The Perfect Atom: Bound States of Supersymmetric Quantum Electrodynamics
aa r X i v : . [ h e p - t h ] J un arXiv:0912.0733PUPT-2327 The Perfect Atom:
Bound States of Supersymmetric Quantum Electrodynamics
Christopher P. Herzog a and Thomas Klose ba Joseph Henry Laboratories and b Princeton Center for Theoretical SciencePrinceton University, Princeton, NJ 08544, USA cpherzog,[email protected]
Abstract
We study hydrogen-like atoms in N = 1 supersymmetric quantum electrody-namics with an electronic and a muonic family. These atoms are bound statesof an anti-muon and an electron or their superpartners. The exchange of aphotino converts different bound states into each other. We determine the en-ergy eigenstates and calculate the spectrum to fourth order in the fine structureconstant. A difference between these perfect atoms and non-supersymmetricones is the absence of hyperfine structure. We organize the eigenstates intosuper multiplets of the underlying symmetry algebra. ontents l = 0-states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3 l > l = 0-states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 l > D.1 l = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24D.2 l > E Second Order Perturbation Theory 26 Introduction
Supersymmetry is often invoked to resolve a number of theoretical difficulties with the stan-dard model of particle physics. The symmetry can control quantum corrections to the Higgsmass, thus providing a solution to the hierarchy problem. The symmetry suggests the strongforce, the weak force, and electrogmagnetism are unified at high energy scales. Moreover, thelightest supersymmetric partner is a candidate for dark matter. Despite these theoretical ad-vantages, we see no direct evidence for supersymmetry at low energies; supersymmetry mustbe broken, and most studies of supersymmetry are devoted to investigating methods for andconsequences of the breaking. In this paper, we take a different tack and look at the energyspectrum of an anti-muon electron bound state in a theory with unbroken supersymmetry,supersymmetric quantum electrodynamics (SQED).We were initially inspired to write this paper by work on gauge/gravity duality. Agauge/gravity duality is a map between a field theory and a string theory. The dualityis useful because when the field theory is strongly interacting, the string theory is weaklyinteracting and vice versa. Both the field theory and the string theory are typically su-persymmetric. Often one is faced with the following awkward situation: A calculation onthe gravity side has revealed some property of the strongly interacting field theory, and thecorresponding property of the field theory at weak coupling has not yet been studied. Oneprime example of such a situation was the computation of the viscosity of N = 4 SU( N )super Yang-Mills theory at strong coupling in ref. [1]. Only five years later was the viscositycalculated in the perturbative limit [2]. In the case of supersymmetric atoms, ref. [3, 4] stud-ied hydrogenic bound states at strong coupling in N = 4 SU( N ) super Yang-Mills modifiedby the addition of two massive N = 2 hypermultiplets. No corresponding study at the timehad been made of such bound states at weak coupling. Moreover, the interesting observationwas made that these bound states exhibited no hyperfine structure [4]. The absence of suchstructure is an almost trivial consequence of N = 2 supersymmetry [5], but in this paperwe will see that hydrogenic atoms of N = 1 SQED also lack hyperfine structure. It should,however, be emphasized that the energy spectra of N = 1 and N = 2 SQED remain notice-ably different. The energy levels of N = 2 hydrogenic atoms are independent of the spinof both the electron and the proton, while fine structure effects remain evident for N = 1atoms.A second motivation for this paper is pure intellectual curiosity. Although the 1s state ofsupersymmetric positronium was considered almost thirty years ago [6], no one to our knowl-edge has studied anti-muon electron bound states in SQED. The way in which the boundstates organize themselves into supermultiplets is surprising and intricate. Similar to whathappens for supersymmetric positronium, both degenerate and second order perturbationtheory contribute at the same order in the fine structure constant α .We hope that these super atoms may be useful in particle physics, perhaps as a candidatefor dark matter, perhaps in a hidden sector, perhaps for neutrino physics.The paper is organized as follows. In section 2, we review SQED. In section 3, we presentthe relevant scattering amplitudes necessary for computing the energy spectrum to order α . In section 4, we reduce the energy spectrum computation from field theory to time Given the strongly interacting nature of the bound states, heavy-light or hybrid meson is perhaps moreappropriate terminology.
The energy spectrum of the hydrogen atom is usually described order by order in the finestructure constant α . The rest mass of the atom is M + m where M is the mass of the protonand m the mass of the electron. The binding energy is of order α µ where µ = M m/ ( M + m )is the reduced mass and we work in units where the speed of light c = 1. Fine structureeffects are of order α µ and involve relativistic corrections along with spin-orbit couplingsof the electron’s spin to its orbital angular momentum. Hyperfine structure is of order α µm/M and involves spin-spin coupling of the electron and proton. There are higher ordercorrections, for example the Lamb shift at order α , but in this paper we work only to order α . The proton is a composite object in the real world, and its compositeness has subtleeffects on the hydrogen spectrum that do not interest us for the purposes of this paper.Thus, we replace the proton with a fundamental particle of positive charge, an anti-muon.Although in the case of the hydrogen atom M is much larger than m , the results we presentare valid for arbitrary values of m and M .We consider SQED in 3+1 dimensions with four super charges. The electron and muonhave super partners, the selectron and smuon. Because the electron and muon have bothcharge and a Dirac mass, they need to be Dirac fermions and as such will each have twocomplex scalar field super partners. In other words, there are two selectrons and two smuons.The super partner of the photon is a Majorana spinor, the photino.The existence of these super partners leads to some interesting effects. An electron anti-muon bound state can mutate into a selectron anti-smuon bound state and back throughphotino exchange. There are also fermionic bound states: an electron anti-smuon or a se-lectron anti-muon which can mutate into each other. Because of photino exchange, theeigenstates of the effective Hamiltonian describing our super atom are actually linear super-positions of these different types of bound states.The total angular momentum is a good quantum number for the bound states and is auseful organizing principle for the energy spectrum. Consider a hydrogenic wave functionwith principal quantum number n and orbital angular momentum l >
0. Let V l be a 2 l + 1dimensional representation of the SO (3) rotation group. A bound state of an electron andan anti-muon will transform as V l ⊗ V / ⊗ V / = V l − ⊕ V l ⊕ V l +1 (1.1)under SO (3). As there are two selectrons and two smuons, there are four fermionic boundstates consisting either of an electron anti-smuon or selectron anti-muon. These bound statestransform as 4( V l ⊗ V / ) = 4 V l − / ⊕ V l +1 / . (1.2)Finally, there are four bosonic bound states consisting of a selectron and anti-smuon, alltransforming as V l . 3iven supersymmetry, the energy spectrum must organize itself into super multipletsarising from the four super charges in 3+1 dimensions. If j is a total angular momentumquantum number, then for j >
0, a massive super multiplet consists of the four representa-tions of the rotation group R j = V j − / ⊕ V j ⊕ V j +1 / . From the analysis in the previousparagraph, we see that to each hydrogenic wave function of principle quantum number n and orbital angular momentum l >
0, we can associate the four super multiplets j R l − / R l R l R l +1 / l − l − l l + l + 1 1 (1.3)Our results for the first few atomic energy levels of supersymmetric hydrogen are givenin Figure 1. More generally, we find that the two multiplets R l are degenerate in energy.The multiplet R l +1 / is higher in energy by an amount ∆E = µα l + 1)(2 l + 1) n , (1.4)while the multiplet R l − / is lower in energy by an amount ∆E = − µα l (2 l + 1) n . (1.5)In the case j = 0, the R − / multiplet of course does not exist and the R multiplets do notcontain the representation V − / : j R R R / R multiplets are degenerate in energy, and the R / multiplet is higher in energyby an amount µα / n .There is also an overall shift in the energies at order α , see (4.6). For the total energyof a state in a super multiplet R j , we find E n ( R j ) = − µα n − µα n (cid:18) n j + 1 −
38 + µ M m (cid:19) + O ( α ) . (1.7)As is also true in QED, this expression can be written purely in terms of j . In other words,the super multiplet R l +1 / that comes from an | nl i state and the R ( l +1) − / super multipletthat comes from an | n, l + 1 i state are degenerate in energy.An important observation about these energy splittings (1.4) and (1.5) is the absenceof hyperfine structure; there is no energy splitting between two states of order α µm/M inthe case m ≪ M . This absence is an effect of supersymmetry. In pure QED, we find the4 Sfrag replacements ± − µα × − µα × − µα × − µα n − µα × − µα × − µα × − µα × − µα × − µα × + δE nl + δE + δE + δE + δE + δE + δE + µα l +1)(2 l +1) n + µα × + µα × + µα × + µα × + µα × + µα × − µα l (2 l +1) n − µα × − µα × − µα × × + × × × × + × × × × × × n = 1 n = 2 n = 3 l = 0 l = 1 l = 2 ± E ( R / ) E ( R / ) E ( R / )2 × R R / × R R / × R R / × R R / R − / × R R / R − / × R R / R − / n = 1 n = 2 n = 3 l = 0 l = 1 l = 2 Figure 1:
Spectrum.
Splitting of the energy levels n = 1 , , l = 0 , , − µα n are shifted due to arelativistic correction to the kinetic energy by δE nl as given in (4.6), and then split due to variousinteractions into two or three levels. V l ⊕ V l +1 ≡ A l +1 / ( l = 0 , , , ... ) and V l − ⊕ V l ≡ A l − / ( l = 1 , , ... ). In thelimit m ≪ M , there is a fine structure splitting between A l +1 / and A l − / , and a furtherhyperfine splitting between the V l and V l ± in A l ± / .In cases with more supersymmetry, the corresponding multiplets are even larger andthe splitting (1.4) and (1.5) will disappear as well [5]. For example in a theory with 8supercharges, a massive multiplet with j ≥ V j − ⊕ V j − / ⊕ V j ⊕ V j +1 / ⊕ V j +1 under the rotation group. To get such a large multiplet, we need to combine the fourmultiplets in table (1.3). Similarly, in the case j = 0, we would need to combine the threemultiplets in table (1.6).It is also instructive to compare our results for muonium to the splitting of the groundstate energy of N = 1 positronium found in [6]. The two computations differ in the respectthat the latter one involves two particles of the same mass that moreover can annihilate.Setting M = m and n = 1 in (1.7), we find that the two levels at l = 0 differ in energyby ∆E = mα which is half the value for the splitting between the ortho and para states ofpositronium [6]. This difference is a consequence of the absence of the annihilation diagrams. We write N = 1 SQED for two families of matter particles which we refer to as “electronic”and “muonic.” The electron e − , its superpartners the selectrons ˜ e −± , and their antiparticles e + and ˜ e + ± , are collectively represented by two chiral superfields Φ e ± with U(1) charge ± e and mass m . Similarly, we write Φ m ± for the muon ( µ − , µ + ) and the smuons (˜ µ −± , ˜ µ + ± ) whichare assigned mass M . The U(1) gauge superfield containing the photon γ and the photino˜ γ is denoted by V . Employing the superspace conventions of Wess and Bagger [7], theLagrangian has the form L SQED = 14 (cid:16)
W W (cid:12)(cid:12) θ + W W (cid:12)(cid:12) ¯ θ (cid:17) + (cid:16) Φ † e + e e V Φ e + + Φ † e − e − e V Φ e − (cid:17)(cid:12)(cid:12) θ ¯ θ + m (cid:16) Φ e + Φ e − (cid:12)(cid:12) θ + Φ † e + Φ † e − (cid:12)(cid:12) ¯ θ (cid:17) + (cid:16) Φ † m + e e V Φ m + + Φ † m − e − e V Φ m − (cid:17)(cid:12)(cid:12) θ ¯ θ + M (cid:16) Φ m + Φ m − (cid:12)(cid:12) θ + Φ † m + Φ † m − (cid:12)(cid:12) ¯ θ (cid:17) , (2.1)where the super fieldstrength is defined by W α = − ¯ D D α V and W ˙ α = − D ¯ D ˙ α V . Afterintegrating out the auxiliary fields, the Lagrangian can be written as a kinetic term for thegauge fields L gauge = − F µν F µν + i λγ µ ∂ µ λ , (2.2)a part that contains the electronic particles L electron = ¯ ψ e (cid:0) iγ µ D µ + m (cid:1) ψ e + φ † e + (cid:0) D − m (cid:1) φ e + + φ † e − (cid:0) D − m (cid:1) φ e − + √ ie (cid:0) φ e + ¯ ψ e P − λ − φ † e + ¯ λP + ψ e − φ e − ¯ λP − ψ e + φ † e − ¯ ψ e P + λ (cid:1) , (2.3)an analogous one for the muons, L muon , which is obtained by replacing the labels e by m ,and a part with contact interactions between the two families L contact = − e (cid:0) | φ e + | − | φ e − | + | φ m + | − | φ m − | (cid:1) . (2.4)6or our notation and conventions, see App. A. In order to find the spectrum of bound states of a particle of the electronic family and ananti-particle of the muonic family in Sec. 5, we first compute the potential between anytwo of these particles. We deduce the potential energies from the non-relativistic limit ofthe tree-level scattering amplitudes which we compute from SQED Feynman rules. Theamplitudes will allow us to calculate the bound state spectrum including all effects up toorder α in the fine structure constant α = e π .In the next subsection, we will explicate the derivation of the potential from the ampli-tudes for the scattering of an electron and an anti-muon. This scattering process is the onlyone that would exist for pure QED. The amplitudes and results for all other cases are listedin the two subsequent subsections. At tree-level the only diagram describing the scattering of an electron and an anti-muoninvolves the exchange of a photon: µ + e − µ + e − = e ¯ u e ( q ) γ µ u e ( p ) ˜ ∆ µν ( p − q ) ¯ v m ( p ′ ) γ ν v m ( q ′ ) . (3.1)The in-going electron and anti-muon momenta are p and p ′ respectively. The outgoingmomenta are q and q ′ . The spinors are u e ( p ) = (cid:18) √ σ · p ξ ei √ ¯ σ · p ξ ei (cid:19) , u e ( q ) = (cid:18) √ σ · q ξ eo √ ¯ σ · q ξ eo (cid:19) , (3.2) v m ( p ′ ) = (cid:18) √ σ · p ′ η mi −√ ¯ σ · p ′ η mi (cid:19) , v m ( q ′ ) = (cid:18) √ σ · q ′ η mo −√ ¯ σ · q ′ η mo (cid:19) . (3.3)We will work in the center of mass frame where p = ( p m + ~p , ~p ) , q = ( p m + ~q , ~q ) , (3.4) p ′ = ( p M + ~p , − ~p ) , q ′ = ( p M + ~q , − ~q ) . (3.5)The first few terms in a non-relativistic expansion of the scattering amplitude are i M = i M me ( ~p − ~q ) ξ † eo η † mi (cid:20) ~p + ~q ) (cid:18) m + 1 M (cid:19) − ( ~p − ~q ) ( ~p − ~q ) M m (3.6) − i ~p × ~q ) · ~σ e (cid:18) m + 1 M m (cid:19) + i ~p × ~q ) · ~σ m (cid:18) M + 1 M m (cid:19) + 14
M m ( ~p − ~q ) ~σ e · ~σ m − M m ( ~p − ~q ) · ~σ e ( ~p − ~q ) · ~σ m + . . . (cid:21) η mo ξ ei . We are grateful to Tomas Rube and Jay Wacker for pointing out a mistake in this and similar formulasin a previous version of the manuscript. Before we worked in Feynman gauge where it would have beennecessary to compute a one-loop diagram to fix an ambiguity in the non-relativistic potential [8]. ~p and ~q . Theplane wave states in quantum field theory are normalized to the Lorentz invariant quantity: h ~p | ~q i = 2 p m + ~q (2 π ) δ (3) ( ~p − ~q ) . (3.7)In non-relativistic quantum mechanics, in contrast, these plane wave states are typicallynormalized to (2 π ) δ (3) ( ~p − ~q ). To take into account the different normalizations, we definethe non-relativistic scattering amplitude M NR ≡ M (cid:2) ( m + ~p )( m + ~q )( M + ~p )( M + ~q ) (cid:3) / . (3.8)Taking into account the change in normalization, we find that i M NR = ie ( ~p − ~q ) ξ † eo ξ † mo (cid:20) M m (cid:18) ~p + ( ~p − ~q ) − ( ~p · ( ~p − ~q )) ( ~p − ~q ) (cid:19) − (cid:18) m + 1 M (cid:19) ( ~p − ~q ) − i (cid:18) M m + 12 m (cid:19) ( ~p × ~q ) · ~σ e − i (cid:18) M m + 12 M (cid:19) ( ~p × ~q ) · ~σ m − M m ( ~p − ~q ) ~σ e · ~σ m + 14 M m ( ~p − ~q ) · ~σ e ( ~p − ~q ) · ~σ m + . . . (cid:21) ξ mi ξ ei . (3.9)We have changed the spinor η of the anti-muon into a spinor ξ as if it described a muon, η = iσ ξ ∗ .We would like to compare this scattering amplitude with the Born approximation resultfor a particle of position ~r and momentum ~p in non-relativistic quantum mechanics scatteringoff of a potential V ( ~r, ~p ). The Born approximation says that M NR = − Z d ~r e − i~q · ~r V ( ~r, ~p ) e i~p · ~r , (3.10)for plane wave initial and final states. We now Fourier transform the amplitude M NR withrespect to ~p − ~q , keeping ~p as a variable. We find FT( M NR ) = − V ( ~r, ~p ) and V ( ~r, ~p ) = e π (cid:20) − r − M m (cid:18) ~p r + ( ~r · ~p ) r + πδ (3) ( ~r ) (cid:19) + π δ (3) ( ~r ) (cid:18) m + 1 M (cid:19) + ~L · ~S e r (cid:18) m + 1 M m (cid:19) + ~L · ~S m r (cid:18) M + 1 M m (cid:19) + 1
M m (cid:18) π ~S e · ~S m δ (3) ( ~r ) + 3ˆ r · ~S e ˆ r · ~S m − ~S e · ~S m r (cid:19) + . . . (cid:21) . (3.11)All terms are understood to be normal ordered, i.e. when ~p and ~L are replaced by operatorsthen they do not act on the coordinate dependence of the potential. The result (3.11) isfamiliar up to subleading corrections in 1 /M . The first term is the Coulomb attraction. Thesecond term is the orbit-orbit, also referred to as the current-current, coupling. The thirdterm is the Darwin term. The fourth term is the spin-orbit coupling of the electron. The fifthterm is the spin orbit coupling of the muon. The last term is the hyperfine coupling betweenthe spin of the electron and the spin of the muon. For a hydrogenic orbital, the expectationvalues of h /r i and h ~p i scale as αµ . Thus, this non-relativistic expansion of the effectivepotential is also an expansion in the fine structure constant. The Coulomb interaction is oforder α µ and the other terms are suppressed by an additional power of α .8 .2 Bosonic amplitudes In SQED, an electron anti-muon bound state mixes with a selectron anti-smuon boundstate through photino exchange. To calculate the energy spectrum, there are a number ofadditional scattering diagrams that must be computed. • e − µ + → ˜ e −± ˜ µ + ± µ + e − ˜ µ + ± ˜ e −± = − ie ¯ v m ( p ′ )( /p − /q ) P ± u e ( p )( p − q ) (3.12) i M NR = ie ( ~p − ~q ) η † mi (cid:20) − ( ~p − ~q ) · ~σ √ M m ∓ M − m ( M m ) / ( ~p − ~q ) ∓ i M + m ( M m ) / ( ~p × ~q ) · ~σ (cid:21) ξ ei V = − e π ξ T mi iσ (cid:20) − i √ M m ~r · ~σr ± π M − m ( M m ) / δ (3) ( ~r ) ± M + m ( M m ) / ~L · ~σr (cid:21) ξ ei • ˜ e −± ˜ µ + ± → e − µ + ˜ µ + ± ˜ e −± µ + e − = 2 ie ¯ u e ( q )( /p − /q ) P ± v m ( q ′ )( p − q ) (3.13) i M NR = ie ( ~p − ~q ) ξ † eo (cid:20) ( ~p − ~q ) · ~σ √ M m ∓ M − m ( M m ) / ( ~p − ~q ) ∓ i M + m ( M m ) / ( ~p × ~q ) · ~σ (cid:21) η mo V = − e π ξ † eo (cid:20) − i √ M m ~r · ~σr ∓ π M − m ( M m ) / δ (3) ( ~r ) ∓ M + m ( M m ) / ~L · ~σr (cid:21) iσ ξ ∗ mo ˜ e −± ˜ µ + ± → ˜ e −± ˜ µ + ± ˜ µ + ± ˜ e −± ˜ µ + ± ˜ e −± = e ( p + q ) µ ˜ ∆ µν ( p − q )( p ′ + q ′ ) ν (3.14) ˜ µ + ± ˜ e −± ˜ µ + ± ˜ e −± = − ie (3.15) i M NR = ie ( ~p − ~q ) (cid:20) M m (cid:18) ~p + ( ~p − ~q ) − ( ~p · ( ~p − ~q )) ( ~p − ~q ) (cid:19) − M m ( ~p − ~q ) (cid:21) V = − e π (cid:20) r + 1 M m (cid:18) ~p r + ( ~r · ~p ) r + πδ (3) ( ~r ) (cid:19) − πM m δ (3) ( ~r ) (cid:21) • ˜ e −± ˜ µ + ∓ → ˜ e −± ˜ µ + ∓ ˜ µ + ∓ ˜ e −± ˜ µ + ∓ ˜ e −± = e ( p + q ) µ ˜ ∆ µν ( p − q )( p ′ + q ′ ) ν (3.16) ˜ µ + ∓ ˜ e −± ˜ µ + ∓ ˜ e −± = ie (3.17) i M NR = ie ( ~p − ~q ) (cid:20) M m (cid:18) ~p + ( ~p − ~q ) − ( ~p · ( ~p − ~q )) ( ~p − ~q ) (cid:19)(cid:21) V = − e π (cid:20) r + 1 M m (cid:18) ~p r + ( ~r · ~p ) r + πδ (3) ( ~r ) (cid:19)(cid:21) .3 Fermionic amplitudes In addition to bosonic bound states in SQED, there are fermionic bound states involving anelectron and anti-smuon or selectron and anti-muon. • e − ˜ µ + ± → e − ˜ µ + ± ˜ µ + ± e − ˜ µ + ± e − = e ¯ u e ( q ) γ µ ˜ ∆ µν ( p − q )( p ′ + q ′ ) ν u e ( p ) (3.18) i M NR = ie ( ~p − ~q ) ξ † eo (cid:20) M m (cid:18) ~p + ( ~p − ~q ) − ( ~p · ( ~p − ~q )) ( ~p − ~q ) (cid:19) − (cid:18) M m + 12 m (cid:19) ( ~p − ~q ) − i (cid:18) M m + 12 m (cid:19) ( ~p × ~q ) · ~σ (cid:21) ξ ei V = − e π ξ † eo (cid:20) r + 1 M m (cid:18) ~p r + ( ~r · ~p ) r + πδ (3) ( ~r ) (cid:19) − π (cid:18) M m + 12 m (cid:19) δ (3) ( ~r ) − (cid:18) M m + 12 m (cid:19) ~L · ~σr (cid:21) ξ ei • ˜ e −± µ + → ˜ e −± µ + µ + ˜ e −± µ + ˜ e −± = e ¯ v m ( p ′ ) γ µ ˜ ∆ µν ( p − q )( p + q ) ν v m ( q ′ ) (3.19) i M NR = ie ( ~p − ~q ) η † mi (cid:20) M m (cid:18) ~p + ( ~p − ~q ) − ( ~p · ( ~p − ~q )) ( ~p − ~q ) (cid:19) − (cid:18) M m + 12 M (cid:19) ( ~p − ~q ) + i (cid:18) M m + 12 M (cid:19) ( ~p × ~q ) · ~σ (cid:21) η mo V = − e π ξ † mo (cid:20) r + 1 M m (cid:18) ~p r + ( ~r · ~p ) r + πδ (3) ( ~r ) (cid:19) − π (cid:18) M m + 12 M (cid:19) δ (3) ( ~r ) − (cid:18) M m + 12 M (cid:19) ~L · ~σr (cid:21) ξ mi e − ˜ µ + ± → ˜ e −∓ µ + ˜ µ + ± e − µ + ˜ e −∓ = 2 ie u T e ( p ) C ( /p − /q ) P ± v m ( q ′ )( p − q ) (3.20) i M NR = ie ( ~p − ~q ) ξ T ei iσ (cid:20) ± ( ~p − ~q ) · ~σ √ M m + 18 M + m ( M m ) / ( ~p − ~q ) − i M + m ( M m ) / ( ~p × ~q ) · ~σ (cid:21) η mo V = − e π ξ † mo (cid:20) ∓ i √ M m ~r · ~σr − π M + m ( M m ) / δ (3) ( ~r ) − M + m ( M m ) / ~L · ~σr (cid:21) ξ ei • ˜ e −± µ + → e − ˜ µ + ∓ µ + ˜ e −± ˜ µ + ∓ e − = − ie ¯ u e ( q )( /p − /q ) CP ± ¯ v T m ( p ′ )( p − q ) (3.21) i M NR = ie ( ~p − ~q ) ξ † eo (cid:20) ∓ ( ~p − ~q ) · ~σ √ M m + 18 M + m ( M m ) / ( ~p − ~q ) + i M + m ( M m ) / ( ~p × ~q ) · ~σ (cid:21) iσ η ∗ mi V = − e π ξ † eo (cid:20) ∓ i √ M m ~r · ~σr − π M + m ( M m ) / δ (3) ( ~r ) − M + m ( M m ) / ~L · ~σr (cid:21) ξ mi In the previous section we have derived the non-relativistic expansions of the potential en-ergy V ( ~r, ~p ) between an electron and an anti-muon or their super partners in terms of therelative coordinate ~r and the relative momentum ~p . To find the effective quantum mechanicaldescription of this system, we also need to expand the kinetic energy of these particles E kin = p m + p e + p M + p m − ( m + M ) (4.1)to the same order, i.e. to fourth order in the momenta. Then the Hamiltonian becomes H = ~p µ − (cid:16) m + 1 M (cid:17) ~p + V ( ~r, ~p ) = ~p µ − αr + H int ( ~r, ~p ) , (4.2)where V and H int are matrices acting on the various “spin” states | s e s m i where s ∈ {↑ , ↓ , + , −} . The components of V are the several potentials given in Sec. 3. In (4.2) we havesingled out the non-relativistic kinetic energy and the Coulomb potential, and denote allother terms by H int . 12e are interested in the bound state spectrum of this system. Without the interactions H int , the solutions would be the familiar hydrogenic bound states | nlm l i with the Bohrenergies E n = − µα / n , see App. B. Our task now is to determine the α correctionsto this spectrum, which have two different sources. The first one is first order degenerateperturbation theory. Most of the terms in the scattering amplitudes are of order α and leadto mixing between the states in the highly degenerate levels of a given n and l .There are a handful of terms in the scattering amplitudes that are of order α , namelythe first terms in (3.12), (3.13), (3.20), and (3.21). Naively, these terms should dominatethe α contributions. However, it turns out that D n, l, m l (cid:12)(cid:12)(cid:12) ~rr (cid:12)(cid:12)(cid:12) n, l ′ , m ′ l E = 0 . (4.3)Thus, these terms do not contribute at the level of first order perturbation theory. However,as was noted in [6], they can and do contribute at second order. Recall the formula for theenergy corrections E (2) i = X j = i |h i | H int | j i| E (0) i − E (0) j , (4.4)where E (0) i are the eigen-energies of the bare Hamiltonian. For h i | H int | j i of order α and E (0) i of order α the second order corrections will be of order α . The sum in (4.4) shouldbe carried over both discrete and continuum states of the hydrogen atom. To carry out thesum, we will make use of Schwinger’s Coulomb Green’s function [9].At first glance, the diagonalization problem of the | nlm l i states seems formidable. For agiven n , we have n different l ’s, for each l , we have 2 l + 1 different m l ’s, and for each m l , wehave 16 different “spins” | s e s m i all of the same energy. As it turns out, states of different l do not mix. Moreover, the total angular momentum in the z direction is a good quantumnumber. The largest matrix we will need to diagonalize is 6 × z -component of angular momentum is conserved is obvious, but that statesof different l do not mix is surprising. Both the second order perturbative corrections andthe hyperfine interaction have the potential to mix an l state with an l + 2 state. For thehyperfine interaction, one can check explicitly that the overlap integral D n, l + 2 , m l (cid:12)(cid:12)(cid:12) r i r j r (cid:12)(cid:12)(cid:12) n, l, m ′ l E = 0 (4.5)vanishes. Another integral, which we discuss in Appendix E, guarantees that there is nomixing of states with different l in second order perturbation theory.In the appendices, we discuss separately the contributions from first order degenerateperturbation theory to the l = 0 and l > l = 0 stateswhile the spin-orbit interactions only contribute when l >
0. In the appendices, we also willcalculate the contribution from second order perturbation theory. Below, we present thefinal result for the mixing matrices. 13 .1 Overall shift
The relativistic correction to the kinetic energy as well as a term ∼ Mm in the potential V do not depend on the spins of the particles. Therefore, these terms lead to an overall shift ofthe levels specified by n and l . We can compute this shift independently from the splitting.It is given by the expectation values of the following terms in the | nlm l i basis: δE nl = − αM m (cid:28) ~p r + ( ~r · ~p ) r + πδ (3) ( ~r ) (cid:29) − (cid:18) m + 1 M (cid:19) h ~p i = − µα n (cid:20) n l + 1 −
38 + µ M m (cid:21) . (4.6) l = 0-states There is an eight dimensional space of bosonic bound states with l = 0: | in i ∈ n |↑↑i , |↓↑i , |↑↓i , |↓↓i , | ++ i , |−−i , | + −i , |− + i o . (4.7)(Because the l = 0 sector is already relatively small, we do not take advantage of the factthat angular momentum in the z -direction is a good quantum number.) The first entry ofthe state describes the electronic portion of the bound state, whether the electron is spinup or down, or whether the selectron comes from the + or − chiral superfield. The secondentry describes the muonic portion. The Hamiltonian to order α for these states takes theform H = E n + δE n + M b . Assembling the contributions from both first and second orderperturbation theory, the mixing matrix for these states is M b = µα n A B T B C
00 0 0 (4.8)with A = M + m ( M + m ) Mm ( M + m ) Mm ( M + m ) M + m ( M + m )
00 0 0 1 , (4.9) B = √ M m ( M − m )( M + m ) (cid:18) − − (cid:19) , (4.10) C = 2 M m ( M + m ) (cid:18) − − (cid:19) . (4.11)A curious observation is that M b = µα n M b . Note that the | + −i and |− + i states decouplefrom the other six states; it remains to diagonalize a 6 × l = 0: | in i ∈ n |↑ + i , |− ↑i , |↓ + i , |− ↓i , |↑ −i , | + ↑i , |↓ −i , | + ↓i o , (4.12)where the Hamiltonian takes the form H = E n + δE n + M f + O ( α ). The mixing matrix inthis case breaks up into a bunch of 2 × M f = µα n (cid:18) D D (cid:19) (4.13)with D = 1 M + m M √ M m √ M m m M √ M m √ M m m . (4.14)Note, for example, that the |↑ + i state mixes only with the |− ↑i state. Again we have M f = µα n M f . l > As explained above, there is no mixing between states with different l . Therefore, we fixthe orbital angular momentum to some l >
0. Furthermore, it is convenient to split thisspace, which contains 8 × (2 l + 1) bosonic states and 8 × (2 l + 1) fermionic states, intoclosed subspaces of states with given z -component, m j , of the total angular momentum.The bosonic sector of such a subspace is spanned by the states: | in i ∈ n | l m l − , ↑↑i , | l m l , ↓↑i , | l m l , ↑↓i , | l m l + 1 , ↓↓i , | l m l , ++ i , | l m l , −−i , | l m l , + −i , | l m l , − + i o . (4.15)There are 2 l + 3 such subspaces labeled by m j = m l = − l − , − l, . . . , l + 1 where j = l − , l ,or l . States in the set (4.15) with magnetic quantum number outside the range − l, ..., l areunderstood to be absent. Thus the dimensions of these subspaces are 1 , , , , . . . , , , α as H = E n + δE nl + M b .The mixing matrix takes the form M b = µα l ( l + 1)(2 l + 1) n A B T B C
00 0 0 . (4.16)In the leptonic sector it is given by A = m l − MM + m c l, − m l mM + m c l, − m l MM + m c l, − m l − M − mM + m m l mM + m c lm l mM + m c l, − m l M − mM + m m l MM + m c lm l mM + m c lm l MM + m c lm l − m l − (4.17)15nd the mixing between leptons and sleptons is given by B = √ mMM + m − c l, − m l m l m l c lm l c l, − m l − m l − m l − c lm l ! (4.18)where c lm l = p ( l − m l )( l + m l + 1). There is no interaction among the sleptons, C = 0.In the fermionic sector, the states have half-integer total magnetic quantum number m j = m l + where the range of m l is − l − , − l, . . . , l . The corresponding 2 l + 2 subspacesfor j = l ± / , , , . . . , , | in i ∈ n | l m l , ↑ + i , | l m l , − ↑i , | l m l + 1 , ↓ + i , | l m l + 1 , − ↓i , | l m l , ↑ −i , | l m l , + ↑i , | l m l + 1 , ↓ −i , | l m l + 1 , + ↓i o . (4.19)For the mixing matrix in this sector, we find M f = µα l ( l + 1)(2 l + 1) n (cid:18) D D (cid:19) (4.20)with D = MM + m m l √ MmM + m m l MM + m c lm l √ MmM + m c lm l √ MmM + m m l mM + m m l √ MmM + m c lm l mM + m c lm l MM + m c lm l √ MmM + m c lm l − MM + m ( m l + 1) − √ MmM + m ( m l + 1) √ MmM + m c lm l mM + m c lm l − √ MmM + m ( m l + 1) − mM + m ( m l + 1) . (4.21) At order α in the fine structure constant, the energy spectrum is given by the 16 n -folddegenerate Bohr levels E nlm l s e s m = E n = − µα n . (5.1)They receive a spin independent shift δE nl at order α , which we have already computed inSec. 4.1. In this section we calculate the additional splittings of these levels and find theenergy eigenstates. The splitting energies and the eigenstates are given by the eigenvaluesand eigenvectors of the mixing matrices M b and M f computed above. Because sphericallysymmetric states ( l = 0) and asymmetric ones ( l >
0) split up differently into two and threelevels, respectively (see Fig. 1 on page 5), we discuss these two cases separately.We organize the eigenstates that remain degenerate at order α into multiplets of theunderlying supersymmetry algebra[ J a , J b ] = iǫ abc J c , [ J a , Q α ] = ( σ a ) αβ Q β , [ J a , Q † α ] = − ( σ a ) αβ Q † β , (5.2)[ J a , H ] = 0 , [ Q α , H ] = 0 , { Q α , Q † β } = Hδ αβ , (5.3)16 + i|↓i − Q ✲ e − |↑i ✛ Q |−i Q ✲ ✛ Q |−i|↓i Q ✲ µ + |↑i ✛ − Q | + i − Q ✲ ✛ − Q Figure 2:
Action of supercharges.
The action of the Q α is indicated by the arrows. Additionallythere is factor of √ m or √ M when acting on electrons or muons, respectively. The action of Q † α isthe inverse of the action of Q α . where ~J = ~L + ~S e + ~S m is the total angular momentum operator, Q α , α = 1 ,
2, are thesupercharges, and H is the Hamiltonian. The action of the supercharges on states to zerothorder in α is depicted in Fig. 2. To this order they anti-commute to the rest energy m + M .We denote super multiplets by R j where j = 0 , , , . . . refers to the total SU(2) spin of thehighest submultiplet, i.e. the one whose states are annihilated by the supercharges Q † α . Interms of spin- j multiplets V j of SU(2), the super multiplet R j is built from V j − / ⊕ V j ⊕ V j +1 / for j > / V ⊕ V / for j = 0. The dimension of R j is (8 j + 4).The energy eigenstates depend on the mass ratio τ ≡ mM . l = 0-states The mixing matrices that need to be diagonalized in this case are given in (4.8) and (4.13)for the bosonic and fermionic bound states, respectively. We find that there are 4 bosonicand 4 fermionic states with eigenvalue ∆E = 0, and 4 bosonic and 4 fermionic states witheigenvalue ∆E = µα n . It turn out that the former states constitute two super multiplets R ,while the latter ones fill one R / . See the l = 0 column of Fig. 1 where these results arevisualized.The energy eigenstates in the first R with ∆E = 0 are given by V : |− + i (5.4) V / : q τ τ |↑ + i − q τ |− ↑i , q τ τ |↓ + i − q τ |− ↓i (5.5) V : √ τ τ (cid:16) |↑↓i − |↓↑i (cid:17) + τ (cid:16) τ | ++ i + |−−i (cid:17) (5.6)the ones in the second R also with ∆E = 0 are V : − √ τ τ (cid:16) |↑↓i − |↓↑i (cid:17) + τ (cid:16) | ++ i + τ |−−i (cid:17) (5.7) V / : q τ τ |↑ −i − q τ | + ↑i , q τ τ |↓ −i − q τ | + ↓i (5.8) V : | + −i (5.9)17nd the ones in R / with ∆E = µα n are V / : q τ |↑ + i + q τ τ |− ↑i , q τ |↓ + i + q τ τ |− ↓i (5.10) V : − − τ τ √ (cid:16) |↑↓i − |↓↑i (cid:17) − √ τ τ √ (cid:16) | ++ i − |−−i (cid:17) (5.11) V : |↑↑i , √ (cid:16) |↑↓i + |↓↑i (cid:17) , |↓↓i (5.12) V / : q τ |↑ −i + q τ τ | + ↑i , q τ |↓ −i + q τ τ | + ↓i (5.13) l > The relevant mixing matrices are (4.16) and (4.20). They correspond to the subsector ofstates with fixed principal quantum number n , fixed orbital angular momentum l , and fixed z -component of the total angular momentum m j . As we argued in Sec. 4, there is no mixingwith other subsectors even though sectors with different l are degenerate at order α .Super multiplets, however, can only be formed by grouping together states with all possi-ble values for m j . The reason for this is that although a sector with fixed ( n, l, m j ) is closedunder the action of the Hamiltonian, it is not closed the action of the angular momentumoperator ~J nor the supercharges Q α . These latter generators carry spin themselves, andtherefore can change the m j -value of the state they act on. Thus, we look at all 4 · · (2 l + 1)states with a given l -value at once. We find that they form two super multiplets R l with ∆E = 0, one super multiplet R l +1 / with ∆E = + µα l +1)(2 l +1) n , and one super multiplet R l − / with ∆E = − µα l (2 l +1) n .The states of the first unperturbed R l are given by V l − : √ l +1 hp l − m l (cid:16)q τ τ | m l , ↑ + i − q τ | m l , − ↑i (cid:17) − p l + m l + 1 (cid:16)q τ τ | m l + 1 , ↓ + i − q τ | m l + 1 , − ↓i (cid:17)i (5.14) V l : | m l , − + i (5.15) V l : √ τ τ (cid:16) | m l , ↑↓i − | m l , ↓↑i (cid:17) + τ (cid:16) τ | m l , ++ i + | m l , −−i (cid:17) (5.16) V l + 12 : √ l +1 hp l + m l + 1 (cid:16)q τ τ | m l , ↑ + i − q τ | m l , − ↑i (cid:17) + p l − m l (cid:16)q τ τ | m l + 1 , ↓ + i − q τ | m l + 1 , − ↓i (cid:17)i (5.17) As discussed in Sec. 4.3, the states are labeled by m l = − l − , − l, . . . , l + 1, and kets that end up witha magnetic quantum number outside the interval [ − l, l ] are defined to vanish. V l − : √ l +1 hp l − m l (cid:16)q τ τ | m l , ↑ −i − q τ | m l , + ↑i (cid:17) − p l + m l + 1 (cid:16)q τ τ | m l + 1 , ↓ −i − q τ | m l + 1 , + ↓i (cid:17)i (5.18) V l : | m l , + −i (5.19) V l : − √ τ τ (cid:16) | m l , ↑↓i − | m l , ↓↑i (cid:17) + τ (cid:16) | m l , ++ i + τ | m l , −−i (cid:17) (5.20) V l + 12 : √ l +1 hp l + m l + 1 (cid:16)q τ τ | m l , ↑ −i − q τ | m l , + ↑i (cid:17) + p l − m l (cid:16)q τ τ | m l + 1 , ↓ −i − q τ | m l + 1 , + ↓i (cid:17)i (5.21)The states in R l +1 / which receive a positive energy shift by ∆E = + µα l +1)(2 l +1) n are V l : √ l +1)(2 l +1) hp ( l + m l )( l − m l + 1) | m l − , ↑↑i− p ( l − m l )( l + m l + 1) | m l + 1 , ↓↓i− ( l + m l +1) − ( l − m l +1) τ τ | m l , ↑↓i + ( l − m l +1) − ( l + m l +1) τ τ | m l , ↓↑i− l +1) √ τ τ (cid:16) | m l , ++ i − | m l , −−i (cid:17)i (5.22)2 V l + 12 : √ l +1 hp l + m l + 1 (cid:16)q τ | m l , ↑ ±i + q τ τ | m l , ∓ ↑i (cid:17) + p l − m l (cid:16)q τ | m l + 1 , ↓ ±i + q τ τ | m l + 1 , ∓ ↓i (cid:17)i (5.23) V l +1 : √ l +1)(2 l +1) hp ( l + m l )( l + m l + 1) | m l − , ↑↑i + p ( l − m l )( l − m l + 1) | m l + 1 , ↓↓i + p ( l + m l + 1)( l − m l + 1) (cid:16) | m l , ↓↑i + | m l , ↑↓i (cid:17)i (5.24)19nd the ones in R l − / whose energy is lowered by ∆E = − µα l (2 l +1) n have the form V l − : √ l (2 l +1) hp ( l − m l )( l − m l + 1) | m l − , ↑↑i + p ( l + m l )( l + m l + 1) | m l + 1 , ↓↓i− p ( l + m l )( l − m l ) (cid:16) | m l , ↓↑i + | m l , ↑↓i (cid:17)i (5.25)2 V l − : √ l +1 hp l − m l (cid:16)q τ | m l , ↑ ±i + q τ τ | m l , ∓ ↑i (cid:17) − p l + m l + 1 (cid:16)q τ | m l + 1 , ↓ ±i + q τ τ | m l + 1 , ∓ ↓i (cid:17)i (5.26) V l : √ l (2 l +1) hp ( l + m l )( l − m l + 1) | m l − , ↑↑i− p ( l − m l )( l + m l + 1) | m l + 1 , ↓↓i + ( l − m l ) − ( l + m l ) τ τ | m l , ↑↓i − ( l + m l ) − ( l − m l ) τ τ | m l , ↓↑i + l √ τ τ (cid:16) | m l , ++ i − | m l , −−i (cid:17)i (5.27) A comprehensive summary of the results of our computation is given at the end of theintroduction in Sec. 1.1. Here we discuss some consequences and applications thereof.
Oscillations
Because of the energy splitting, there is an oscillation between different “fla-vors”. Say we prepare an atom in the flavor state | ++ i with l = 0, then it can oscillate into |−−i and √ (cid:0) |↑↓i − |↓↑i (cid:1) . The probabilities of finding the atom in one of these states attime t after the atom was purely | ++ i are P | ++ i = 1 + 6 τ + τ + 4 τ (1 + τ ) cos ∆E t (1 + τ ) (6.1) P |−−i = 8 τ (1 − cos ∆E t )(1 + τ ) (6.2) P |↑↓−↓↑i = 4 τ (1 − τ ) (1 − cos ∆Et )(1 + τ ) (6.3)where ∆E = µα n . If we plug in the actual mass of the electron m ≈ . M ≈
100 MeV, then ∆E ≈ . n = 1). In the limit m ≪ M , theoscillations have a frequency of ω ≈ Hz (cid:16) m
500 keV (cid:17) (137 · α ) . (6.4)20 upersymmetry We can write the mixing matrices in terms of a superpotential W as M b = W † W and M f = W W † . The matrix W can easily be constructed from the eigenstatesgiven in Sec. 5 as follows. Let Λ = diag( E , . . . , E ) be the eigenvalues of M b and M f in somefixed order, and let V and U be matrices whose columns are the corresponding eigenvectors.Then the superpotential is given by W = U √ Λ V † . If we pair up the eigenvectors in U and V appropriately, we can set W = Q + Q † or W = Q + Q † . The prerequisite that all E i ≥ l = 0 sector, and in the other sectors we can achieve this requirement byadding the identity matrix times the smallest eigenvalue to the mixing matrices. Bose condensate
Ignoring interactions between these hydrogenic atoms, including anyinstability to form molecules, what happens if we place a large number of these atoms in abox? If the atoms are fermionic, then only one fermionic atom can rest in the single particleground state of the box. (More generally, a small but finite number of fermionic atoms canrest in the ground state if the ground state has a small but finite degeneracy.) In contrast,there is no limit to the number of bosonic atoms that can exist in the single particle groundstate. Through emission of a photino, a fermionic atom can convert into a bosonic atom.Given our assumptions about the absence of interactions, the multiparticle ground statewill contain at most one fermionic atom. There should be no Fermi sea for these “perfectatoms”. It would be interesting to see what changes if any occur to this qualitative picturewhen interactions between the atoms are considered.
Supersymmetric chemistry
For bound states of higher charge nuclei and more than oneelectron, the Pauli exclusion principle will play a much weaker role then it does in traditionalatomic physics. An electron in an excited orbital can always reduce its interaction energywith the nucleus by converting into a selectron and moving into a lower orbital at the possibleprice of increasing its interaction energy with other orbiting selectrons. A supersymmetricperiodic table should look quite different from the periodic table we are used to. There maybe additional interesting effects related to these atoms’ ability to form molecules. We leavea study of such effects for the future.
Acknowledgments
We would like to thank Silviu Pufu, Stefan Stricker, Aleksi Vuorinen, and Lian-Tao Wangfor discussions. We also thank Tomas Rube and Jay Wacker, who independently calculatedthe spectrum of supersymmetric hydrogen [10], for bringing our attention to a mistake in anearlier version of this manuscript. The work of C.H. was supported in part by the US NSFunder Grant Nos. PHY-0756966 and PHY-0844827.
A Notation and conventions
We use the metric η µν = diag( − , + , + , +) and the Levi-Civita symbol ǫ = − ǫ = 1.Implicit contractions of two-dimensional spinor indices are defined as ψχ ≡ ψ α χ α , ¯ ψ ¯ χ ≡ ¯ ψ ˙ α ¯ χ ˙ α , and complex conjugation acts as ( ψ α ) † = ¯ ψ ˙ α , ( ψ α ) † = ¯ ψ ˙ α , ( ψ α χ α ) † = ¯ χ ˙ α ¯ ψ ˙ α . We21aise and lower spinor indices from the left: ψ α = ǫ αβ ψ β , ψ α = ǫ αβ ψ β . We employ the Paulimatrices σ µα ˙ α = ( − , ~σ ) , ¯ σ µ ˙ αα = ǫ ˙ α ˙ β ǫ αβ σ µβ ˙ β = ( − , − ~σ ) (A.1)to define the Dirac matrices as γ µ = (cid:18) σ µ ¯ σ µ (cid:19) , γ = γ γ γ γ = (cid:18) − i i (cid:19) . (A.2)The chiral projectors P ± = ( ± iγ ) have the matrix representation P + = diag(1 ,
0) and P − = diag(0 , σ µ ¯ σ ν + σ ν ¯ σ µ ) αβ = − η µν δ αβ ,(¯ σ µ σ ν + ¯ σ ν σ µ ) ˙ α ˙ β = − η µν δ ˙ α ˙ β , { γ µ , γ ν } = − η µν . Superfields and components
The vector superfield V = − θσ µ ¯ θ A µ ( x ) + i θ ¯ θ ¯ χ ( x ) − i ¯ θ θχ ( x ) + θ ¯ θ D ( x ) contains the photon A µ , the gaugino χ , and the auxiliary real scalar D . The chiral superfields Φ ± = φ ± ( y ) + √ θψ ± ( y ) + θ F ± ( y ) (where y µ = x µ + iθσ µ ¯ θ )contain the slepton φ ± , the leptons ψ ± , and the auxiliary complex scalars F ± . We introducea Dirac spinor for the leptons ψ = (cid:18) ψ + α ¯ ψ ˙ α − (cid:19) , ¯ ψ = ψ † γ = (cid:0) − ψ α − − ¯ ψ + ˙ α (cid:1) , (A.3)and a Majorana spinor for the photino λ = (cid:18) χ α ¯ χ ˙ α (cid:19) , ¯ λ = λ † γ = (cid:0) − χ α − ¯ χ ˙ α (cid:1) , (A.4)subject to the condition λ = λ C ≡ C ¯ λ T with the charge conjugation matrix C = iγ γ = − C T = − C † = C ∗ = − C − = diag( iσ , − iσ ). The sign in the gauge covariant derivative D µ X = ∂ µ X + iqA µ X is determined by the U(1) charge q of X . The field strength is F µν = ∂ µ A ν − ∂ ν A µ . B Bound states in Coulomb potential
The free Hamiltonian is given by H = ~p µ − αr . We use the constants µ = mMm + M for thereduced mass, α = e π for the fine structure constant, and a B = µα for the Bohr radius. Thewave functions ψ nlm ( ~r ) = h ~r | nlm i for the bound states are given by ψ nlm ( ~r ) = 1 p a B n s ( n − l − n + l )! (cid:18) rna B (cid:19) l L l +1 n − l − (cid:16) rna B (cid:17) exp (cid:16) − rna B (cid:17) Y lm ( θ, ϕ ) . (B.1)Their energies are E n = − µα n . In Mathematica one has to write L ba ( x ) = LaguerreL [ a , b , x ]and Y lm ( θ, ϕ ) = SphericalHarmonicY [ l , m , θ, ϕ ].22 xpectation values h nlm | r | nlm i = µαn , h nlm | r | nlm i = µ α n ( l + ) , h n | r i r j r | n i = µαδ ij n (B.2) h nlm | ~p r | nlm i = µ α n (cid:20) nl + − (cid:21) , h nlm | ~p | nlm i = µ α n (cid:20) nl + − (cid:21) (B.3) h n ′ l ′ m ′ | δ (3) ( ~r ) | nlm i = µ α πn δ nn ′ δ l δ l ′ δ m δ m ′ (B.4) h nl ′ m ′ | r | nlm i = µ α n l ( l + )( l + 1) δ ll ′ δ mm ′ for l, l ′ > h nl ′ m ′ | ( ~r · ~p ) r + 2 πδ (3) ( ~r ) | nlm i = µ α n (cid:20) nl + − (cid:21) δ ll ′ δ mm ′ (B.6) Angular integrals
Define components of the unit position vector ˆ ~r as ˆ r ≡ zr = cos θ andˆ r ± ≡ x ± iyr = sin θe ± iϕ . h l ′ m ′ | ˆ r | lm i = s ( l ′ + m ′ )( l ′ − m ′ )(2 l ′ + 1)(2 l + 1) δ l ′ ,l +1 δ m ′ ,m + s ( l + m )( l − m )(2 l ′ + 1)(2 l + 1) δ l ′ ,l − δ m ′ ,m (B.7) h l ′ m ′ | ˆ r ± | lm i = ∓ s ( l ′ ± m ′ − l ′ ± m ′ )(2 l ′ + 1)(2 l + 1) δ l ′ ,l +1 δ m ′ ,m ± ± s ( l ∓ m − l ∓ m )(2 l ′ + 1)(2 l + 1) δ l ′ ,l − δ m ′ ,m ± (B.8) h lm | ˆ r ˆ r | lm i = 2 l ( l + 1) − m − l + 3)(2 l − , h lm | ˆ r + ˆ r − | lm i = 2 l ( l + 1) + 2 m − l + 3)(2 l − , (B.9) h l m + 1 | ˆ r + ˆ r | lm i = − (2 m + 1) c lm (2 l + 3)(2 l − , h l m + 1 | ˆ r + ˆ r + | l m − i = − c lm c l, − m (2 l + 3)(2 l − , (B.10)where here and below we use c lm ≡ p ( l − m )( l + m + 1) . (B.11) C Feynman Rules
Fields and particles
Field φ + φ † + φ − φ †− ψ + ¯ ψ + ψ − ¯ ψ − χ ¯ χ creates ˜ e ++ ˜ e − + ˜ e −− ˜ e + − e ++ e − + e −− e + − ˜ γ + ˜ γ − annihilates ˜ e − + ˜ e ++ ˜ e + − ˜ e −− e − + e ++ e + − e −− ˜ γ − ˜ γ + The U(1) e charges of the particles are encoded in the labels as Q (˜ e qh ) = q , Q ( e qh ) = q , Q (˜ γ h ) = 0, and the U(1) R are given by R (˜ e qh ) = qh , R ( e qh ) = 0, R (˜ γ h ) = h .23 ropagators The momentum p flows from y to x . For photons in Coulomb gauge h A µ ( x ) A ν ( y ) i → ˜ ∆ ( p ) = i~p , ˜ ∆ i ( p ) = 0 , ˜ ∆ ij ( p ) = − ip (cid:18) δ ij − p i p j ~p (cid:19) . (C.1)Coulomb gauge is better suited for taking a non-relativistic limit of the SQED amplitudes.For a discussion of the complications that arise when combining Feynman gauge with anon-relativistic limit, see for example [8]. For photinos h λ α ( x )¯ λ β ( y ) i → ˜ ∆ αβ ( p ) = (cid:18) − i/p + iε (cid:19) αβ = i/p αβ p − iε . (C.2) Electron wave functions ψ ( x ) = Z d p (2 π ) p E ~p h b s~p u s ( p ) e ipx + d s † ~p v s ( p ) e − ipx i (C.3)Contractions with external particles are given by h e − ( p, s ) | ¯ ψ = ¯ u s ( p ) e − ipx ψ | e − ( p, s ) i = u s ( p ) e ipx (C.4) h e + ( p, s ) | ψ = v s ( p ) e − ipx ¯ ψ | e + ( p, s ) i = ¯ v s ( p ) e ipx (C.5)It is useful to know the identities √ σ · p = σ · p + m p p + m ) , √ ¯ σ · p = ¯ σ · p + m p p + m ) . (C.6)In the quantum mechanical setting we convert the spinor η for an anti-particle into a spinorfor a particle using the relation η = Cξ ∗ where the charge conjugation matrix is C = iσ . D Degenerate Perturbation Theory
D.1 l = 0 The only non-zero matrix elements of the interaction Hamiltonian H deg = H int + ~p (cid:18) m + 1 M (cid:19) + e π ~p M mr , (D.1)for s-wave states | n s e s m i are h n ξ eo ξ mo | H deg | n ξ ei ξ mi i = µ α n ξ † mo ξ † eo (cid:20) (cid:18) m + 1 M (cid:19) + 23 M m ~σ e · ~σ m (cid:21) ξ ei ξ mi , (D.2) h n ±±| H deg | n ξ ei ξ mi i = ∓ µ α n M − m ( M m ) / ξ T mi iσ ξ ei , (D.3) h n ξ eo ξ mo | H deg | n ±±i = ± µ α n M − m ( M m ) / ξ † eo iσ ξ ∗ mo , (D.4) h n ±±| H deg | n ±±i = µ α n M m (D.5)24or bosonic atoms, and h n ξ eo ±| H deg | n ξ ei ±i = µ α n (cid:18) M m + 12 m (cid:19) ξ † eo ξ ei , (D.6) h n ± ξ mo | H deg | n ± ξ mi i = µ α n (cid:18) M m + 12 M (cid:19) ξ † mo ξ mi , (D.7) h n ∓ ξ mo | H deg | n ξ ei ±i = 12 µ α n M + m ( M m ) / ξ † mo ξ ei , (D.8) h n ξ eo ∓| H deg | n ± ξ mi i = 12 µ α n M + m ( M m ) / ξ † eo ξ mi (D.9)for fermionic ones. D.2 l > States of different n and l do not mix, and we fix n and l . We begin with the bosonic states(4.15). The only non-zero matrix elements of the interaction Hamiltonian are h nlm ′ l ξ eo ξ mo | H deg | nlm l ξ ei ξ mi i = C (cid:28) lm ′ l (cid:12)(cid:12)(cid:12)(cid:12) ξ † mo ξ † eo (cid:18) ~L · ~σ e (cid:18) m + 1 M m (cid:19) + (D.10)+ ~L · ~σ m (cid:18) M + 1 M m (cid:19) ++ 12
M m (3ˆ r · ~σ e ˆ r · ~σ m − ~σ e · ~σ m ) (cid:19) ξ ei ξ mi (cid:12)(cid:12)(cid:12)(cid:12) lm l (cid:29) , h nlm ′ l ±±| H deg | nlm l ξ ei ξ mi i = ∓ C M + m ( M m ) / D lm ′ l (cid:12)(cid:12)(cid:12) ~L (cid:12)(cid:12)(cid:12) lm l E · (cid:0) ξ T mi iσ ~σξ ei (cid:1) , (D.11) h nlm ′ l ξ eo ξ mo | H deg | nlm l ±±i = ± C M + m ( M m ) / D lm ′ l (cid:12)(cid:12)(cid:12) ~L (cid:12)(cid:12)(cid:12) lm l E · (cid:0) ξ † eo ~σ iσ ξ ∗ mo (cid:1) , (D.12)where C ≡ µ α n l ( l + )( l + 1) . (D.13)The nonzero matrix elements of the interaction Hamiltonian for the fermionic states (4.19)are h nlm ′ l ξ eo ±| H deg | nlm l ξ ei ±i = C (cid:18) M m + 12 m (cid:19) h lm ′ l ξ eo ±| ~L · ~σ | lm l ξ ei ±i , (D.14) h nlm ′ l ± ξ mo | H deg | nlm l ± ξ mi i = C (cid:18) M m + 12 M (cid:19) h lm ′ l ± ξ mo | ~L · ~σ | lm l ± ξ mi i , (D.15) h nlm ′ l ∓ ξ mo | H deg | nlm l ξ ei ±i = C M + m ( M m ) / h lm ′ l | ~L | lm l i · ξ † mo ~σξ ei , (D.16) h nlm ′ l ξ eo ∓| H deg | nlm l ± ξ mi i = C M + m ( M m ) / h lm ′ l | ~L | lm l i · ξ † eo ~σξ mi . (D.17)25hus to determine the energy corrections from first order degenerate perturbation theory,we need the matrix elements of ~L · ~σ , ~σ e · ~σ m , and ˆ r · ~σ e ˆ r · ~σ m : h ~L · ~σ i | m l ↑i | m l + 1 ↓ih m l ↑ | m l c lm l h m l + 1 ↓ | c lm l − ( m l + 1) , (D.18) h ~σ e · ~σ m i | m l − ↑↑i | m l ↓↑i | m l ↑↓i | m l + 1 ↓↓ih m l − ↑↑ | h m l ↓↑ | − h m l ↑↓ | − h m l + 1 ↓↓ | , (D.19) h ˆ r · ~σ e ˆ r · ~σ m i | m l − ↑↑i | m l ↓↑i | m l ↑↓i | m l + 1 ↓↓ih m l − ↑↑ | cos θ sin θ cos θ e − iφ sin θ cos θ e − iφ sin θ e − iφ h m l ↓↑ | sin θ cos θ e iφ − cos θ sin θ − sin θ cos θ e − iφ h m l ↑↓ | sin θ cos θ e iφ sin θ − cos θ − sin θ cos θ e − iφ h m l + 1 ↓↓ | sin θ e iφ − sin θ cos θ e iφ − sin θ cos θ e iφ cos θ . (D.20)In evaluating this last matrix, the integrals (B.9–B.10) are useful. We also find ±h iσ ~L · ~σ i | m l − ↑↑i | m l ↓↑i | m l ↑↓i | m l + 1 ↓↓ih m l + + | c l, − m l − m l − m l − c lm l h m l − −| − c l, − m l m l m l c lm l . (D.21) E Second Order Perturbation Theory
We are interested in computing the correction to the energy of a state | nlm, s e s m i at secondorder in perturbation theory. (For ease of notation, we remove the subscript l from m l inthis subsection and replace m with m e .) These second order corrections will not mix statesof different n and l , n because the energies are different and l because of the vanishing of anintegral we discuss below. We need to compute the matrix ∆E ( n, l, m, m ′ , s e , s ′ e , s m , s ′ m ) = X i ′ h nlm, s e s m | H int | i ih i | H int | nlm ′ , s ′ e s ′ m i E n − E i , (E.1)where the ′ on the sum means we should omit the states with E i = E n . (We can omit thesestates in the sum because of eq. (4.3).) This sum involves both the discrete and continuumhydrogenic states.The Hermitian matrix ∆E has a block diagonal form. The four bosonic states | m − ↑↑i , | m ↓↑i , | m ↑↓i , | m + 1 ↓↓i (E.2)mix among themselves according to2 ∆E − + ( m − − ∆E − ( m ) − ∆E − ( m ) − ∆E −− ( m + 1) − ∆E ( m − ∆E ( m ) ∆E ( m ) ∆E − ( m + 1) − ∆E ( m − ∆E ( m ) ∆E ( m ) ∆E − ( m + 1) − ∆E ++ ( m − ∆E +0 ( m ) ∆E +0 ( m ) ∆E + − ( m + 1) . (E.3)26he factor of two comes from the sum over scalar intermediate states ++ and −− . We willdefine ∆E ij ( m ) presently. The coefficients of ∆E vanish for the other four bosonic states | m, ++ i , | m, −−i , | m, + −i , | m, − + i , (E.4)provided l >
0. In the special case l = 0, the states | m, ++ i , | m, −−i mix as2 (cid:18) ∆E r ∆E r ∆E r ∆E r (cid:19) (E.5)where ∆E r = ∆E + ( ∆E + − + ∆E − + ) /
2. For the eight fermionic states, ∆E reduces tofour 2 × | m, ↑ ±i , | m + 1 , ↓ ±i and the states | m, ± ↑i , | m + 1 , ± ↓i eachhave the same second order mixing matrix (cid:18) ∆E ( m ) + ∆E − + ( m ) ∆E − ( m + 1) − ∆E − ( m + 1) ∆E ( m ) − ∆E +0 ( m ) ∆E ( m + 1) + ∆E + − ( m + 1) (cid:19) . (E.6)The orbital part of the interaction that contributes at second order in perturbation theoryhas the schematic form ( p − q ) i / ( ~p − ~q ) . We will work in a basis where p = p z and p ± = ( p x ± ip y ).To compute the sum (E.1), we make use of a beautiful result of Schwinger [9] for theCoulomb Green’s function in momentum space: (cid:18) E − ~p µ (cid:19) G ( ~p, ~p ′ ; E ) + α π Z d ~p ′′ ( ~p − ~p ′′ ) G ( ~p ′′ , ~p ′ ; E ) = δ (3) ( ~p − ~p ′ ) . (E.7)Introducing a p such that E = − p / µ , the construction makes use of a mapping of thefour momentum ( p , ~p ) to an S : ~ξ = − p ~p + p ~p , ξ = p − ~p p + ~p , ~ξ + ξ = 1 . (E.8)Let Ω = ( ψ, θ, φ ) be a set of angular coordinates on the S . The matrix elements in theHamiltonian at second order in perturbation theory can be built from the expression: ∆E ij ( m ) = 4 π α M m e Z d ~p d ~p ′ d ~q d ~q ′ (2 π ) φ ∗ n,l,m + i + j ( p ) ( p − p ′ ) i ( ~p − ~p ′ ) G ( p ′ , q ′ ; E ) ( q − q ′ ) j ( ~q − ~q ′ ) φ nlm ( q ) . (E.9)where G ( p ′ , q ′ ; E ) = − µp ( p + ~p ′ ) ( p + ~q ′ ) X n ′ l ′ m ′ Y n ′ l ′ m ′ (Ω p ′ ) Y ∗ n ′ l ′ m ′ (Ω q ′ )1 − αµ/n ′ p . (E.10)The conserved energy is E = − µα / n . The φ nlm are the hydrogen wave functions inmomentum space: φ nlm ( p ) = 4( p ) / ( ~p + p ) Y nlm (Ω) ; Y nlm (Ω) = Z nl ( ψ ) Y lm ( θ, ϕ ) . (E.11)27he Y nlm (Ω) are spherical harmonics on the momentum S . The Y nlm and Y lm satisfy similarorthonormality and completeness relations. More explicitly, Z nl ( ψ ) = N nl sin l ψ C l +1 n − l − (cos ψ ) , (E.12)where the C ln ( x ) are the Gegenbauer polynomials and N nl = (cid:20) n ( n − l − n + l )! 2 l +1 ( l !) π (cid:21) / . (E.13)Making use of the relation14 π ξ − ξ ′ ) = X nlm n Y nlm (Ω) Y ∗ nlm (Ω ′ ) , (E.14)the expression (E.9) simplifies to ∆E ij ( m ) = − µ α M m e n X n ′ l ′ m ′ n ′ n ′ − n S n ′ l ′ m ′ ,m + i + ji S n ′ l ′ m ′ ,mj ∗ , (E.15)where S n ′ l ′ m ′ ,mi = n ′ − n nn ′ Z d Ω Y ∗ nlm (Ω) ξ i ξ Y n ′ l ′ m ′ (Ω) . (E.16)To perform this integral, we use the definition of the Y nlm (Ω): S n ′ l ′ m ′ ,mi = n ′ − n nn ′ h lm | ˆ r i | l ′ m ′ i Z Z ∗ nl ( ψ ) sin ψ ψ Z n ′ l ′ ( ψ ) dψ . (E.17)The integral h lm | ˆ r i | l ′ m ′ i vanishes unless l = l ′ ± ∆E ij ( m ) = − µ α M m e n X n ′ l ′ [ c − δ l ′ ,l − + c + δ l ′ ,l +1 ] n ′ − n n ′ (cid:12)(cid:12)(cid:12)(cid:12)Z Z ∗ nl ( ψ ) sin ψ ψ Z n ′ l ′ ( ψ ) dψ (cid:12)(cid:12)(cid:12)(cid:12) , (E.18)where c ± depends on i , j , l , and m but not on n ′ , l ′ , or m ′ .Let’s define I ( n, n ′ , l, l ′ ) ≡ Z π (sin ψ ) l + l ′ +3 ψ C l ′ +1 n ′ − l ′ − (cos ψ ) C l +1 n − l − (cos ψ ) dψ . (E.19)For positive integers n and n ′ , we find that | I ( n, n ′ , l, l + 1) | = (cid:16) π ( n + l )!2 l +1 l !( l +1)!( n − l − (cid:17) , n < n ′ , , n > n ′ . (E.20)28iven these results, we can evaluate the sum for l > ∆E ij ( m ) = − µ α M m e n " c − n ( n − l − n + l )! n − X n ′ =1 ( n ′ − n ) ( n ′ + l − n ′ − l )!+ c + n ( n + l )!( n − l − ∞ X n ′ = n +1 ( n ′ − n ) ( n ′ − l − n ′ + l + 1)! = − µ α M m e n (cid:20) c + l + 1)(2 l + 2) − c − l (2 l + 1) (cid:21) . (E.21)(If l = 0, the coefficient of c − will vanish because the state l ′ = l − I , we deduce that there is no mixing between l and l + 2 states atsecond order in perturbation theory. Note that I ( n, n ′ , l, l ′ ) = I ( n ′ , n, l ′ , l ). To get mixingbetween these states we need some amplitude to scatter from an nl state to an n ′ , l + 1 stateand back to a n, l + 2 state. In other words, the product I ( n, n ′ , l, l + 1) I ( n ′ , n, l + 1 , l + 2)should not vanish. However, I ( n ′ , n, l + 1 , l + 2) will vanish unless n ′ < n while I ( n, n ′ , l, l + 1)will vanish unless n < n ′ .To evaluate the ∆E ij completely, we need l ′ X m ′ = − l ′ h lm | ˆ r i | l ′ m ′ ih l ′ m ′ | ˆ r i | lm i = 12 l + 1 h l δ l ′ ,l − + ( l + 1) δ l ′ ,l +1 i , (E.22) l ′ X m ′ = − l ′ h lm | ˆ z | l ′ m ′ ih l ′ m ′ | ˆ z | lm i = 12 l + 1 (cid:20) l − m l − δ l ′ ,l − + ( l + 1) − m l + 3 δ l ′ ,l +1 (cid:21) , (E.23) l ′ X m ′ = − l ′ h l, m ± | ˆ r ± | l ′ m ′ ih l ′ m ′ | ˆ r ∓ | l, m ± i == ( l ± m )( l ± m + 1)(2 l + 1)(2 l − δ l ′ ,l − + ( l ∓ m )( l ∓ m + 1)(2 l + 1)(2 l + 3) δ l ′ ,l +1 , (E.24) l ′ X m ′ = − l ′ h lm | ˆ z | l ′ m ′ ih l ′ m ′ | ˆ r ∓ | l, m ± i = l ′ X m ′ = − l ′ h l, m ± | ˆ r ± | l ′ m ′ ih l ′ m ′ | ˆ z | lm i = c l, ± m l + 1 " ∓ l ± m l − δ l ′ ,l − ± l ∓ m + 12 l + 3 δ l ′ ,l +1 , (E.25) l ′ X m ′ = − l ′ h l, m ± | ˆ r ± | l ′ m ′ ih l ′ m ′ | ˆ r ± | l, m ∓ i = − c lm c l, − m l + 1 " δ l ′ ,l − l − δ l ′ ,l +1 l + 3 . (E.26)The coefficients of δ l ′ ,l − and δ l ′ ,l +1 are c − and c + respectively. Finally, we find ∆E ( m ) = ( − l ( l + 1) + 3 m ) N , (E.27) ∆E ∓± ( m ) = ( l ( l + 1) − m ± (2 l − l + 3) m ) N , (E.28) ∆E ± ( m ∓
1) = ∆E ∓ ( m ) = ( ∓ l ( l + 1) + 3 m ) c l, ∓ m N , (E.29) ∆E ++ ( m −
1) = ∆E −− ( m + 1) = 3 c lm c l, − m N , (E.30)29here N = − µ α M m e n l ( l + 1)(2 l − l + 1)(2 l + 3) . (E.31)In the special case l = 0, we find that ∆E ±∓ (0) = 2 ∆E (0) = − µ α M m e n
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