On hydrogenlike bound states in \( \mathcal{N} \) = 4 super YangMills
JHE
= 4 super
0 Box 516 , SE75120 Uppsala , Sweden
1 Yusuke Sakata
2 Department of Physics and Astronomy, Uppsala University
3 Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo
Using relativistic quantum mechanics, we study the spectrum of a nonBPS twoparticle bound state in the massive phase of N = 4 super YangMills, in the limit when one of the particles is infinitely heavier than the other. We find that the spectrum shows the exact n2 degeneracy for each principal quantum number n, just as in the strict nonrelativistic limit. This is in line with the findings of CaronHuot and Henn, who studied the same system in the large N limit with the technique of integrability and the dual conformal
Extended Supersymmetry; Supersymmetric Gauge Theory; Supersymmetry

HJEP09(217)5
YangMills
symmetry.
Breaking
ArXiv ePrint: 1706.07587
1 Introduction
2
3
4
5
Relativistic quantummechanical analysis
Extracting the KleinGordon equation
Comments on further corrections
Discussions
A Structure of massive nonBPS bound state
1
Introduction
The quantummechanical spectrum of two particles bound by the Coulomb force
H =
p~
2µ − r
α
,
shows an n2 degeneracy for each principal quantum number n. This can be understood from
the existence of an accidental conserved charge [1] which extends the rotational symmetry
so(3) to so(4). One way to see this is as follows. The states at a given principal quantum
number n transform as
V0 ⊕ V1 ⊕ · · · ⊕ Vn = Vn/2 ⊗ Vn/2
where Vl is the spinl representation of so(3). The right hand side has a natural action of
the extended so(4) ≃ su(2) × su(2) symmetry, where the first su(2) acts on the first Vn/2
and the second acts on the second Vn/2. For more details, see e.g. the extensive review
article [2] and references therein. Of course this extended symmetry is broken in the real
world, due to various corrections. For a detailed account of these corrections, see e.g. [3].
Recently, it was shown by CaronHuot and Henn [4] that in N = 4 super YangMills
theory, there is a twoparticle bound state where relativistic corrections still preserve the
extended so(4) symmetry. The analysis there was carried out using sophisticated techniques
of integrability of the N = 4 theory, developed during the last decade and a half.
CaronHuot and Henn interpreted their results as a manifestation of the dual conformal
symmetry in the massive phase of the theory in the large N limit. Let us comment on this
point. The N = 4 super YangMills theory in the massless phase is long established to be
symmetric under the superconformal group acting on the spacetime xμ and its
superpartpμ and its superpartners, see e.g. the textbook [
5
] and references therein. This dual
superconformal symmetry has been generalized to some amplitudes in the massive phase [
6–9
].
Turning back to the hydrogen atom, already in 1935, Fock realized that the so(4)
symmetry of the hydrogen spectrum is a subgroup of the conformal transformation acting
on the momentum vector p~ ∈ R3 [2, 10]. More specifically, he mapped the momentum space
R3 by an inverse stereographic projection to S3, and showed that the Coulomb interaction
written in this manner is invariant under the natural so(4) action on this S3. Therefore
it is indeed natural to attribute this so(4) symmetry to be the unbroken part of the dual
conformal symmetry so(4, 2) in the presence of the rest mass of the bound state.
In this note, we consider the same bound state in the same N = 4 super YangMills
theory, from a totally elementary point of view. Our analysis goes as follows. First,
we consider the standard KleinGordon equation of a particle in the Coulomb potential,
modified to include the effect of the massless scalar exchange. We show that it can be easily
solved exactly, and that it preserves the n
2 degeneracy. Second, we perform a schematic
analysis of Feynman diagrams contributing to the twoparticle bound states, and show
that in the limit where one of the particles is infinitely heavier, the spectrum can indeed
be found by solving the modified KleinGordon equation discussed in the first part.
Our analysis only scratches the surface of the work done by CaronHuot and Henn:
in their work the same system was studied in the large N limit at arbitrary coupling
including full effects of quantum field theory, while in this note we only treat it at the
level of relativistic quantum mechanics, without taking the large N limit. Our intention
is purely educational: our elementary analysis would be understandable to anyone with
a basic knowledge of quantum field theory, and would not require the mastery of the
techniques of integrability of N = 4 super YangMills. Hopefully this short note would
serve as an introduction to this interesting work of CaronHuot and Henn.
Before proceeding, let us mention that N = 1 and N = 2 supersymmetric analogues of
hydrogen atoms have been considered since the 80’s [
11, 12
]. More recently, the N = 1
version was revisited in [13, 14]. They have found that the split of the degeneracy due to
the relativistic correction is milder with more supersymmetry. The result of [4] and ours can
be thought of as a confirmation of the continuation of this trend to N = 4 supersymmetry.1
The rest of the note is organized as follows. In section 2, we study the KleinGordon
equation of a particle coupled to both the Coulomb field and to the classical massless scalar
field, which captures the main modification due to the N = 4 supersymmetry. The solution
shows the n2 degeneracy. In section 3, we show that this modified KleinGordon equation
does arise from the N = 4 super YangMills. In section 4, we briefly comment on further
corrections to the spectrum. Finally in section 5, we conclude with a brief discussion of
further directions of research.
1We also note that there is an approach trying to find a supersymmetric structure in the relativistic
Hamiltonian of the standard hydrogen atom [15].
– 2 –
In the nonrelativistic limit, we have
En,l = m 1 − 2n2 − 2n4 l + 1/2 − 4
+ · · ·
α
2
α
4
n
3
which clearly shows the relativistic splitting of the n2 degeneracy.
The main effect of having N = 4 supersymmetry is that i) the mass of a charged particle
is given by a vacuum expectation value (vev) of another scalar φ˜ which is a superpartner of
the photon, and that ii) this massless scalar is also exchanged between the charged particles.
When we consider two particles of like charges, the attractive potential coming from the
massless scalar exactly cancels the repulsive potential coming from the photon. Here we
consider two particles of unlike charges, for which the KleinGordon equation now reads
h−(∂t − iAt)2 + △ − φ˜2i φ = 0,
At =
α
r
where the form of φ˜ is chosen so that the particle φ has mass m and the leading 1/r
potential exactly cancels in the case of like charges. A slight rewrite using ∂t = −iE gives
whose eigenvalues are easily seen to be
(m + E)
m − E − r
2α
− △ φ = 0
2α2
En,l = m 1 − n2 + α2
= m 1 − n2 +
2α2
2α4
n4 + · · ·
Let us first consider the KleinGordon equation of a charged particle in a Coulomb field.
It is given by
−(∂t − iAt)2 + △ − m2 φ = 0,
At = α/r.
The energy eigenvalues can be obtained by the standard separation of variables (see e.g.
Chap. 21 of [16]); one obtains
En,l = m
which indeed shows the complete n2 degeneracy kept intact.
3
Extracting the KleinGordon equation
Now we would like to show that the modified KleinGordon equation (2.4) actually governs
the spectrum of a bound state in the N = 4 super YangMills in a suitable limit.
The Lagrangian of the N = 4 U(N ) YangMills was originally constructed in [17] and
its bosonic part has the following wellknown form
L = tr
(
1
− 4 Fμν F μν
1
− 2
DμΦiDμΦi +
X [Φi, Φj ]2
g
We are interested in the limit me/mp → 0.
(anti)BPS vector multiplets have the following content
VBPS ≃ VantiBPS ≃ (1, 5)φ ⊕ (2, 4)ψ ⊕ (3, 1)A.
Due to the cancellation of the exchanges of the massless scalar and vector, the ‘electron’
and the ‘antiproton’ do not bind, while the ‘electron’ and the ‘proton’ form a bound state.
This is the bound state we want to analyze.
The vev (3.2) breaks the su(4) ≃ so(6) Rsymmetry to usp(4) ≃ so(5). The massive
where each field is an N × N matrix and we use the convention that the coupling constant
enters in the definition of the covariant derivative.
We give a nonzero vev to Φ1 of the form
hΦ1i = diag(m1, . . . , mN )
while we keep all other scalars Φ2,...,6 to be zero. The (i, j)component of the fields obtains
the mass ∝ mi − mj , while the diagonal components remain massless.
We can take
m1 : m2 : m3 ∼ 0 : 1 : 2000 say, and call the (1, 2) component to be the ‘electron’ while
(3, 1) component to be the ‘proton’. In the following we use the notation
where ~k is the momentum transfer. The first two amplitudes are insensitive to the
Rsymmetry representations, while the third one from the direct fourparticle interaction
depends on it. In the representation 14 we find that it is given by
iM1 ≃ iM2 ≃
4mempg2
~k2
iM3 = −2g2.
– 4 –
under the little group so(3) times the unbroken Rsymmetry group so(5). Then, the states
with the principal quantum number n and the orbital angular momentum l transform as
Vl ⊗ VBPS ⊗ VantiBPS
where Vl is the spinl representation of so(3). For more details, see appendix A, where we
review the structure of general massive nonBPS states of N = 4 supersymmetry.
We note that in the massive nonBPS multiplet (3.5), the symmetrictraceless
representation 14 of so(5) appears exactly once, in the scalarscalar bound state. Since every
state in this nonBPS multiplet has the same energy, we can just study the energy of the
bound state in this representation 14. This allows us to restrict the two particles in the
bound state to be the scalar component within the (anti)BPS multiplets. The processes
which contribute to the bound state in this so(5) representation are shown in figure 1,
where we denoted the ‘scalar electron’ by ˆe and the ‘scalar proton’ by pˆ.
The corresponding amplitudes to the leading nonrelativistic order are given as follows.
Both the first amplitude corresponding to a photon exchange and the second amplitude
corresponding to a massless scalar exchange are given by
me := m1 − m2,
mp := m1 − m3.
HJEP09(217)5
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
In the relativistic quantum field theory the singleparticle states are usually normalized
so that hpqi = 2p(m2 + p~2)δ3(p~ − ~q) while in the nonrelativistic quantum mechanics it
is normalized so that hp~~qi = δ3(p~ − ~q). Therefore in the nonrelativistic normalization the
total scattering amplitude is given by
MNR ≃ 4memp
M
.
(3.8)
This means that in the limit me/mp → 0, the direct deltafunction interaction term
iM3 drops out, and the effect of the infinitely heavy particle can be replaced by the
1/r potential created by the massless vector and the massless scalar. Defining α by the
requirement At = α/r, the value of the massless scalar background is of the general form
φ˜(x) = cm − c′α/r where c and c′ are numerical coefficients. They can be fixed by a
computation from the original Lagrangian (3.1) or just by demanding that the field φ to
have mass m and the leading 1/r potential to exactly cancel in the case of like charges.
Taking the latter approach immediately gives c = c′ = 1. In conclusion, we find that
the energy spectrum of the bound state, in this limit, can be obtained by just solving the
modified KleinGordon equation (2.4). This is what we wanted to show in this section.
4
Comments on further corrections
In this paper we only considered corrections to the spectrum at the level of the relativistic
quantum mechanics in the limit where one of the particle is infinitely heavier than the
other. Here we briefly discuss further corrections from various sources.
Hyperfine splitting and recoil effect. For the realworld hydrogen the leading
relativistic correction scaling with µα 4 is called the fine structure, and the corrections of the
form µα 4me/mp are called the hyperfine splitting or the recoil effects. The hyperfine
splitting is the one depending on the spin of the heavier particle, and the recoil effect is the
one independent of it. Note that they vanish in the limit me/mp → 0.
So far we showed that the fine splitting is absent in the bound state under our
considerations. There is no hyperfine splitting, from the simple reason that the bound state
form a single nonBPS multiplet under the supersymmetry. It would be nice to extend our
study to the recoil effect, by making me/mp finite.
In principle we can evaluate the Feynman diagrams such as those shown in figure 1 to
the required order and plug it in to the BetheSalpeter equation determining the bound
– 5 –
ˆe
pˆ
ˆe
pˆ
ˆe
pˆ
ˆe
pˆ
ˆe
pˆ
ˆe
pˆ
state spectrum. However, at the same order α4, we also need to include the contributions
from crossed ladder diagrams such as shown in figure 2.
The basic reason is as follows. To the Hamiltonian of the bound state, the treelevel
diagram contributes a term of the form ∼ α/r + · · · while the oneloop diagrams contribute
terms of the form ∼ α2/r2 + · · · . Recall now that the expectation value of the operator
1/rn in the standard wavefunctions of the nonrelativistic hydrogen atom scales as ∼ αn.
Therefore, the leading contribution from the treelevel diagram is ∼ α2 as it should be, with
a subleading α4 term from the relativistic kinematics, and the leading contribution from
the oneloop diagram is also of order ∼ α4. The straight ladder diagrams are twoparticle
irreducible and therefore do not contribute in the BetheSalpeter equation.
It is known [
18
] that the crossed photon diagram, the first of the three shown in figure 2,
does not contribute in the case of QED in the Coulomb gauge. But there are additional
crossed diagrams in our case, including the crossed gaugino, the crossed scalar, and many
others, and we suspect that they do indeed have a nonzero contribution at this order. To
say anything definite, we need to have a more detailed study of the system.
Lamb shift. In the case of realworld hydrogen, the leading radiative correction is the
famous Lamb shift and is of order α5 log α. In the supersymmetric case, the radiative
gaugino and photon corrections are again of order α5 log α [
14
].
However, the radiative correction from the massless scalar is known to give a correction
of order α3 log α [
19
], as reproduced by the integrability technique in [4]. Note that this
is parametrically larger than the relativistic quantummechanical corrections, which are of
order α4. Therefore, strictly speaking, it is of no use to isolate the relativistic
quantummechanical corrections in the present case, as we did in this paper.
Here we give a rough argument that this effect produces no l dependence; see [4]
for details. Note that in the end any computation of the correction boils down to an
evaluation of the expectation value of the operator ΔH = αnV (~r, p~) modifying the
nonrelativistic Hamiltonian in the standard uncorrected hydrogen wavefunction. A oneloop
correction corresponds to α2 contributions. For the photon exchange, we know V ∼ p~ 2/r
or ∼ δ3(~r), whose expectation values give (µα )3, in total giving a correction of order
α2+3. In general, the expectation value of a dimensiond operator is of order (µα )d. To
produce an α3 shift, the operator V should be of dimension 1, which restricts its form
to be ∼ 1/r, whose expectation value is of the form ∼ (µα )/n2, and cannot produce an
ldependent contribution. The subleading correction from the massless scalar would be of
order α5 log α as in the standard Lamb shift.
– 6 –
The vev (3.2) breaks U(N ) to U(1)N , and we have been considering
the bound state of a particle with gauge charge (q1, q2) = (1, −1) and another with (q1, q3) =
(−1, 1), with the total charge (q2, q3) = (−1, 1). There are many additional states with the
total charge (q2, q3) = (−1, 1) which can mix with our bound state.
The first possibility is that we just have a single particle with the total charge (q2, q3) =
(−1, 1) in the intermediate state. The corresponding Feynman diagram is shown in figure 3.
The diagram itself is of order α, and the process proceeds via two constituent particles
merging into one, therefore with the delta function interaction δ3(~r). In total the correction
is of order α1+3. In the case of the positronium in the real world, this indeed contributes to
the splitting of the orthopositronium and the parapositronium at this order. In our case,
however, the contribution is zero, since the intermediate channel is a massive (anti)BPS
state and there is no component with the representation 14 to mix with our states.
Next, we can have, in the intermediate state, massive nonBPS multiplets which are
composed of the (n, 2)component and the (3, n)component of the gauge multiplet, where
n 6= 1. We can safely neglect the cases when n 6= 2, 3 by taking mn6=1,2,3 very large, since
these will then have masses of order 2mn − m2 − m3 ≫ me + mp.
When n = 2 or = 3, this is a decay channel where ˆe and pˆ merge to form a massive
(anti)BPS state with charge (q2, q3) = (−1, 1), simultaneously emitting a massless vector
multiplet. The diagram itself is of order α, and again this proceeds via a point interaction.
Therefore the decay width is of order α2hδ3(~r)i ∼ α5.
5
Discussions
In this paper we studied the relativistic correction to the spectrum of a Coulombic bound
state of two particles in the massive phase of N = 4 super YangMills, using relativistic
quantum mechanics. We found that in the me/mp → 0 limit, the correction keeps the n2
degeneracy, meaning that it preserves the extended so(4) invariance.
As for future directions, firstly, we would like to analyze the spectrum in the finite
me/mp case. An approach would be to extend the pNRQED framework [20, 21] to include
the presence of the massless scalar, as in [
19
].
Secondly, it would be interesting to understand more fully the structure of the dual
conformal symmetry in the massive phase. The states at a given principal quantum number
n transform as
(V0 ⊕ V1 ⊕ · · · ⊕ Vn) ⊗ VBPS ⊗ VantiBPS = Vn/2 ⊗ Vn/2 ⊗ VBPS ⊗ VantiBPS.
(5.1)
– 7 –
One immediate question is how the extended so(4) ≃ su(2) × su(2) should act on the BPS
and antiBPS linear combinations of the supercharges. It is tempting to think that one
su(2) acts on VBPS and another acts on VantiBPS.
The analyses given in the literature so far typically use an extension to higher
dimensions to regard massive states in four dimensions as massless states in higher dimensions.
The modified KleinGordon equation (2.4) indeed has this structure; it would be interesting
to supersymmetrize it and study its symmetry group. Also, considering the importance of
the central charge Z[ij] which modifies the anticommutator of two supersymmetry
generators in the massive phase, another approach would be to try to add this central charge,
in some way or other, to the Yangian symmetry which combines both the original and the
dual superconformal symmetry [22], possibly following the discussions in [9].
We would like to come back to these questions in the future.
Acknowledgments
The authors thank S. CaronHuot and C. P. Herzog for discussions. YS and TY are
partially supported by the Programs for Leading Graduate Schools, MEXT, Japan, via
the Leading Graduate Course for Frontiers of Mathematical Sciences and Physics. RS
is partially supported by the Jasso scholarship from the Graduate School of Science, the
University of Tokyo. YT is in part supported by JSPS KAKENHI GrantinAid
(WakateA), No.17H04837 and JSPS KAKENHI GrantinAid (KibanS), No.16H06335, and also
supported in part by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo.
A
Structure of massive nonBPS bound state
The structure of representations of nonconformal supersymmetries in general dimensions
were worked out in [23, 24]. Here we give a brief account for the particular case of N = 4
supersymmetry in four dimensions, in the special case where the central charge breaks
su(4) ≃ so(6) Rsymmetry to usp(4) ≃ so(5).
Let us say that the fourmomentum of the total system is (m, 0, 0, 0) with the little
group so(3) and the central charge is ZIJ = zJ IJ where J IJ is the invariant tensor of the
usp(4) with z ∈ R. We would like to understand the structure of the irreducible
representation of the little group so(3), the Rsymmetry so(5) together with the supersymmetry
generators whose anticommutation relations are
{ QIa , QbJ } = ǫabzJ IJ ,
{ QIa , (QbJ )† } = 2mδabδJI .
Here and in the following, the indices a, b are for the little group so(3) and therefore there
is no distinction of the dotted and undotted spinor indices.
It is convenient to take the following linear combination of the supercharges instead:
satisfying the reality condition
1
2
RI±a := √ (QIa
± J IJ ǫab(Q)†Jb)
(RI±a)† = J IJ ǫabRJ±b
– 8 –
(A.1)
(A.2)
(A.3)
and the anticommutation relations
{ RI±a , RJb
± } = −(2m ± z)J IJ ǫab,
{ RI±a , RJb
In this form it is clear that we need the condition 2m ≥ z. Specializing to the nonBPS
case, we assume 2m > z in the following.
By regarding the pair (I, a) as a single index µ running from 1 to 8, we can make a
further rescaling of the operators to make them in the form
μ
{ γ± , γ±ν } = 2δμν ,
μ
{ γ± , γ∓ν } = 0.
The so(3) × so(5) rotation is inside the so(8) rotation acting on the indices µ, ν . So in the
μ
rest of this summary we consider the representation of so(8) rotation M μν together with γ±.
The operators
satisfy the following relations:
1
4
S±μν =
[ γ±μ , γ±ν ]
[ S±μν , γ±ρ ] = [ M μν , γ±ρ ],
[ S±μν , S±ρσ ] = [ M μν , S±ρσ ].
Therefore, the operators
T μν := M μν
− S+μν − S−μν
satisfy the so(8) commutation relations, and commute with γ±μ’s. Furthermore, two sets of
operators {M μν , γ±μ} and {T μν , γ±μ} generate the same algebra. Those who know the coset
construction in 2d conformal field theories would recognize this as a baby version of it.
The structure of the irreducible representation of the latter is now clear: it is of the
states. M μν then acts as a diagonal subgroup.
form V ⊗VBPS⊗VantiBPS, where V is an irreducible representation of so(8) generated by T μν
and V(anti)BPS is the irreducible representation of γ±μ isomorphic to the N = 4 (anti)BPS
Open Access.
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(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
HJEP09(217)5
[1] W. Pauli, U¨ber das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik (in
German), Z. Phys. 36 (1926) 336 [INSPIRE].
Rev. Mod. Phys. 38 (1966) 330 [INSPIRE].
[2] M. Bander and C. Itzykson, Group theory and the hydrogen atom,
[3] M.I. Eides, H. Grotch and V.A. Shelyuto, Theory of light hydrogenlike atoms,
Phys. Rept. 342 (2001) 63 [hepph/0002158] [INSPIRE]. An updated version is also available
in the book form as Theory of light hydrogenic bound states by the same authors,
Springer Tracts Mod. Phys. 222, Springer, Germany, (2007).
[4] S. CaronHuot and J.M. Henn, Solvable relativistic hydrogenlike system in supersymmetric
YangMills theory, Phys. Rev. Lett. 113 (2014) 161601 [arXiv:1408.0296] [INSPIRE].
– 9 –
YangMills amplitudes, JHEP 01 (2011) 140 [arXiv:1010.5874] [INSPIRE].
maximal super YangMills amplitudes in 6d and 4d and their hidden symmetries,
JHEP 01 (2015) 098 [arXiv:1405.7248] [INSPIRE].
dimensions — supercharge and the generalized JohnsonLippmann operator,
[5] H. Elvang and Y.T. Huang, Scattering amplitudes, Cambridge University Press, Cambridge U.K., ( 2015 ) [arXiv: 1308 .1697] [INSPIRE].
[6] L.F. Alday , J.M. Henn , J. Plefka and T. Schuster , Scattering into the fifth dimension of N = 4 super YangMills , JHEP 01 ( 2010 ) 077 [arXiv: 0908 .0684] [INSPIRE].
[7] S. CaronHuot and D. O'Connell , Spinor helicity and dual conformal symmetry in ten [10] V. Fock , On the theory of the hydrogen atoms , Z. Phys . 98 ( 1935 ) 145 [INSPIRE].
[11] W. Buchmu ¨ller, S.T. Love and R.D. Peccei , Supersymmetric bound states, Nucl. Phys. B 204 ( 1982 ) 429 [INSPIRE] . Phys. Lett. B 155 ( 1985 ) 427 [INSPIRE].
[12] P. Di Vecchia and V. Schuchhardt , N = 1 and N = 2 supersymmetric positronium , [13] C.P. Herzog and T. Klose , The perfect atom: bound states of supersymmetric quantum electrodynamics , Nucl. Phys. B 839 ( 2010 ) 129 [arXiv: 0912 .0733] [INSPIRE].
[14] T. Rube and J.G. Wacker, The simplicity of perfect atoms: degeneracies in supersymmetric hydrogen , J. Math. Phys. 52 ( 2011 ) 062102 [arXiv: 0912 .2543] [INSPIRE].
[18] I. Lindgren , Gauge dependence of interelectronic potentials , J. Phys. B 23 ( 1990 ) 1085 .
[19] A. Pineda , The static potential in N = 4 supersymmetric YangMills at weak coupling,