting by \(M_h^2\) the corrected light Higgs on–shell mass, and by \(m_h^2\) the tree level mass given in (7.50), then including leading logarithms in \(\alpha _s\) and \(y_t\) up to 3 loops on finds

    $$\begin{aligned}M_h^2= & {} m_h^2+\hat{v}^2\,\hat{y}_t^4\,\left[ 12\,L\,\kappa _L -12\,L^2\,\kappa _L^2 \left( 16\,\hat{g}^2_3-3\,\hat{y}^2_t\right) \right. \nonumber \\&\left. + 4\,L^3\,\kappa _L^3\,\left( 736\hat{g}_3^4-240\,\hat{g}_3^2\,\hat{y}_t^2 -99 \,\hat{y}_t^4\right) +\cdots \right] \,, \end{aligned}$$

    with \(L=\ln M_{\text {SUSY}}/M_t\), \(\hat{v}=v^\mathrm{SM}(M_t)\), \(\hat{g}_3=g_3^\mathrm{SM}(M_t)\), \(\hat{y}_t=y_t^\mathrm{SM}(M_t)\) and \(\kappa _L=1/(16\pi ^2)\). The 3–loop term is scheme dependent and depends on specific approximations made [161].

  20. 20.

    The highest power in \(\tan \beta \) at a given order L in the loop expansion is \(\alpha ^L\,\tan ^L \beta \). As a correction only the leading one of order \(\alpha ^2 \,\tan ^2 \beta \) is numerically significant.

  21. 21.

    For scenarios beyond the CMSSM see [169, 195].

  22. 22.

    It resembles the VMD type II Lagrangian (5.72), which describes the effective interaction of the neutral \(\rho \) meson with the photon. The role of the quarks is assumed to be played by new charged very heavy Fermions F.

  23. 23.

    One of the biggest unsolved problems of the SM is the non-observation of strong CP violation which would be provided by a non-vanishing \(\frac{\varTheta }{32\pi ^2}\,G_{\mu \nu }\tilde{G}^{\mu \nu }\) term supplementing the QCD Lagrangian with \(G_{\mu \nu }\) the gluon field strength tensor and \(\tilde{G}^{\mu \nu }\) its dual. For non-zero quark masses this term predicts observable CP violation in strong interactions “the strong CP problem”. A fairly convincing answer could be provided by the Peccei-Quinn [223,224,225] extension of the SM by a U(1) approximate global symmetry, which is spontaneously broken at some low scale \(f_a\). The axion a is the pseudo Nambu-Goldstone boson of this symmetry of mass \(m_a \ll \varLambda _\mathrm{QCD}\).

  24. 24.

    The Barr-Zee diagram Fig. 7.9b, typically found in 2HDMs, here appears reduced to a one–loop diagram, where the lepton/quark (\(\tau ,b\)) triangle in the heavy mass limit is shrunk to a point, now the \(g_{a\gamma \gamma }\) effective coupling. The h, A muon coupling here is \(y_{a\ell }\).

  25. 25.

    Not to forget the role of QFT for other systems of infinite (large) numbers of degrees of freedom: condensed matter physics and critical phenomena in phase transitions (Ken Wilson 1971). The Higgs mechanism as a variant of the Ginzburg-Landau effective theory of superconductivity (1950) and the role QFT and the renormalization group play in the theory of phase transitions are good examples for synergies between elementary particle physics and condensed matter physics.

  26. 26.

    The KLOE and BaBar measurements have been obtained via the radiative return method which is a next to leading order approach. On the theory side one expects that the handling of the photon radiation requires one order in \(\alpha \) more than the scan method for obtaining the same accuracy. Presently a possible deficit is on the theory side. What is urgently needed are full \(O(\alpha ^2)\) QED calculations, for Bhabha luminosity monitoring, \(\mu \)–pair production as a reference and test process, and \(\pi \)–pair production in sQED as a first step and direct measurements of the final state radiation from hadrons. The CMD-3 and SND measurements take data at the same accelerator (same luminosity/normalization uncertainties) and use identical radiative corrections, such that for that part they are strongly correlated and this should be taken into account appropriately in combining the data. The present state-of-the-art event generator is PHOKHARA [256] for radiative return events and BABAYAGA [257] for the Bhabha channel (see also [258, 259]).

  27. 27.

    Of course such questions have been carefully investigated, and a sophisticated magnetic probe system has been developed by the E821 collaboration.

  28. 28.

    which is not always true, for example if we read a newspaper or if you read this book.

  29. 29.

    An energy or an equivalent mass may always be translated into a temperature by means of the Boltzmann constant k which relates \(1^\circ \mathrm{K} \equiv 8.6 \times 10^{-5} \mathrm{eV} \). Thus \(T=E/k\) is the temperature of an event at energy E. As we know the universe expands and thereby cools down, thus looking at higher temperatures means looking further back in the history of the universe. By solving Friedmann’s cosmological equations with the appropriate equations of state backwards in time, starting from the present with a cosmic microwave background radiation temperature of \(2.728^\circ \mathrm{K}\) and assuming the matter density to be the critical one \(\varOmega _{\mathrm {tot}}=1\), one may calculate the time at which temperatures realized at LEP with 100 to 200 GeV of center of mass energy where realized. This time is given by \(t=2.4/\sqrt{N(T)} \, \left( 1~\text {MeV}/kT \right) ^2 \mathrm{sec}\), with \(N(T)=\sum _{\mathrm{bosons}\,B} g_B(T) + \frac{7}{8} \sum _{\mathrm{fermions}\,F} g_F(T)\), the effective number of degrees of freedom excited at temperature T (see Eq. (19.43) in [266]). For LEP energies \(m_b \ll kT \simeq M_W\) the numbers \(g_{B/F}(T)\) counting spin, color and charge of bosonic/fermionic states in the massless limit include all SM particles except \(W^{\pm },Z,H\) and t one obtains \(N(T)= 345/4\). Thus LEP events happened to take place in nature \(t \sim 0.3 \times 10^{-10} \mathrm{\ sec}\) after the Big Bang for \(T\sim 100~\text {GeV}\,.\) With the LHC we reach \(t_{\mathrm {LHC}}\sim 1.66 \times 10^{-15}~\mathrm {sec}\).

  30. 30.

    A 95% CL lower bound of \(77.5~\text {GeV}\) had been estimated by the LEP Collaborations  [268] at that time.

  31. 31.

    Some analyses [270,271,272] claim a failure of vacuum stability i.e. \(\lambda (\mu )\) has a zero and gets negative at about \(10^{9}~\text {GeV}\), and find a metastable vacuum instead, just missing stability. This have been questioned in [273] later. A final answer depend on the precise knowledge of the top Yukawa coupling and related problems have been analyzed in [274].

  32. 32.

    As we have consider the SM to exhibit at cutoff of the size of the Planck mass, we can also calculate the vacuum energy \(V(0)\equiv \langle V(H)\rangle \) as a large but finite number. At high energies near the Planck scale the SM is in the symmetric phase i.e. \(\langle H \rangle =0\), while \(\langle H^2 \rangle \) and \(\langle H^4 \rangle \) are non-vanishing. This requires a Wick reordering of the potential [60] which is shifting the effective mass such that the coefficient proportional to \(\lambda \) changes from 2 to 5 / 2.

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  1. Institut für Physik, Humboldt-Universität zu Berlin, Berlin, Germany

    Friedrich Jegerlehner

  2. Deutsches Elektronen-Synchrotron (DESY), Zeuthen, Germany

    Friedrich Jegerlehner

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Jegerlehner, F. (2017). Comparison Between Theory and Experiment and Future Perspectives. In: The Anomalous Magnetic Moment of the Muon. Springer Tracts in Modern Physics, vol 274. Springer, Cham. https://doi.org/10.1007/978-3-319-63577-4_7

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