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Quant. Phys. Lett. 7, No. 3, 47-50 (2018) 47 Letters An International Journal

http://dx.doi.org/10.18576/qpl/070302

Supersymmetry and the

Simon Davis Research Foundation of Southern California 8861 Villa La Jolla Drive #13595 La Jolla, CA 92039, USA

Received: 2 Sep. 2018, Revised: 12 Nov. 2018, Accepted: 24 Nov. 2018 Published online: 1 Dec. 2018

Abstract: The relation between the presence of and the value of the magnetic moment of the will be examined. A method is given for the evaluation of the magnetic moment to arbitrary large orders in a series expansion.

Keywords: Dalgaard-Strulik model, energy, economic growth, time delay, limit cycle

1 1 Introduction the multiplicative factor qg = 1 + for g 2 resulting g 1 ≥ from quantum surfaces in the− coefficients of the The magnetic moment of the electron predicted by the supermoduli space integrals, and ( 1) for loops, − of the to the magnetic field in the evaluation of kg would yield an estimate of the terms in non-relativistic limit of the is e . Given g αg 2m the series A0 + c0 ∑g=1( 1) kgqg for the magnetic the relation µ = g e , the introduction of supersymmetry − g! 2m moment, where κe.m. is set equal to α after the is sufficient to produce the classical value of g = 2. The group flow of the coupling and the precise measurement of the magnetic moment of the multiplicative constant c0 can be calculated from the electron gives a value which is predicted by quantum experimental value. electrodynamics to twelve decimal places [1]. Since the exact value of g differs from 2, the relation between supersymmetry and the magnetic moment can be clarified. Although the classical supersymmetric 2 The Superysmmetric Point-Particle Action point-particle model does not have this value of g, it remains to be determined if supersymmetry can be Consider the Lagrangian preserved in the field theory after radiative corrections. 1 2 i A review of the calculation of the magnetic moment L = mx˙ + fa f˙a eAax˙a µSaBa (1) in supersymmetric theories in §3 indicates that 2 2 −− − perturbative corrections would vanish as a result of spin i where fa is a Grassmann variable and Sa = εabc fb fc sum rules. This conclusion then forms the basis of the − 2 order estimates of the series expansion of the vertex [2]. The equations of motion are amplitude with an electromagnetic coupling that exists only in the phase of broken supersymmetry. mx¨a eA˙a = µS j∂iB j f˙a + µεabc faBb = 0 (2) − − The terms in a superstring perturbation series will be The precession of the spin follows from sufficient to establish in §4 the higher-order terms in the expansion of the squared absolute value of the summed i S˙a = εabc( f˙b fc + fb f˙c) = iεabc( µεbde fd Be) fc = iµ fa fcBc = µεabcBbSc. amplitude. The coefficients that compensate for the − 2 − − − division by a factorial are determined by a combinatorial The value of µ will be fixed by invariance under series derived from each of the components of the supersymmetry. Under the supersymmetry compactification divisor at genus g. The alternation of the transformations signs will yield cancellations between the terms and reduce the amplitude from an exponential function of the f κg a g e.m xa xa iε fa fa + ε√mx˙b, (3) genus to an expression of the form ( 1) kg . Given − g! → − √m →

∗ Corresponding author e-mail: [email protected] c 2018 NSP Natural Sciences Publishing Cor. 48 S. Davis: Supersymmetry and Magnetic Moment

moment again can evaluated. Any change from g = 2 e µSaBa µSaBa + iµ√mεεabcBax˙a fc eAax˙a eAax˙a + i εAa f˙a, would render the classical point-particle model to be not − → − − → − √m (4) invariant under supersymmetry transformations. The and, since removal of infinities and supersymmetry breaking may be considered precisely within the supersymmetric field iµ√mεεabcBax˙b fc = iµ√mε(∂bAc ∂cAb)x˙b fc, theory. There are renormalization theorems, for example, − − which prevent supersymmetry breaking in sQED. The renormalization theorems tend to imply that couplings or

dtiµ√mε(∂bAc ∂cAb)x˙b fc (5) relations between parameters in the Lagrangian are Z − unaffected by radiative corrections. The contributions of = dt iµ√mε[∂b(Acxb f˙c) 2Ac f˙c ∂c(Abxb f˙c) + Abδbc f˙c]. 1 −Z − − the and the paired spin- 2 field cancel in the evaluation of the perturbative expansion in Integration of the divergence terms will vanish at spatial supersymmetric quantum electrodynamics[?]. Therefore, infinity, leaving it remains to be determined whether a soft breaking of supersymmetry can accommodate a change in the value 3 3 3 of g by a factor of 1 O(10− ). The summability of the d xdt iµ√m( Ac f˙c)= d xdt iµ√mAc f˙c, (6) ± Z − −Z perturbation series once supersymmetry is broken is e affected by a class of bubble diagrams [7]. A study of the which will cancel the variation from eAax˙a if µ = or − m electron magnetic dipole moment in a minimal g = 2. supersymmetic with other vector bosons has yielded conditions on the parameters [8]. Sum rules for magnetic moments of charged particles in supermultiplets receive corrections if supersymmetry-breaking terms are included [?]. While 3 Supersymmetry Breaking and the the anomalous magnetic moments of the charged particles have the same signs in the standard model and the Magnetic Moment minimal supersymmetric extension, based on the most general form of the WWγ vertex, the coefficient of the The point-particle limit of the Lorentz model [3] is known term qα gβ µ qβ gαµ have opposite signs for the two to coincide with the non-relativistic limit of the Dirac theories, and− the condition of the same sign requires a equation. The self-energy infinities are evident in the upper bound for the of the [10]. It equations of motion of the point-particle theory [4] with has been suggested that calculations of the the action 1 µ 1 µν magnetic dipole moment, where theoretical and τ 2 4 µ µν I = 2 d x˙ + d x A j 16π F F and gµ 2 9 µ − µ − experimental values for − differ by (4.3 1.6) 10 j (x)=Rq δ 4(x R x(τ))x˙ (τ)dτ,  2 ± × − − only, could confirm supersymmetry, and the R 9 supersymmetric contributions are found to be O(10− ) ν µ µ ex¨µ + e˙x˙µ = qFµν x˙ ∂ Fµν = 4π j (7) [11] Much of the difference may be traced to hadronic gµ 2 a polarization effects [?]. The expansion of 2− consists of where eµ is the world-line density. The reaction of the terms resulting from broken supersymmetry of radiation to the accelerated motion of the causes a O(2 10 10) at two loops in the leading logarithm divergence in the solution for the field. These infinities × − 2 approximation [13], and the viability of the model would can be partially removed whenx ˙ = 0 and entirely be determined by a sum over the higher-loop terms. If it is removed if the particle has a mass through the inclusion included within a superstring calculation, the convergence of counterterms in the effective action [?]. The effect of can be ensured. the renormalization of the massive equations of motion is the occurrence of new terms that may be added to Eq.(2), which includes the coupling of the spin to the magnetic 4 Magnetic Moment in field. The generalization of the Dirac theory to Finite ultraviolet and infrared cut-offs for quantum renormalizable theory of quantum electrodynamics electrodynamics yields a convergent perturbation allows the systematic removal of all divergences from the expansion, even though the number of diagrams increases perturbative calculation of the magnetic moment of the at a factorial rate [14][?]. This diagrammatic expansion electron. The radiative corrections alter the value of g may be derived from a superstring perturbation from 2 to a slightly larger value. expansion, since an exponential bound is derived yielding The self-energy infinities are eliminated in a convergent series for sufficiently low values of the supersymmetric quantum electrodynamics, which has an coupling [16]. invariance generated by a superalgebra with an abelian The electric magnetic moment has been given by the e bosonic sector. The radiative corrections of the magnetic coefficient of 2m hs¯ eˆs in the coupling to the magnetic field

c 2018 NSP Natural Sciences Publishing Cor. Quant. Phys. Lett. 7, No. 3, 47-50 (2018) / www.naturalspublishing.com/Journals.asp 49

at zero orbital angular in the renormalized integration ranges for the coordinates the light-cone Lagrangian of quantum electrodynamics, where s is the worldsheet [20] and the Riemann surfaces and equal to intrinsic spin quantum number ande ˆs is the unit vector in 1 − direction of the spin vector. The coefficients of the n I n √2 ∑ e− ih Ci i  = √2(0.7880624660869638) 1.1144886275. ≃ diagrams in this expansion decrease with the order n of n π (S6) n { i}∈ 9 perturbation series as α where α is the fine structure   (10) constant. The problem of the factorial divergence of the This factor will be cancelled in the ratio in Eq.(9). The th number of diagrams at n order can be resolved within series for the then can be superstring perturbation theory. Initial calculations evaluated to an arbitrary large order. indicate that correction to the magnetic moment at one loop vanishes when supersymmetry is preserved [?]. The magnetic moment can be calculated through a perturbation series derived from since there 5 Perspective may be contributions from higher loops. With a perturbation series of the form Classical supersymmetry is sufficient to derive a of 2. It also may be found through the ∞ π2 κg spin coupling term in the Dirac equation. When g kg e.m. A0 + c0 ∑ ( 1) , supersymmetry is present, spin sum rules tend to cause − 4 g! g=1 the higher-order terms in the perturbative series for the magnetic moment to vanish. Therefore, the computation where A0 is the leading-order vertex amplitude, the factor g of the electron and muon magnetic moments through a of ( 1) arises from the superstring measure, κe.m is the gauge− coupling at electromagnetic scales, set equal to α, perturbation expansion will be valid only when supersymmetry is broken. It follows that order estimates kg represents a compensating factor that is from a combinatorial series related to the components of the of each term in the scattering would contribute to compactification divisor [18] at each genus the magnetic moment. The superstring perturbation series provides order 1 g estimates for each term in a nonvanishing amplitude. 2 ( 2 +1) k 1 g 1 1 ( 1) − 2 1 2k 2 g e / 2πg ∑ − · ζ(2k) 1 ... 1 − 2  1 + 1 − 2 (2π)2k  − g   − g  There is a factor of ( 1) the holomorphic part of the p g k=1 − g superstring measure that yields alternating signs in the 1 ( 1) g 2 2 − 2 k π g 2k+2 ( 1) − 2(2 ) − 1 2g 3 g+4k 3 series expansion of the amplitude. When the + ∑ − · (g 2k + 2) 1 ... 1 − g− − ζ −  − g   − g   k=1 supersymmetry is present, the value of the effective string coupling is close to the unified gauge coupling of the α 1 α2 and the relative weighting for the terms is k1 : 2! k2 : minimal supersymmetric standard model. The 1 α3 1 α4 − conventional value of the electromagnetic coupling, the 3! k3 : 4! k4 : .. . − Quantum deformations of surfaces in the string path fine structure constant, is different from the unified gauge integral have been found to contribute a factor coupling, and therefore, it results from the 1 flow after supersymmetry qg = 1 + for g 2 [19]. Since the amplitudes then g 1 ≥ α differ from− the superstring perturbation expansion, the breaking. With the value of in place of the effective vanishing of radiative contributions by supersymmetry is string coupling, a phenomenologically valid estimate of no longer valid. There is a factor of 1 for each fermion the magnetic moment can be given to arbitrary precision. It therefore can be calculated with a finer resolution than loop. Setting q = 1, the perturbative− series is 1 the current theoretical estimates of quantum ∞ 2 g electrodynamics. π kgqgκe.m. A0 + c0 ∑ (8) g=1 4 g! References after cancellation of the factors of ( 1)g. The coefficient c can be selected such that − 0 [1] G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio and B. αg Odom, New Determination of the Constant ∑3 g π2 kgqg A0 + c0 g=1( 1) 4 g! from the Electron g Value and QED, Phys. Rev. Lett. 97 − = 1.0011596521 (9) (2006) 030802. A0 [2] A. P. Balachandran, G. Marmo, B.-S. Skagerstam and A. when κe.m. is set equal to the fine structure constant α. Stern, Gauge Symmetries and Fibre Bundles, Lect. Notes The observed amplitudes in Lorentzian space with a Phys. 188, Springer-Verlag, Berlin, 1983. G2/SU(3) compactification differ from the Euclidean [3] H. A. Lorentz, Theory of Electron, Teubner, Leipzig,1906, amplitudes in R10 by a factor that receives contributions reprinted, Dover, New York, 1952. from the space-time resulting from windings [4] P. A. M. Dirac, Classical Theory of Radiating Elctrons, Proc. around the compact space and the transformation of the R. Soc. 167 (1938) 148-169.

c 2018 NSP Natural Sciences Publishing Cor. 50 S. Davis: Supersymmetry and Magnetic Moment

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