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arXiv:1411.5243v2 [hep-ph] 30 Nov 2014
SLAC–PUB–16130
Baryon Spectrum from Superconformal Quantum Mechanics and
its Light-Front Holographic Embedding
Guy F. de T´eramond
Universidad de Costa Rica, San Jos´e, Costa Rica∗
Hans G¨unter Dosch
Institut f¨ur Theoretische Physik, Philosophenweg 16, D-6900 Heidelberg, Germany†
Stanley J. Brodsky
SLAC National Accelerator Laboratory,
Stanford University, Stanford, California 94309, USA‡
(Dated: December 8, 2014)
Abstract
We describe the observed light-baryon spectrum by extending superconformal quantum mechan-
ics to the light front and its embedding in AdS space. This procedure uniquely determines the
confinement potential for arbitrary half-integer spin. To this end, we show that fermionic wave
equations in AdS space are dual to light-front supersymmetric quantum mechanical bound-state
equations in physical space-time. The specific breaking of conformal invariance explains hadronic
properties common to light mesons and baryons, such as the observed mass pattern in the radial
and orbital excitations, from the spectrum generating algebra. The holographic embedding in AdS
also explains distinctive and systematic features, such as the spin-Jdegeneracy for states with the
same orbital angular momentum, observed in the light baryon spectrum.
∗gdt@asterix.crnet.cr
†h.g.dosch@thphys.uni-heidelberg.de
‡sjbth@slac.stanford.edu
1
I. INTRODUCTION
The classical Lagrangian of QCD is invariant under scale and conformal transforma-
tions in the limit of massless quarks [1,2]. However, meson and baryon bound-states have
well-defined ground states and towers of excited states with well defined and measurable
properties such as mass and spin. A simple but fundamental question in hadron physics is
thus to understand the mechanism which endows a nominally conformal theory with a mass
scale, as well as to explain the remarkably similar linear Regge spectroscopy of both mesons
and baryons.
In the quest for semiclassical equations to describe bound states, in the large distance
strongly coupled regime of QCD, one can start by reducing the strongly correlated multi-
parton light-front Hamiltonian dynamical problem to an effective one-dimensional quantum
field theory [3]. This procedure is frame-independent and leads to a semiclassical, relativistic
light-front (LF) wave equation for the valence state (the lowest Fock state), analogous to the
Schr¨odinger and Dirac equations in atomic physics. The complexities arising from the strong
interaction dynamics of QCD and an infinite class of Fock components are incorporated in
an effective potential U, but its determination from first principles remains largely an open
question.
Thus, a second central problem in the theoretical search for a semiclassical approximation
to QCD is the construction of the effective LF confining potential Uwhich captures the
underlying dynamics responsible for confinement, the emergence of a mass scale as well as
the universal Regge behavior of mesons and baryons. Since our light-front semiclassical
approach is based on a one-dimensional quantum field theory, it is natural to extend the
framework introduced by V. de Alfaro, S. Fubini and G. Furlan (dAFF) [4] to the frame-
independent light-front Hamiltonian theory, since it gives important insight into the QCD
confinement mechanism [5]. Remarkably, dAFF show that a mass scale can appear in the
Hamiltonian without breaking the conformal invariance of the action.
The dAFF construction [4] begins with the study of the spectrum of a conformally in-
variant one-dimensional quantum field theory which does not have a normalizable ground
state. A new Hamiltonian is defined as a superposition of the generators of the conformal
group and consequently it leads to a redefinition of the corresponding evolution parameter
τ, the range of which is finite. This choice determines the quantum mechanical evolution of
2
the system in terms of a compact operator with normalizable eigenstates and a well defined
ground state. A scale appears in the Hamiltonian while retaining the conformal invariance
of the action [4]. This remarkable result is based on the isomorphism of the algebra of
the one-dimensional conformal group Conf(R1) to the algebra of generators of the group
SO(2,1). One of the generators of this group, the rotation in the 2-dimensional space, is
compact. As a result, the form of the evolution operator is fixed and includes a confining
harmonic oscillator potential, thus equally spaced eigenvalues [6,7]. Since the generators of
Conf(R1) have different dimensions, their relations with the generators of SO(2,1) imply a
scale, which according to dAFF may play a fundamental role [4,5].
A third important feature in the construction of semiclassical equations in QCD, is the
correspondence between the equations of motion for arbitrary spin in Anti–de Sitter (AdS)
space and the light-front Hamiltonian equations of motion for relativistic light hadron bound-
states in physical space-time [3,8]. This approach is inspired by the AdS/CFT correspon-
dence [9] where, in principle, one can compute physical observables in a strongly coupled
gauge theory in terms of a weakly coupled classical gravity theory defined in a higher dimen-
sional space [9–11]. In fact, an additional motivation for using AdS/CFT ideas to describe
strongly coupled QCD follows from the vanishing of the β-function in the infrared, which
leads to a conformal window in this regime [12–14].
The procedure, known as light-front holography [3,15,16], allows one to establish a
precise relation between wavefunctions in AdS space and the LF wavefunctions describing
the internal structure of hadrons. As a result, the effective LF potential Uderived from
the AdS embedding is conveniently expressed, for arbitrary integer spin representations, in
terms of a dilaton profile which is determined by the dAFF procedure described above [5,8].
The result is a light-front wave equation which reproduces prominent aspects of hadronic
data, such as the mass pattern observed in the radial and orbital excitations of the light
mesons [16], and in particular a massless pion in the chiral limit.
The light-front holographic embedding for baryons is not as simple as for mesons, since
a dilaton term in the AdS fermionic action can be rotated away by a redefinition of the
fermion fields in AdS [16,17], and therefore it has no dynamical effects on the spectrum. In
practice, one can introduce an effective interaction in the fermion action, a Yukawa term,
which breaks the maximal symmetry in AdS and consequently the conformal symmetry in
Minkowski space. This leads to a linear confining interaction in a LF Dirac equation for
3
baryons whose eigensolutions generate a baryonic Regge spectrum [18,19]. The confining
interaction term can be constrained by the condition that the square of the Dirac equation
leads to a potential which matches the form of the dilaton-induced potential for integer spin,
but this procedure appears to be ad-hoc.
There are some striking similarities between the spectra of the observed light mesons and
baryons: they are of similar mass, the slope and spacing of the quantum orbital excitations
in Land their daughter spacing in n, the radial quantum number, is the same. This behavior
in the meson sector is related to the introduction of a scale within the framework of the
conformal algebra. This procedure leaves the action invariant [4,5]. Since supersymmetry
is related with boson-fermion symmetry, it is compelling to examine the properties of the
supersymmetric algebra and its superconformal extension to describe baryons in complete
analogy to the bosonic case, where the confining potential was determined by the confor-
mal algebra of one-dimensional quantum field theory [4,5]. In fact, it is straightforward to
translate a quantum mechanical model into its supersymmetric (SUSY) counterpart by fol-
lowing Witten’s construction [20]. Superconformal quantum mechanics, the supersymmetric
extension [21,22] of conformal quantum mechanics [4], then follows from the properties of
the superconformal algebra.
We shall show in this article that the structure of supersymmetric quantum mechanics
is encoded holographically in the AdS equations for arbitrary half-integer spin for any su-
perpotential. Most important for the present discussion, we will show that superconformal
quantum mechanics [22] has an elegant representation on the light front and its holographic
embedding in AdS space. Remarkably, this procedure uniquely determines the form of the
confinement potential for arbitrary half-integer spin. If one extends with Fubini and Rabi-
novici [22], the method of de Alfaro, Fubini and Furlan [4] to the superconformal algebra,
the form of the potential in the new evolution equations is completely fixed. We will also
discuss in this article how the different embeddings of mesons and baryons in AdS space [8]
lead to distinct systematic features of meson and baryon spectroscopy. In particular, we will
show that the integrability methods used to construct baryonic light-front equations [19] are
the light-front extension of the usual formulation of supersymmetric Hamiltonian quantum
mechanics [20,23]. In fact, a possible indication of a supersymmetric connection was already
mentioned in Ref. [19], but a proof was not actually given there [24].
This article is organized as follows: In Sec. II we review for convenience light-front con-
4
formal quantum mechanics and its holographic embedding in AdS space. In Sec. III we
extend supersymmetric quantum mechanics to the light-front and describe its embedding in
AdS space. We show in particular that properly taking the square root of the light-front
Hamiltonian operator leads to a linear relativistic invariant Dirac equation. In Sec. IV
superconformal quantum mechanics is extended to light-front holographic QCD. The appli-
cation of the method to the complex patterns observed in baryon spectroscopy is discussed
in Sec. V. Some final comments and conclusions are given in Sec. VI. In Appendix we
discuss briefly the specific action of the supercharges.
II. LIGHT-FRONT CONFORMAL QUANTUM MECHANICS AND ITS HOLO-
GRAPHIC EMBEDDING
Following Ref. [4] one starts with the one-dimensional action
S[x] = 1
2Zdt ˙x2−g
x2,(1)
where x(t) is a field operator, the constant gis dimensionless, and thas dimensions of length
squared. The action (1) is invariant under conformal transformations in the variable t, thus in
addition to the Hamiltonian Hthere are two more invariants of motion, namely the dilatation
operator Dand the operator of special conformal transformations K, corresponding to the
generators of the conformal group Conf(R1) with commutation relations
[H, D] = i H, [H, K] = 2 i D, [K, D] = −i K. (2)
Specifically, if one introduces the new variable τdefined through the relation
dτ =dt
u+v t +w t2,(3)
it then follows that the operator
G=u H +v D +w K, (4)
generates the quantum mechanical unitary evolution in τ[4]
G|ψ(τ)i=id
dτ |ψ(τ)i.(5)
5
One can show that Gis a compact operator for 4uw −v2>0 [4]. In terms of the fields x
and p= ˙xthe new Hamiltonian Gis given by
G(x, p) = 1
2up2+g
x2−1
4v(xp +px) + 1
2wx2,(6)
at t= 0. In the Schr¨odinger representation x(0) is represented by the position operator and
p→ −id
dx . The Hamiltonian is [4]
G=1
2u−d2
dx2+g
x2+i
4vxd
dx +d
dxx+1
2wx2,(7)
=uH +vD +wK,
with
H=1
2−d2
dx2+g
x2,(8)
D=i
4xd
dx +d
dxx,(9)
K=1
2wx2,(10)
the superposition of the ‘free’ Hamiltonian H, the generator of dilatations Dand the gen-
erator of special conformal transformations Kin one dimension.
We now compare the dAFF Hamiltonian with the light-front Hamiltonian in the semiclas-
sical approximation described in [3]. A physical hadron in four-dimensional Minkowski space
has four-momentum Pµand invariant hadronic mass squared HLF =PµPµ=M2[25,26].
In the limit of zero quark masses the longitudinal modes decouple and the LF eigenvalue
equation HLF |φi=M2|φiis a light-front wave equation for φ[3]
−d2
dζ2−1−4L2
4ζ2+U(ζ, J)φ(ζ) = M2φ(ζ),(11)
a relativistic single-variable LF Schr¨odinger equation. The boost-invariant transverse-impact
variable ζ[15] measures the separation of quark and gluons at equal light-front time [27],
and it also allows one to separate the bound-state dynamics of the constituents from the
kinematics of their LF internal angular momentum Lin the transverse light-front plane [3].
The effective interaction Uis instantaneous in LF time and acts on the lowest state of
the LF Hamiltonian. To actually compute Uin the semiclasscal approximation one must
systematically express higher Fock components as functionals of the lower ones. This method
has the advantage that the Fock space is not truncated and the symmetries of the Lagrangian
are preserved [28].
6
Comparing the Hamiltonian G(7) with the light-front wave equation (11) and identifying
the variable xwith the light-front invariant variable ζ, we have to choose u= 2, v = 0 and
relate the dimensionless constant gto the LF orbital angular momentum,
g=L2−1
4,(12)
in order to reproduce the light-front kinematics. Furthermore w= 2λ2
Mfixes the form of the
confining light-front potential to that of a harmonic oscillator in the LF transverse plane [5],
U∼λ2
Mζ2.(13)
In contrast to the baryonic case, which is discussed below, one can perform a constant level
shift by adding an arbitrary constant, with dimension mass squared, to the confining term
in the light front potential.
A. Light-Front Holographic Embedding
The next step is to determine the arbitrary constant term in the LF effective potential for
arbitrary integer spin representations. Following Ref. [8] this constant term in the potential
is determined by the embedding of the LF Hamiltonian equations in AdS space. To this end
it is convenient to consider an effective action for a spin-Jfield in AdSd+1 space represented
by a totally symmetric rank-Jtensor field ΦN1...NJ, where M, N are the indices of the
d+ 1 higher dimensional AdS space with coordinates xM= (xµ, z). The coordinate zis the
holographic variable and the xµare Minkowski flat space-time coordinates. In the presence
of a dilaton background ϕthe effective action in [8] is
Seff =Zddx dz √g eϕ(z)gN1N′
1···gNJN′
JgMM′DMΦ∗
N1...NJDM′ΦN′
1...N′
J
−µ2
eff (z) Φ∗
N1...NJΦN′
1...N′
J,(14)
where √g= (R/z)d+1 and DMis the covariant derivative which includes the affine connec-
tion (Ris the AdS radius). The dilaton ϕ(z) effectively breaks the maximal symmetry of
AdS, and the zdependence of the effective AdS mass µef f allows a clear separation of kine-
matical and dynamical effects. It is determined by the precise mapping of AdS to light-front
physics [8].
7
In order to map to the LF Hamiltonian, one considers hadronic states with momentum
Pand a z-independent spinor ǫν1···νJ(P) with polarization components along the physical
Minkowski coordinates. In holographic QCD such a state is described by the product of
a free state with moment P, propagating in physical space-time, and z-dependent wave
function ΦJ
Φν1···νJ(x, z) = eiP ·xǫν1···νJ(P) ΦJ(z),(15)
with invariant hadron mass PµPµ≡ηµν PµPν=M2. Variation of the action leads to the
wave equation
−zd−1−2J
eϕ(z)∂zeϕ(z)
zd−1−2J∂z+(µ R)2
z2ΦJ=M2ΦJ,(16)
where (µ R)2= (µeff (z)R)2−Jz ϕ′(z) + J(d−J+1) is a constant determined by kinematical
conditions in the light front [8]. Variation of the AdS action also gives the kinematical con-
straints required to eliminate the lower spin states J−1, J −2,··· from the fully symmetric
AdS tensor field Φν1...νJ[8]:
ηµν Pµǫνν2···νJ= 0, ηµν ǫµνν3···νJ= 0.(17)
We now perform the AdS mapping to light-front physics in physical space-time. To this
end we factor out the scale (1/z)J−(d−1)/2and dilaton factors from the AdS field writing
ΦJ(z) = (R/z)J−(d−1)/2e−ϕ(z)/2φJ(z).(18)
Upon the substitution of the holographic variable zby the light-front invariant variable ζ
and replacing ΦJinto the AdS eigenvalue equation (16), we obtain for d= 4 the QCD
light-front frame-independent wave equation (11) with the effective LF potential [8,29]
U(ζ, J) = 1
2ϕ′′(ζ) + 1
4ϕ′(ζ)2+2J−3
2ζϕ′(ζ).(19)
The fifth dimensional AdS mass µin (16) is related to the light-front internal orbital angular
momentum Land the total angular momentum Jof the hadron according to
(µR)2=−(2 −J)2+L2,(20)
where the critical value L= 0 corresponds to the lowest possible stable solution [30].
From the holographic relation (19) it follows that the harmonic potential is holographi-
cally related to a unique dilaton profile, ϕ=λMz2provided that ϕ(z)→0 as z→0. From
(19) we find the effective LF potential (13)
U(ζ, J) = λ2
Mζ2+ 2λM(J−1).(21)
8
The term λ2
Mζ2is determined uniquely by the underlying conformal invariance of the one-
dimensional effective theory, and the constant term 2λM(J−1) is determined by the spin
representations in the embedding space.
For the effective potential (21) equation (11) has eigenfunctions
φn,L(ζ) = |λM|(1+L)/2s2n!
(n+L)! ζ1/2+Le−|λM|ζ2/2LL
n(|λM|ζ2),(22)
and eigenvalues
M2
n,J,L = 4λMn+J+L
2,(23)
for λM>0. The spectral predictions explain the essential features of the observed light
meson spectrum [16], including a zero pion mass in the chiral limit, and Regge trajectories
with the same slope in the quantum numbers nand L. The solution for λM<0 leads to
a meson spectrum in clear disagreement with observations. Since the effective interaction
is determined from the conformal symmetry of the effective one-dimensional quantum field
theory, which is not severely broken for small quark masses, the method can be successfully
extended to describe, for example, the Kand K∗excitation spectrum [16,31].
III. LIGHT-FRONT SUPERSYMMETRIC QUANTUM MECHANICS AND ITS
HOLOGRAPHIC EMBEDDING
Supersymmetric quantum mechanics is a simple realization of a graded Lie algebra which
contains two fermionic generators, the supercharges, Qand Q†, and a bosonic generator,
the Hamiltonian H, which are operators in a state space [20]. It closes under the graded
algebra sl(1/1):
1
2{Q, Q†}=H, (24)
{Q, Q}={Q†, Q†}= 0,(25)
[Q, H] = [Q†, H] = 0.(26)
It is useful to write down the SUSY formulation of quantum mechanics in terms of anti-
commuting spinor operators χ. A minimal realization of the group generators is given in
terms of the one-dimensional representation
Q=χd
dx +W(x),(27)
9
and
Q†=χ†−d
dx +W(x),(28)
where W(x) is called the superpotential in the context of supersymmetric theories. The
spinor operators χand χ†satisfy the anti-commutation relation
{χ, χ†}= 1.(29)
Using a representation in terms of 2 ×2 Pauli-spin matrices we have
χ=σ1+iσ2
2, χ†=σ1−iσ2
2,(30)
and
[χ, χ†] = σ3.(31)
Thus the Hamiltonian is
H=1
2{Q, Q†}=1
2−d2
dx2+W2(x) + σ3W′(x).(32)
It can be written in matrix form:
H=1
2
H+0
0H−
=1
2
−d2
dx2+V+(x) 0
0−d2
dx2+V−(x)
,(33)
with effective potential
V±(x) = W2(x)±W′(x).(34)
Since Hcommutes with Qand Q†(26), it follows that the eigenvalues of H+and H−are
identical.
A. Supersymmetric Quantum Mechanics in the Light-Front
To give a relativistic formulation of supersymmetric quantum mechanics it is convenient
to write the anti-commuting spinor operators in terms of a 4 ×4 matrix representation of
the Clifford algebra, which acts on four-dimensional physical space where the LF spinors are
defined. We use the Weyl representation where the chirality operator γ5is diagonal, and
define the matrices αand βby
iα =
0I
−I0
, β =
0I
I0
,(35)
10
where Ia two-dimensional unit matrix. The matrices αand βare hermitian and anti-
commuting
α†=α, α2= 1,(36)
β†=β, β2= 1,(37)
{α, β}= 0.(38)
From the product of αand βwe construct a third matrix γ5, which corresponds to the usual
chirality operator: γ5=iαβ
γ5=
I0
0−I
.(39)
The matrix γ5is also hermitian and anti commutes with αand β
γ†
5=γ5, γ2
5= 1,(40)
{γ5, α}={γ5, β}= 0.(41)
The SUSY LF Hamiltonian HLF is given by the sl(1/1) algebra
{Q, Q†}=HLF ,(42)
{Q, Q}={Q†, Q†}= 0,(43)
but the supercharges Qand Q†are now represented by 4×4 matrices. Furthermore, since the
Hamiltonian operator HLF =PµPµ=M2is invariant, it implies that HLF should depend
on a frame independent variable. In impact space the relevant invariant variable is ζ, and
thus the representation:
Q=ηd
dζ +W(ζ),(44)
and
Q†=η†−d
dζ +W(ζ),(45)
where the spinor operators ηand η†satisfy the anti-commutation relation
{η, η†}= 1,(46)
and are given by
η=β+iα
2, η†=β−iα
2,(47)
11
in the 4 ×4 matrix representation defined above. We also have
[η, η†] = γ5.(48)
The LF Hamiltonian is thus expressed as
HLF ={Q, Q†}=−d2
dζ2+W2(ζ) + γ5W′(ζ),(49)
which is frame independent.
B. A Linear Dirac Equation from Supersymmetric Quantum Mechanics in the
Light-Front
Since γ2
5= 1, the LF Hamiltonian (49) can be conveniently expressed as HLF =BB†
where
B=d
dζ +γ5W(ζ),(50)
and
B†=−d
dζ +γ5W(ζ).(51)
The next step is to take the ‘square root’ of the Hamiltonian HLF . For this purpose
we write HLF as a product of Hermitian operators which we label DLF ; thus HLF =D2
LF .
Using the relation iαB =−iB†αand equations (50) and (51), we have
DLF =−iα −d
dζ +γ5W(ζ),(52)
and thus the invariant Dirac equation [19]
(DLF −M)ψ(ζ) = 0,(53)
where ψ(ζ) is a LF Dirac spinor. Premultiplying the linear Dirac wave equation (53) by the
operator DLF +Mand using the properties of the Dirac matrices given above, we recover
the LF eigenvalue equation
HLF ψ=D2
LF ψ=M2ψ, (54)
where HLF is given by (49). We thus reproduce the results obtained in Ref. [19] using an
operator construction of the light-front Hamiltonian and the Dirac equation, but starting
from light-front supersymmetric quantum mechanics [32].
12
It is convenient to separate the kinematic and dynamic contributions to the superpoten-
tial. We write
W(ζ) = ν+ 1/2
ζ+u(ζ),(55)
where νis a dimensionless parameter representing the LF orbital angular momentum, and
the dynamical effects are encoded in the function u(ζ). From (52) we can express the
LF-invariant Dirac equation (53) for the superpotential (55) as a system of coupled linear
differential equations
−d
dζ ψ−−ν+1
2
ζψ−−u(ζ)ψ−=Mψ+,
d
dζ ψ+−ν+1
2
ζψ+−u(ζ)ψ+=Mψ−,(56)
where the chiral spinors are defined by ψ±=1
2(1 ±γ5)ψ.
C. Holographic Embedding
We can now determine the LF superpotential u(ζ) in (55) for arbitrary half-integer spin
by embedding the LF results in AdS space. We start with an effective action for Rarita-
Schwinger (RS) spinors in AdS space [ΨN1···NT]α, which transform as symmetric tensors of
rank Twith indices N1. . . NT, and as Dirac spinors with index α[33]. In presence of an
effective interaction V(z) the effective action is given by [8]
Seff =1
2Zddx dz √g gN1N′
1···gNTN′
T
h¯
ΨN1···NTiΓAeM
ADM−µ−V(z)ΨN′
1···N′
T+h.c.i,(57)
where √g=R
zd+1 and eM
Ais the inverse vielbein, eM
A=z
RδM
A. The covariant derivative
DMincludes the affine connection and the spin connection. The tangent-space Dirac ma-
trices obey the usual anti-commutation relation ΓA,ΓB= 2ηAB . We have not included a
dilaton factor eϕ(z)in (57) since it can be absorbed by redefining the RS spinor according
to ΨT→eϕ(z)/2ΨT[8,17]. This is a consequence of the linear covariant derivatives in the
fermion action, which also prevents a mixing between dynamical and kinematical effects,
and thus, in contrast to the effective action for integer spin fields (14), the AdS mass µ
in Eq. (57) is constant. Since a dilaton factor has no dynamical consequences, one must
13
introduce an effective confining interaction V(z) in the fermion action to break conformal
symmetry and generate a baryon spectrum [18,19].
It is shown below that the potential V(z), which has been introduced hitherto ad hoc,
is precisely related to the superpotential u(55). Furthermore, in Sec. IV it is shown that,
in analogy with the boson case [5], the form of uis determined in the framework of the
superconformal algebra.
A physical baryon has plane-wave solutions with four-momentum Pµ, invariant mass
PµPµ=M2, and polarization indices along the physical coordinates. Factoring out the
four-dimensional plane-wave and spinor dependence, as well as the scale factor (1/z)T−d/2,
we have
Ψ±
ν1···νT(z) = eiP ·xu±
ν1···νT(P)R
zT−d/2
Ψ±
T(z),(58)
where T=J−1
2and the fully symmetric RS chiral spinor u±
ν1...νT=1
2(1 ±γ5)uν1...νTsatisfies
the four-dimensional chirality equations
γ·P u±
ν1...νT(P) = Mu∓
ν1...νT(P), γ5u±
ν1...νT(P) = ±u±
ν1...νT(P).(59)
Variation of the AdS action (57) leads for d= 4 to the Dirac equation
−d
dz Ψ−
T−µR
zΨ−
T−R
zV(z)Ψ−
T=MΨ+
T,
d
dz Ψ+
T−µR
zΨ+
T−R
zV(z)Ψ+
T=MΨ−
T,(60)
and the Rarita-Schwinger condition [33] in physical space-time [8]
γνΨνν2... νT= 0.(61)
By identifying the holographic variable zwith the invariant LF variable ζand the AdS
LF spinors by the mapping Ψ±
T(z)→ψ±(ζ), we can compare (60) with (56). Provided that
the AdS mass µis related to the parameter νby
µR =ν+1
2,(62)
the specific LF mapping gives a relation between the effective interaction V(z) in the AdS
action (57) and the function u(ζ) in the LF superpotential (55)
u(ζ) = R
ζV(ζ).(63)
14
In fact they are identical (modulo a kinematic factor), and this relation thus leads to a J-
independent potential. This is a remarkable result, since independently of the specific form
of the potential, the value of the baryon masses along a given Regge trajectory depends
only on the LF orbital angular momentum L[34]. Thus, in contrast with the vector mesons
(21), there is no spin-orbit coupling, in agreement with the observed near-degeneracy in the
baryon spectrum [36,37].
IV. LIGHT-FRONT SUPERCONFORMAL QUANTUM MECHANICS
In order to fix the superpotential u(55) we follow Fubini and Rabinovici in Ref. [22],
and consider a one-dimensional quantum field theory invariant under conformal and super-
symmetric transformations. Imposing conformal symmetry leads to a unique choice of the
superpotential W(27), namely
W(x) = f
x,(64)
in order for fto be a dimensionless constant. In this case the graded-Lie algebra has, in
addition to the Hamiltonian Hand the supercharges Qand Q†, an additional generator S
which is the square root of the generator of conformal transformations K. The enlarged
algebraic structure is the superconformal algebra of Haag, Lopuszanski and Sohnius [22,38,
39]. Using the one-dimensional quantum-mechanical representation of the operators
Q=χd
dx +f
x,(65)
Q†=χ†−d
dx +f
x,(66)
S=χ x, (67)
S†=χ†x, (68)
it is simple to verify that the algebraic structure of the enlarged algebra is fulfilled. We find
1
2{Q, Q†}=H, 1
2{S, S†}=K, (69)
1
2{Q, S†}=f
2+σ3
4−D, (70)
1
2{Q†, S}=f
2+σ3
4+D, (71)
15
where the operators
H=1
2−d2
dx2+f2−σ3f
x2,(72)
K=1
2x2,(73)
D=i
4d
dxx+xd
dx.(74)
satisfy the conformal algebra (2). The anticommutation of all other generators vanish:
{Q, Q}={Q†, Q†}={Q, S}=···= 0.
In analogy with the dAFF procedure [4], we now define, following Fubini and Rabi-
novici [22], a new supercharge Ras a linear combination of the generators Qand S
R=√u Q +√w S, (75)
and compute a new Hamiltonian G
G=1
2{R, R†}.(76)
We find
G=uH +wK +1
2√uw (2f+σ3),(77)
which is a compact operator for uw > 0.
The quantum mechanical evolution operator G(77) obtained by this procedure is analo-
gous to the Hamiltonian (6) obtained by the procedure of de Alfaro, Fubini and Furlan [4].
Remarkably, in the superconformal case there appears beside the confining term w K also
a constant term 1
2√uw(2f±1) in G, which, as we will describe below, plays a key role in
explaining the correct phenomenology.
A. Superconformal Quantum Mechanics in the Light-Front
The light-front extension of the superconformal results follows from the LF superpotential
W(ζ) = ν+ 1/2
ζ,(78)
which corresponds to a kinematic term in the LF Hamiltonian. We now extend the new
Hamiltonian G(77) to a relativistic LF Hamiltonian by the method described in Sec. III A.
16
This amounts to replace the Pauli matrix σ3in (77) with γ5in (48). We obtain:
HLF ={R, R†}
=−d2
dζ2+ν+1
22
ζ2−ν+1
2
ζ2γ5+λ2
Bζ2+λB(2ν+ 1) + λBγ5,(79)
where the arbitrary coefficients uand win (77) are fixed to u= 1 and w=λ2
B. Thus the
supercharge Ris the superposition
R=Q+λBS. (80)
In 2 ×2 block-matrix form the light-front Hamiltonian (79) can be expressed as
HLF =
−d2
dζ2−1−4ν2
4ζ2+λ2
Bζ2+ 2λB(ν+ 1) 0
0−d2
dζ2−1−4(ν+1)2
4ζ2+λ2
Bζ2+ 2λBν
.(81)
The light-front eigenvalue equation HLF |ψi=M2|ψihas eigenfunctions
ψ+(ζ)∼ζ1
2+νe−λBζ2/2Lν
n(λBζ2),(82)
ψ−(ζ)∼ζ3
2+νe−λBζ2/2Lν+1
n(λBζ2),(83)
and eigenvalues
M2= 4λB(n+ν+ 1).(84)
As a consequence of parity invariance, the eigenvalues for the chirality plus and minus
eigenfunctions are identical. One can also show that the probabilities for both components
ψ+and ψ−are the same (See appendix )
Zdζ ψ2
+(ζ) = Zdζ ψ2
−(ζ).(85)
For λB<0 no solution is possible.
V. SYSTEMATICS OF THE BARYON SPECTRUM
To determine how well the superconformal light-front holographic model encompasses
the systematics of the baryon spectrum, we list in Table Ithe confirmed (3-star and 4-star)
baryon states from the Particle Data Group [40]. The internal spin, light-front internal
17
orbital angular momentum and radial quantum number assignment of the Nand ∆ exci-
tation spectrum is found from the total angular momentum-parity PDG assignment using
the conventional SU(6) ⊃SU(3)f lavor ×SU(2)spin multiplet structure [41], but other model
choices are also possible [42]. Further details can be found in [16].
The lowest possible stable state, the nucleon N1
2
+(940), corresponds to n= 0 and ν=
0. This fixes the scale √λB=MP/2. The resulting predictions for the spectroscopy of
the positive-parity spin-1
2light nucleons are shown in Fig. 1(a) for the parent Regge
trajectory for n= 0 and ν= 0,2,4,···, L, where Lis the relative LF angular momentum
between the active quark and the spectator cluster. Thus the dimensionless constant fin
the superpotential (64) is f=L+1
2for the plus parity nucleon trayectory. The predictions
for the daughter trajectories for n= 1, n= 2,··· are also shown in this figure. Only
confirmed PDG [40] states are shown. The Roper state N1
2
+(1440) and the N1
2
+(1710) are
well described in this model as the first and second radial excited states of the nucleon.
The newly identified state, the N3
2
+(1900) [40] is depicted here as the first radial excitation
of the N3
2
+(1720). The model is successful in explaining the J-degeneracy for states with
the same orbital angular momentum observed in the light baryon spectrum, such as the
L= 2 plus parity doublet N3
2
+(1720) −N5
2
+(1680), which corresponds to and J=3
2and 5
2
respectively (See Fig. 1(a)).
In Fig. 1(b) we compare the positive parity spin-1
2parent nucleon trajectory with
the negative parity spin-3
2nucleon trajectory. As it is shown in this figure, the gap scale 4λ
determines not only the slope of the nucleon trajectories, but also the spectrum gap between
the plus-parity spin-1
2and the minus-parity spin-3
2nucleon families, as indicated by arrows
in this figure. This means the respective assignment ν=Land ν=L+ 1 for the lower and
upper trajectories in Fig. 1(b), or f=L+1
2and f=L+3
2respectively. The degeneracy
of states with the same orbital quantum number Lis also well described, as for example
the degeneracy of the L= 1 minus-parity triplet N1
2
−(1650), N3
2
−(1700), and N5
2
−(1675),
which corresponds respectively to J=1
2,3
2and 5
2(See: Fig. 1(b)).
Baryons with negative parity and internal spin S=1
2, such as the N1
2
−(1535), as well as
baryon states with positive parity and internal spin S=3
2, such as the ∆3
2
+(1232) are well
described by the assignment ν=L+1
2, or f=L+ 1. This means, for example, that the
positive and negative-parity ∆ states are in the same trajectory consistent with experimental
observations, as depicted in Fig. 1(d). The newly found state, the N3
2
−(1875) [40], depicted
18
TABLE I. Classification of confirmed baryons listed by the PDG [40]. The labels L,Sand nrefer
to the internal orbital angular momentum, internal spin and radial quantum number respectively.
The even-parity baryons correspond to the 56 multiplet of SU (6) and the odd-parity to the 70.
SU (6) S L n Baryon State
56 1
20 0 N1
2
+(940)
3
20 0 ∆3
2
+(1232)
56 1
20 1 N1
2
+(1440)
3
20 1 ∆3
2
+(1600)
70 1
21 0 N1
2
−(1535) N3
2
−(1520)
3
21 0 N1
2
−(1650) N3
2
−(1700) N5
2
−(1675)
1
21 0 ∆1
2
−(1620) ∆3
2
−(1700)
56 1
20 2 N1
2
+(1710)
1
22 0 N3
2
+(1720) N5
2
+(1680)
3
22 0 ∆1
2
+(1910) ∆3
2
+(1920) ∆5
2
+(1905) ∆7
2
+(1950)
70 3
21 1 N1
2
−N3
2
−(1875) N5
2
−
3
21 1 ∆5
2
−(1930)
56 1
22 1 N3
2
+(1900) N5
2
+
70 1
23 0 N5
2
−N7
2
−
3
23 0 N3
2
−N5
2
−N7
2
−(2190) N9
2
−(2250)
1
23 0 ∆5
2
−∆7
2
−
56 1
24 0 N7
2
+N9
2
+(2220)
3
24 0 ∆5
2
+∆7
2
+∆9
2
+∆11
2
+(2420)
70 1
25 0 N9
2
−N11
2
−
3
25 0 N7
2
−N9
2
−N11
2
−(2600) N13
2
−
in Fig. 1(c) is well accounted as the first radial excitation of the N3
2
−(1520). The degeneracy
of the L= 1 minus-parity doublet N1
2
−(1535) −N3
2
−(1520) for J=1
2and 3
2is also well
described. Likewise, the ∆(1600) corresponds to the first radial excitation of the ∆(1232)
19
LL
1-2014
8844A5
M2
(GeV2)
(c) (d)
n=2 n=1 n=0
n=2n=3 n=1 n=0
Δ(1950)
Δ(1920)
Δ(1700)
Δ(1620)
Δ(1910)
Δ(1905)
Δ(1232)
Δ(1600)
Δ(2420)
N(1875)
N(1535)
N(1520)
0 2 4
1
3
5
7
20 4
0
2
4
6
M2
(GeV2)
(a) (b)
n=2n=3 n=1 n=0
N(1710)
N(1700)
4λ
N(1720)
N(1680)
N(1675)
N(2250)
N(2600)
N(2190)
N(2220)
ν=L
ν=L+1
N(1650)
N(940)
N(1440)
N(940)
N(1900) N(2220)
N(1720)
N(1680)
0 2 4
0
2
4
6
0 2 4 6
0
4
8
FIG. 1. Orbital and radial baryon excitation spectrum. Positive-parity spin-1
2nucleons (a) and
spectrum gap between the negative-parity spin- 3
2and the positive-parity spin-1
2nucleons families
(b). Minus parity spin-1
2N(c) and plus and minus parity spin-1
2and spin-3
2∆ families (d). We
have used in this figure the value √λB= 0.49 GeV for nucleons and 0.51 GeV for the Deltas.
as shown in Fig. 1(d). The model explains the degeneracy of the L= 2 plus-parity
quartet ∆1
2
+(1910), ∆3
2
+(1920), ∆5
2
+(1905), and ∆ 7
2
+(1950) which corresponds to J=1
2,
3
2,5
2and 7
2respectively (See: Fig. 1(d)). Our results for the ∆ states agree with those of
Ref. [43]. “Chiral partners” such as the N1
2
+(940) and N1
2
−(1535) nucleons with the same
total angular momentum J=1
2, but with different orbital angular momentum and parity
are non-degenerate from the onset. To recapitulate, the parameter f, has the internal spin
Sand parity Passignment given in Table II, which is equivalent to the assignment given
in [44].
20
TABLE II. Orbital quantum number assignment for the superpotential parameter ffor baryon
trajectories according to parity Pand internal spin S.
S=1
2S=3
2
P = + f=L+1
2f=L+ 1
P = – f=L+ 1 f=L+3
2
This particular assignment successfully describes the full light baryon orbital and radial
excitation spectrum, and in particular the gap between trajectories with different parity and
internal spin [44]. The assignment ν=Lfor the lowest trajectory, the proton trajectory,
is straightforward and follows from the stability of the ground state, the proton, and the
mapping of AdS to light-front physics. The assignment for other spin and parity baryons
states, given in Table II, is motivated by the observed spectrum. It is hoped that further
analysis of the different quark configurations and symmetries of the baryon wave function [36,
45,46] will indeed explain the assignment of the dimensionless parameter f.
If we follow the non-SU (6) quantum number assignment for the ∆ 5
2
−(1930) given in
Ref. [36], namely S= 3/2, L= 1, n= 1, we find with the present model the value
M∆(1930) = 4√λB= 2MP, also consistent with the experimental result 1.96 GeV [40]. An
important feature of light-front holography and supersymmetric LF quantum mechanics is
the fact that it predicts a similar multiplicity of states for mesons and baryons, consistent
with experimental observations [36]. This property is consistent with the LF cluster de-
composition of the holographic variable ζ, which describes a system of partons as an active
quark plus a cluster of n−1 spectators [37]. From this perspective, a baryon with 3 quarks
looks in light-front holography as a quark–diquark system.
Another interesting consequence of the supersymmetric relation between the plus and
minus chirality states, is the equal equal probability expressed by (85). This remarkable
equality means that in the light-front holographic approach described here the proton’s spin
Jz=Lz+Szis carried by the quark orbital angular momentum: hJzi=hLz
qi=±1/2 since
hSz
qi= 0.
21
VI. CONCLUSIONS AND OUTLOOK
In this article we have shown how superconformal quantum mechanics [21,22] can be
extended to the light-front and how it can be precisely mapped to holographic QCD. We have
also examined the higher half-integer spin representations of the model by embedding the
resulting Dirac invariant light-front wave equation in AdS space. This procedure introduces
a scale in the Hamiltonian equations and completely fixes the light-front potential in the
Dirac equation introduced in Refs. [18,19]. In this approach the main features of the
observed light-baryon spectrum are described.
The construction procedure is similar to that of bosons [4,5]. Both are based on the
breaking of conformal invariance within the algebraic structure, by a redefinition of the
quantum mechanical evolution in terms of a superposition of the operators of the conformal
or superconformal algebras. Since the generators have different dimensions this amounts
to the introduction of a scale in the Hamiltonian while maintaining a conformal action.
Compared with the holographic construction for baryons, this unified approach is more
satisfactory. In contrast to the meson case, the dilaton in the fermion action has no effect
on the baryon spectrum. Consequently, a Yukawa potential must be introduced by hand to
break conformal invariance. Here, the same underlying principle is used to introduce a mass
scale and generate the masses for mesons and baryons from a spectrum generating algebra.
For baryons the quantum mechanical evolution is determined from a supercharge which is
a superposition of elements of the superconformal algebra [22]. In fact, the introduction of
the generator S(the square root of the generator of conformal transformations K) is the
key step for extending the dAFF [4,5] procedure for obtaining a confining potential in the
LF Dirac equation for baryons.
Mapping the results to light-front bound-state equations leads to a linear potential in
the light-front Dirac equation and to a harmonic potential with additional constants in the
quadratic Hamiltonian for fermions. In contrast to the case of mesons, there is no possibility
to shift the energy levels by adding a constant to the linear potential in the light-front Dirac
equation. Therefore superconformal quantum mechanics, together with the introduction of
the scale according to Fubini and Rabinovici [22], fixes completely the fermionic Hamil-
tonian. The equations of motion obtained by following this procedure are equivalent to
the holographic light-front equations obtained from the fermion Lagrangian in AdS5, with a
22
Yukawa coupling providing the effective potential. In the bosonic case light-front holographic
QCD yields a J-dependent constant from the holographic embedding – in addition to the
confining harmonic potential obtained from conformal quantum mechanics [4] – which leads
to a J-dependent level shift [5]. Such a level shift is not possible for fermions, and therefore
there is a spin-Jdegeneracy for states at fixed Land n, an important characteristic which
is actually observed in experiment. The model is also consistent with similar Regge meson
and baryon spectra and similar multiplicity of states for mesons and baryons. In effect, the
light-front Dirac equation for baryons described here is effectively a quark-diquark model.
However, a quark-diquark construction is not imposed, but it is a natural consequence of the
light-front cluster decomposition which follows from the LF embedding in AdS space [37].
In this approach the quark and diquark are both massless.
In this paper we have described a mechanism for the emergence of a confining light-front
Hamiltonian for hadrons. A mass scale √λand confining potentials appear in the light-front
Schr¨odinger and Dirac equations, consistent with the conformal invariance of the action, by
applying the group-theoretical methods of Refs. [4,22]. We have given a relation between
the dimensionless quantities L,for g, and µR occurring in the light-front Hamiltonian, the
quantum mechanical evolution operator in the algebraic approach, and the wave equations in
AdS5, respectively (See Eqs. (12), (20), (62) and Table II). We expect that further analysis
of the different quark configurations and symmetries of the hadron wavefunctions will shed
further light on the detailed relations between these dimensionless parameters.
Even if a supersymmetric connection inspired by the universality of the Regge trajectories
for bosons and baryons was our starting point, in the context of this article the supersym-
metric construction of baryonic states refers to the “supersymmetry” between positive and
negative chirality of light-front spinors. In this case supersymmetry is not broken since
there is a perfect pairing for each baryonic state including the ground state, consistent with
parity invariance. This does not exclude the possible supersymmetric connections between
mesons and baryons which would be manifest as a consequence of confinement dynamics.
In fact, although the form of the potential is fixed in both cases by the dAFF procedure
and its extension to the superconformal algebra, the numerical values of the confining scales
are a priori not related. Nevertheless the values of λfor the coefficient of the confining
potentials come out to be similar in both cases with similar spacing between the orbital and
radial hadronic excitations. This suggests a supersymmetric relation between the underlying
23
dynamics of the observed bosonic and fermionic hadrons. In this case, supersymmetry is
broken since the ground state, the pion, is massless in the chiral limit and is not paired. We
shall treat this subject elsewhere.
ACKNOWLEDGMENTS
We thank Vittorio de Alfaro, Cedric Lorc´e and Michael Peskin for helpful comments.
This work is supported by the Department of Energy contract DE–AC02–76SF00515.
Appendix: Supercharges and Ladder Operators
The supercharge operator R(80) in the light-front quantum mechanical representation
discussed in Sect. IV can be expressed as
R=Q+λS =η b, (A.1)
R†=Q†+λS†=η†b†,(A.2)
where the spinor operators ηand η†in a 4 ×4 matrix representation are
η=
0I
0 0
, η†=
0 0
I0
,(A.3)
with Ia two-dimensional unit matrix, and the operators band b†are given by
bν=d
dζ +ν+1
2
ζ+λζ, (A.4)
b†
ν=−d
dζ +ν+1
2
ζ+λζ. (A.5)
The LF Hamiltonian HLF (81) is conveniently factorized in terms of the linear operators b
Hν
LF ={R, R†}=
bνb†
ν0
0b†
νbν
,(A.6)
and is thus integrable [47,48].
Consider the eigenvalue equation for bνb†
ν
−d2
dx2−1−4ν2
4x2+κ2ζ2+ 2κ(ν+ 1)φν(x) = φν(x),(A.7)
24
where x=ζM and κ=λ/M. Equation (A.7) is equivalent to bνb†
ν|νi=|νi. It is also simple
to verify that b†
ν|νi ∼ |ν+ 1ior
−d
dζ +ν+1
2
ζ+λ ζφν(ζ)∼φν+1(ζ).(A.8)
Likewise, one can show that bν|νi ∼ |ν−1i.
We now construct a new supercharge Tand its adjoint T†as the linear superposition [22]
T=Q†−λS†=η†a, (A.9)
T†=Q−λS =η a†,(A.10)
where
aν=−d
dζ +ν+1
2
ζ−λζ, (A.11)
a†
ν=d
dζ +ν+1
2
ζ−λζ. (A.12)
One can show that the operator (A.11) lowers the radial quantum number nby one unit
and raises νby one unit
a|n, νi ∼ |n−1, ν + 1i.(A.13)
For a given νthe lowest possible state corresponds to n= 0. Consequently the state
|n= 0, νiis annihilated by the action of the operator a,a|n= 0, νi= 0, or equivalently
d
dζ −ν+1
2
ζ+λζφn=0
ν(ζ) = 0,(A.14)
with solution
φn=0
ν(ζ) = Cνζ1/2+νe−λζ2/2,(A.15)
where Cνis a constant. Writing
φν(ζ) = Cνζ1/2+νe−λζ2/2Gν(ζ),(A.16)
and substituting in (A.8) we get
2x Gν(x)−G′
ν(x)∼x Gν+1(x),(A.17)
with x=√λ ζ, a relation which defines the confluent hypergeometric function U(n, ν + 1, x)
in terms of U(n, ν, x) [49]
U(n, ν + 1, x) = U(n, ν, x)−U′(n, ν, x),(A.18)
25
or equivalently
2x U(n, ν + 1, x2) = 2x U(n, ν, x2)−dU ′(n, ν, x2)
dx .(A.19)
Thus the normalizable solution to the eigenvalue equation bb†φ(ζ) = M2φ(ζ):
φn,ν (ζ) = Cνζ1/2+νe−λζ2/2Lν
n(λζ2).(A.20)
The solution also follows from the iterative application of the ladder operators following the
procedure described in [50]. We find
φ(ζ)n,ν ∼ζ1/2−νeλζ2/21
ζ
d
dζ n
ζ2(n+ν)e−λζ2,(A.21)
with eigenvalues
M2= 4λ(n+ν+ 1).(A.22)
Since we know the general solution for the upper component of the spinor wavefunction
φν, it is straightforward to compute the lowest component b†φν, with identical mass, by
applying the supercharge operators. We find
T
φn,ν
0
= 0,(A.23)
R†
φn,ν
0
=Cn,ν
0
φn,ν+1
,(A.24)
with
Cn,ν =rλ
n+ν+ 1.(A.25)
Thus the solution
ψ(ζ) = ψ+u++ψ−u−(A.26)
=Cz 1
2+νe−λζ2/2"Lν
nλζ2u++√λζ
√n+ν+ 1Lν+1
nλζ2u−#,(A.27)
with normalization
Zdζ ψ2
+(ζ) = Zdζ ψ2
−(ζ).(A.28)
Identical results follow by solving directly the Dirac equation (56) for the conformal super-
potential (55) with u=λζ.
26
The light-front quantum mechanical evolution operator Hν
LF (A.6) was constructed in
terms of the supercharges Rand R†. We can also construct a light-front Hamiltonian Hν
LF
in terms of the supercharges Tand T†given by (A.9) and (A.10):
Hν
LF ={R, R†}=
a†
νaν0
0aνa†
ν
.(A.29)
The light-front Hamiltonians HLF (A.29) and HLF (A.6) are related by HLF (λ) = HLF (−λ).
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27
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28
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29
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1−xPn−1
j=1 xjb⊥jwhere x=xnis the
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