Unitarity of Deconstructed Five-Dimensional Yang-Mills Theory

11 Pages·0.17 MB·English

by  R. Sekhar Chivukula

BUHEP-02-02
UTHEP-02-26
hep-ph/0201164
Unitarity of Deconstructed Five-Dimensional
Yang-Mills Theory
2
0 R. Sekhar Chivukulaa† and Hong-Jian He b‡
0
2
a Department of Physics, Boston University,

n
590 Commonwealth Ave., Boston, Massachusetts 02215, USA
a
J
b Center for Particle Physics,
9
University of Texas at Austin, Texas 78712, USA
2

2 January, 2002
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6
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Abstract
p
-
p
e Thelow-energypropertiesofacompactifiedfive-dimensionalgaugetheorycanbereproduced
h
in a four-dimensionaltheory with a replicatedgauge groupandan appropriategaugesymmetry
:
v breaking pattern. The lightest vector bosons in these “deconstructed” or “remodeled” theories
i
X have masses and couplings approximately equal to those of the Kaluza-Klein tower of massive
vectorstates presentin a compactified higher-dimensionalgaugetheory. We analyze the unitar-
r
a ityoflow-energyscatteringofthemassivevectorbosonsinadeconstructedtheory,andexamine
the relationship between the scale of unitarity violation and the scale of the underlying chiral
symmetry breaking dynamics which breaks the replicated gauge groups. As in the case of com-
pactified five-dimensional gauge theories, low-energy unitarity is ensured through an interlacing
cancellation among contributions from the tower of massive vector bosons. We show that the
behavior of these scattering amplitudes is manifest without such intricate cancellations in the
scattering of the would-be Goldstone bosons of the deconstructed theory. Unlike compactified
five-dimensional gauge theories, the amplitude for longitudinal vector boson scattering in de-
constructed theories does grow with energy, though this effect is suppressed by 1/(N +1), with
N +1 being the number of replicated gauge groups.
†Electronic addresses: [email protected]
‡Electronic addresses: [email protected]
Theworldmaybeconsistentlydescribedbyacompactifiedhigherdimensionaltheory,manifested
via additional towers of massive Kaluza-Klein (KK) states at low energies. Recently, it has been
shown [1, 2] that the low-energy properties of a compactified five-dimensional gauge theory may
be reproduced from a four-dimensional theory with a replicated gauge group structure and an
appropriate symmetry breaking pattern1. A simple scheme is illustrated by the “aliphatic moose”
shown in Figure1. In this “moose” or “quiver” diagram [11, 12], the circles represent N + 1
SU(m) gauge groups (labeled by j = 0,1,2, ,N) and the directed lines represent the Goldstone
···
bosons from the spontaneous symmetry breaking of the two adjacent SU(m) groups down to their
diagonalsubgroup. Thus,wehavethegaugesymmetrybreakingpattern, SU(m)N+1 SU(m) ,
diag
→
generating N(m2 1) massive spin-1 vector states.
−
The Goldstone bosons may be collected into SU(m) matrix fields U (j = 1,2, ,N) which
j
···
transform under the adjoining SU(m) SU(m) groups according to
j−1 j
⊗
†
U (x) Ω (x)U (x)Ω (x) , (1)
j → j−1 j j
where Ω is the SU(m) gauge transformation. We thus write U as [13],
j(j−1) j(j−1) j
i2πa(x)Ta
j
U (x) = exp , (2)
j
v
(cid:18) (cid:19)
where Ta are the SU(m) generators (normalized by TrTaTb = δab/2), and v is the analog of the
{ }
QCD pion decay constant f which characterizes chiral symmetry breaking.
π
U U U
1 2
N
0 1 2 N
Figure 1: The “aliphatic moose” model with N +1 replicated SU(m) gauge groups.
Regardless of the underlying dynamics responsible for the gauge symmetry breaking, the low-
energy properties of this model may be most economically described by an effective Lagrangian
with only the gauge and Goldstone degrees of freedom. The leading terms in this description are
1 N v2 N
= FaµνFa + Tr DµU D U† , (3)
L − 4 j jµν 4 j µ j
Xj=0 Xj=1 (cid:16) (cid:17)
aµν
where the F is the field-strength of gauge group SU(m) , and the covariant derivative is,
j j
DµU = ∂µU igAaµ TaU +igAbµU Tb. (4)
j j − j−1 j j j
1Related issues have been considered previously in a variety of contexts,see [3, 4, 5, 6, 7, 8, 9, 10].
1
Following Ref.[1], we will refer the above model as a “deconstructed” theory. In the effective
Lagrangian(3),thereareN(m2 1)massivegaugebosonswhichacquiretheirmassesfromabsorbing
−
thecorrespondingwould-beGoldstonebosonsviatheHiggs mechanism, andnoscalars remain. The
nonlinearsigma model(s)inthedeconstructed theory arenotrenormalizable. Naive powercounting
[14, 15, 16] implies such an effective theory is valid only for scales . 4πv, and the underlying
dynamics of chiral symmetry breaking must become manifest at or below this scale.
Note that we have chosen the gauge couplings (g) and the vacuum expectation values (v) of the
deconstructed theory to be the same for all gauge groups and symmetry breakings. This pattern
was chosen so as to reproduce the low-energy properties of a five-dimensional (5D) SU(m) Yang-
Mills theory in which the fifth dimension is compactified to a line segment 0 x5 πR. This
≤ ≤
compactification can be done consistently by an orbifold projection as follows: restrict the the
gauge fields of the five-dimensional theory AM(xN) [M,N (µ, 5) with µ (0,1,2,3)] to those
∈ ∈
periodic in x5 with period 2πR and further impose a Z symmetry,
2
b
Aµ(xν,x5)= +Aµ(xν, x5), A5(xν,x5)= A5(xν, x5). (5)
− − −
These projections force the gauge-covariant boundary conditions,
b b b b
F5N = FN5 = 0 (6)
atx5 = 0andπR. AnalyzingtheKKmodesofthiscompactifiedtheoryshowsthat,inunitarygauge,
b b
one has an infinite tower of massive SU(m) adjoint vector fields of mass n/R (n = 0,1, ). In the
···
compactified 5D theory, the self-interactions of the zero-mode fields are that of a four-dimensional
(4D) Yang-Mills theory with gauge-coupling g = g /√πR, where g is the 5D Yang-Mills coupling
5 5
with dimension of (mass)−1/2. The interactions of the KK modes amongst themselves and with the
zero mode gauge-bosons are given by Yang-Mills like couplings [2, 17, 18].
Compactified 5D Yang-Mills theory results in an effective 4D KK theory which has the remark-
able property [18] that low-energy unitarity is ensured through an interlacing cancellation among
contributions from the relevant KK levels, and is delayed to energy scales higher than the cus-
tomary limit of Dicus-Mathur and Lee-Quigg-Thacker [19, 20, 21, 22] through the introduction of
additional vector bosons rather than Higgs scalars. In this Letter, we analyze the unitarity of low-
energy massive vector boson scattering in the deconstructed theory, and examine the relationship
between the scale of unitarity violation and the scale of the underlying chiral symmetry breaking
dynamics responsible for spontaneously breaking the replicated gauge groups. We show that the
interactions of the massive vector bosons in the deconstructed theory are, for levels small compared
with N, precisely the same as those of the 4D KK theory. We explicitly show that, up to correc-
tions suppressedby1/N, theinteractions amongthewould-beGoldstone bosonsencoded ineqn.(3)
match exactly with the interactions among the corresponding modes of Aa5 absorbed through the
n
geometrical Higgs mechanism in the compactified 5D gauge theory.
We begin by reviewing the correspondence between the 4D aliphatic moose model [cf. eqn.(3)]
and the orbifold compactification of 5D Yang-Mills theory [2]. Diagonalizing the N +1 by N +1
dimensional mass-squared matrix in the aliphatic theory, we find the mass eigenvalues
nπ
M = gvsin , (7)
n
2(N +1)
with n (0,1,2, ,N), which correspond to eigenstates
∈ ···
1
Aa = (Aa +Aa + +Aa ), (8)
0µ √N +1 0µ 1µ ··· Nµ
e
2
for the remaining massless gauge field, and
N
2 1 nπ
Aa = cos k+ Aa , (n = 1,2, ,N), (9)
nµ N +1 2 N +1 kµ ···
r k=0 (cid:20)(cid:18) (cid:19) (cid:21)
X
e
forthethemassiveadjointvector bosons. Themassless fields Aa belongtotheresidualunbroken
{ 0µ}
gauge group SU(m) with coupling g = g/√N +1.
diag
Comparing the mass spectrum (7) of the deconstructed theeory (for n N +1) with the linear
≪
spectrum n/R in the KK theory, we seeethat the two coincide under the identification [1, 2],
1 πgv πgv
= = , (10)
R 2(N +1) 2√N +1
e
and eqn.(7) can be expanded as,
n π2 n 2 n4
M = 1 +O M [1 δ ], (11)
n R − 24 N +1 N4 ≡ n − n
" #
(cid:18) (cid:19) (cid:18) (cid:19)
where M = n/R and δ = (n2/N2). The identification (10) corresponds to interpreting the
n n
O
Moose diagram itself as a discretized fifth-dimension with a lattice spacing,
πR 2 1+N−1 2√1+N−1
a = = = . (12)
N gv gv√N
(cid:0) (cid:1)
The spectrum of the deconstructed theory approximates the linear spectrum of the 4D KK theory
e
so long as n N +1, i.e., M 1/a.
n
≪ ≪
The correspondence between the deconstructed theory and 4D KK theory may be completed by
identifying the couplings of the unbroken massless gauge group,
1 πR
= , (13)
g2 g2
5
which yields
e
2g
g = . (14)
5 v
r
This correspondence implies that
a 1 1+N−1
= = , (15)
g2 N g2 g2
5
andthe “bare” couplingof the deconstructed theory, g, may beidentified with theeffective strength
e
of the gauge coupling in the 5D Yang-Mills theory underlying the 4D KK theory.
As indicated in eqn.(11), the exact correspondence between the spectra of the deconstructed
theory and the 4D KK theory is realized only for sufficiently large N. We will explicitly show that
the vector-boson scattering amplitudes agree in these theories as well, up to corrections suppressed
by 1/N. The “continuum limit” corresponds to a 0 for fixed R and g, or, equivalently, N
→ → ∞
e
3
with2 g = (√N ). From eqn.(10), therefore, we deduce that v = (√N ) when approaching the
O O
continuum limit.
A 5D gauge theory is nonrenormalizable, and one manifestation of this is the bad high-energy
behavior of massive vector-boson scattering. In Ref.[18], it is shown that tree-level gauge boson
scattering in the 5D SU(m) Yang-Mills theory violates unitarity at an energy scale of the order
96π 1
√s = E Λ= , (16)
cm ≤ 23m g2
5
and therefore this theory is, at best, a low-energy effective theory valid only up to a scale of order
Λ. In the deconstructed theory, from eqn.(13), Λ corresponds to an energy scale of order
96π 1
Λ , (17)
≃ 23m g2Na
which is higher than 1/a so long as
e
3.6
g . . (18)
√mN
When the deconstructed theory is embeddedinto a 4D renormalizable high-energy theory [1, 2], the
e
4D theory provides a “high-energy” completion of thecompactified 5D Yang-Mills theory. From the
considerations above, we see that for weak or moderate coupling and modest N, a deconstructed
theory provides a high-energy completion which respects the bound in eqn.(16).
As noted above, the deconstructed theory itself involves chiral symmetry breaking dynamics
in order to provide the Goldstone-boson “link” fields that allow particles to “hop” in the fifth
dimension. Powercounting[14,15,16]showsthatthenon-linearsigmamodellow-energydescription
must break down at a scale . 4πv. Given the effective lattice spacing eqn.(12), we see that,
1/a < 4πv, provided
√N +1
g . 8π . (19)
N
In this case the non-linear sigma model description can remain valid up to the scale 1/a, at which
e
the model no longer behaves like the compactified effective 4D KK theory. In what follows we will
investigate the scattering of massive vector-bosons and their correspondingGoldstone bosons in the
deconstructed theory for energy scales less than 1/a, therefore we need not be concerned about the
underlying 4D chiral symmetry breaking dynamics.
To analyze the relevant scattering processes, we start by deriving the unitary gauge Lagrangian
of the deconstructed theory in Fig.1. In this gauge all link-fields U are set to the identity
j
{ }
via appropriate SU(m) gauge transformations. Expressing all the vertices in terms of the mass-
eigenstate gauge fields, we derive the interaction Lagrangian
N
= gCabc ∂ Aa AbµAcν +∂ Aa (AbµAcν +AbµAcν)
Lgauge − µ 0ν n n µ nν 0 n n 0
nX=1h i
2Asapracticalmatter,eofcourse,thecoeuplinegofetherepliceatedgeaugeegroupseg iseboundedbyO(4π). Hence,there
is a bound on how large N can be for a fixed size of the low-energy coupling g. This is similar to the bound on the
underlyingscaleofthe5D gaugetheoryrelativetothecompactification scalearising from theconstraint thatthe4D
gauge coupling has a finitesize. e
4
N
g
Cabc ∆ (n,m,ℓ)∂ Aa AbµAcν
−√2 3 µ nν m ℓ
n,m,l=1
X
e
g2 N e e e
CabcCade Ab Ac AdµAeν +all permutations (20)
− 4 0µ 0ν n n
nX=1h i
e
g2 Ne e e e
CabcCade ∆ (n,m,ℓ) Ab Ac AdµAeν +all permutations
−4√2 3 0µ nν m ℓ
e n,mX,ℓ=1 h i
g2 N e e e e
CabcCade ∆ (n,m,ℓ,k)Ab Ac AdµAeν ,
− 8 4 nµ mν ℓ k
n,m,ℓ,k=1
X
e
e e e e
with ∆ and ∆ given by
3 4
∆ (n,m,ℓ) = δ(n+m ℓ)+δ(n m+ℓ)+δ(n m ℓ),
3
− − − −
∆ (n,m,ℓ,k) = δ(n+m+ℓ k)+δ(n+m ℓ+k)+δ(n m+ℓ+k)+ (21)
4
− − −
δ(n+m ℓ k)+δ(n m ℓ+k)+δ(n m+ℓ k)+δ(n m ℓ k),
− − − − − − − − −
so long as3 (n, m, ℓ, k) N+1. Theseinteractions are precisely those found in the 4D KK theory
≪
[18, 17, 2].
Thedeconstructed theory describes asetof massive self-interacting vector bosonswitha charac-
teristiccoupling g. Thetraditionalarguments[19,20,21,22]suggestthatthescatteringamplitudes
of longitudinally polarized vector bosons at level n N +1 would grow with energy and violate
≪
unitarity at an eneergy scale,
4πM 2π2nv 4nπ2 4nπ2g
E⋆ n = = . (22)
∼ g ≈ √N +1 gNa g2
5
This cannot be the case because, for given v, E⋆ could be made muchesmaller than 4πv – the scale
at which the chiral symmetry breaeking dynamics is expeected to enter the deconstructed theory.
Furthermore, E⋆ could be made much smaller than the scale 1/a below which the theory should
correspond to the 4D KK theory. This provides a clue for how to proceed: the vector boson
masses in the 4D KK theory arise through a geometrical Higgs mechanism, and unitarity is ensured
through an interlacing cancellation among the contributions of the relevant gauge KK modes [18].
We therefore expect that there will be similar cancellations in the deconstructed theory. Since the
unitary-gauge Lagrangian (20) is identical to that derived in the compactified 5D theory [18, 17, 2]
for low KK-levels, the deviations of the vector-boson amplitude in the deconstructed theory from
that in the compactified 5D theory can only come from the modification term in the mass-spectrum
(11). After a careful analysis of the gauge amplitudes with the mass expansion of (11) and the
high energy expansion of M /E, we find that the individual (E4) terms completely cancel and
n
O
the non-canceled (E2) terms appear only at the order 1/N. For instance, we derive the following
O
(E2) amplitudes for the elastic scattering of the longitudinal gauge bosons, AanAbn AcnAdn,
O L L → L L
g2δ s 9 11 9 11
AanAbn AcnAdn = n cCabeCcde+ + c CaceCbde+ e ec CaedeCebce ,
T L L → L L M2(N +1) − 2 2 2− 2
h i n (cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
(23)
e e e e
3If (n, m,ℓ, k) are such that n+m+ℓ or n+m+ℓ+k equals 2q(N +1) for q = 1,2,···, the factors ∆3,4 will
havean additional contribution equalto (−1)q.
5
which, as expected, is proportional to the nonlinear modification δ in eqn.(11), and has the coef-
n
ficient g2δ /[M2(N +1)] 1/[6v2(N +1)] suppressed by 1/(N +1). Here, c =cosθ and θ is the
n n ≃
scattering angle. For the inelastic scattering AanAbn AcmAdm (n = m), we arrive at
(cid:0) (cid:1) L L → L L 6
s
AanAbn AcmAdm = 2cCeabeCecde+(e3+5ec)CaceCbde+(3 5c)CadeCbce , (24)
T L L → L L 6v2(N +1) − −
h i h i
whiceh, weith thee aideof Jacobi identity CabeCcde +CaceCdbe +CadeCbce = 0, can be related to the
(E2) elastic amplitude (23) via
O
2
AanAbn AcmAdm AanAbn AcnAdn . (25)
T L L → L L ≃ 3T L L → L L
h i h i
Theunitarity analysisabeoveeisperfeormeedinthevecteor beosonseectoer ofthedeconstructedtheory.
Similar high energy behavior must also arise in the would-be Goldstone boson sector of the decon-
structed theory. Since the deconstructed theory is based on the spontaneous symmetry breaking
SU(m)N+1 SU(m) , in the corresponding R gauge gauge [cf. eqn.(29)] we can derive the
diag ξ
→
equivalence theorem
Aa (p ),Ab (p ), = C πa(p ),πb (p ), + (M /E), (26)
T nL n mL m ··· modT n n m m ··· O n,m,···
h i h i
where the leveles (n,m,·e··) = 1,2,···,N, and eache externeal momentum is put on mass-shell, p2n =
M2, etc. Here the fields πa are the would-be Goldstone bosons “eaten” by the corresponding
n { n}
mass eigenstate vector fields Aa . This is analogous to the traditional equivalence theorem in the
{ n}
Standard Model [23, 21, 24e, 25, 26, 27, 28, 29], and the modification factor Cmod = 1+ (loop)
O
appears at loop level [25, 26, 27e, 28, 29].
To analyze the Goldstone boson scattering, we will derive the complete R gauge Lagrangian for
ξ
the deconstructed theory. From the nonlinear dimension-2 term of Goldstone boson kinetic energy
in eqn.(3), we deduce the following bilinear gauge-Goldstone mixings,
N N
1
= gv Aaµ ∂ πa Aaµ∂ πa = M Aaµ∂ πa. (27)
Lmix −2 j−1 µ j − j µ j − n n µ n
Xj=1 h i nX=1
e e
Here M is given by eqn.(7) and the fields πa are the eigenstates of “eaten” Goldstone bosons
n { n}
defined by the orthogonal rotation,
e
N
2 nkπ
πa = sin πa, (n = 1,2, ,N). (28)
n N +1 N +1 k ···
k=1r (cid:18) (cid:19)
X
e
The bilinear mixing (27) can be eliminated by defining a general R gauge fixing term,
ξ
N
1 2
= ∂µAa ξ M πa , (29)
LGF − 2ξ nµ− n n n
n
Xn=0 (cid:16) (cid:17)
e e
where n = 0 corresponds to the usual gauge-fixing of the unbroken group SU(m) . The would-
diag
be Goldstone bosons πa acquire gauge-dependent masses M2 = ξ M2. The appropriate ghost
n πn n n
Lagrangian can be derived as well, though it is not explicitly needed for the current analysis.
e e
6
The interactions of the Goldstone bosons with the gauge bosons and among themselves arise
from the nonlinear sigma model, `a la Callan-Coleman-Wess-Zumino (CCWZ) [13]. However, the
“geometric” Goldstone sector in compactificted 5D Yang-Mills theory appears very different since
its Goldstone bosons Aa5 , the fifth components of the 5D gauge fields, interact at most bi-
{ n }
linearly with other gauge modes, and have no self-interaction among themselves [18]. How does this
highly nonlinear CCWZ Goldstone sector match with the geometric, linearized Goldstone sector
in compactificted 5D gauge theory? As we now show, the correspondence between the Goldstone
sector of the deconstructed theory and that of the compactified 5D Yang-Mills theory works at
(N0), with the corrections suppressed by powers of 1/N.
O
Using the orthogonal rotations (9) and (28), and collecting the terms of order N0 and N−1, we
arrive at, after a lengthy derivation, the following Goldstone interactions,
N g2 N
LO = +gCabc Abµπc ∂µπa M Aaµ + CabcCade Ab Adµπc πe
LGB 0 n n− n n 2 0µ 0 n n
Xn=1 (cid:16) (cid:17) e nX=1
eg eN e e e e e e e
+ Cabc ∆ (n,m,ℓ)Ab πc ∂µπa M Aaµ
√2 3 nµ m ℓ − ℓ ℓ
n,mX,ℓ=1 (cid:16) (cid:17)
e (30)
g2 e Ne e e ee e e
+ CabcCade ∆ (n,m,ℓ)Ab Adµπc πe
√2 3 0µ n m ℓ
n,m,ℓ=1
e X
g2 N e e e e ee e
+ CabcCade e ∆ (n,m,ℓ,k)Ab Adµπcπe,
4 4 nµ m ℓ k
n,m,ℓ,k=1
X
e
e e e ee e e
N 1 e 2
NLO = ∆ (n,m,ℓ,k) δabδcd+dabedcde
LGB 12(N +1)v2 4π m
n,m,ℓ,k=1 (cid:20) (cid:21)
X
e e
e πa∂µπb πc∂ πd ∂µπa∂ πb πcπd + (g), (31)
× n m ℓ µ k − n µ m ℓ k O
h (cid:16) (cid:17)i
where e ee e e e ee e e e
∆ (n,m,ℓ)= δ(n+m ℓ)+δ(n m+ℓ) δ(n m ℓ),
3
− − − − −
∆ (n,m,ℓ,k)= δ(n+m+ℓ k)+δ(n+m ℓ+k)+δ(n m+ℓ+k)+δ(n m ℓ+k)
4
e e e − e − e − − −
δ(n+m ℓ k) δ(n m+ℓ+k) δ(n m ℓ k), (32)
e e − e − − − − e − − e − − e
∆ (n,m,ℓ,k)= δ(n+m+ℓ k)+δ(n+m ℓ+k)+δ(n+m ℓ k)
4π
− − − −
e e e
δ(n+m ℓ+k) δ(n+m+ℓ k) δ(n m+ℓ+k) δ(n m ℓ k).
e e − e − − e − − − e − − − −
In (31), the factor 2 contains the m of SU(m); and the symmetric d-function is defined by
m e e e e
Ta, Tb = 1δab + dabcTc . The unspecified terms of (g) or smaller in (31) contain at most
{ } m O
one partial derivative and at least one gauge field, and are irrelevant to the (E2) leading behavior
O
of the Goldstone scattering amplitude to be derived shortley. Other contributions suppressed by
1/N2 or higher will also not be needed below.
Itisimportanttonotethattheleadingorder(LO)GoldstoneLagrangian LO in(30)containsat
LGB
7
mosttwoGoldstonefields. Itpreciselymatches4 withthatderivedinthecompactified5DYang-Mills
theory [18], under the identification of πa Aa5. To order N0, the correspondence between the
n ←→ n
CCWZ Goldstone sector of the deconstructed theory and the geometric, linearized Goldstone sector
in the compactificted 5D gauge theoryeis exact. At this order all non-renormalizable interaction
vertices containing of dimension > 4 disappear, and the leading order high-energy behavior of
Goldstone boson scattering matches that of compactified 5D Yang-Mills theory [18].
ThedeviationofthedeconstructedGoldstoneLagrangianfromthatofthecompactified5Dgauge
theory explicitly appearsat the next-to-leading order (NLO)of the 1/N-expansion. Thedimension-
6 quartic Goldstone vertices in (31) contain two partial derivatives, analogous to the usual chiral
Lagrangian of low energy QCD [14]. The additional factor 1/(N + 1) in (31) indicates that the
interactions, and therefore the amplitudes of scattering, among the eigenstate would-be Goldstone
bosons are suppressed by N +1. For the scattering processes πaπb πaπb and πaπb πa πb
n n → n n n n → m m
(n = m), at (1/N) and (E2/v2), we derive
6 O O
e e e e e e e e
3 3
πaπb πaπb = πaπb πa πb = sXab,cd+tXac,bd+uXad,bc , (33)
T n n → n n 2T n n → m m 2(N +1)v2
h i h i
(cid:2) (cid:3)
wheereeXab,cde e2δabδcd +deabeedcde. eForeSU(m) = SU(2), Xab,cd = δabδcd and (33) reduces to the
≡ m
familiar form of the ππ scattering of low energy QCD [14] except an overall factor 1/(N +1).
∼
Making use of the SU(m) identity
2
CabeCcde = δacδbd δadδbc + dacedbde dadedbce , (34)
m − −
(cid:16) (cid:17) (cid:16) (cid:17)
we find that the gauge amplitudes (23)-(24) fully agree to the corresponding Goldstone amplitudes
(33) at the same order of 1/N, satisfying the equivalence theorem (26).
Projecting the elastic amplitude πaπb πaπb to the isospin-singlet and spin-0 channel,
T n n → n n
i.e., [nn;nn]= [3/2(N +1)][s/16π2v2] for SU(2), we readily derive the unitarity bound,5
T00 (cid:2) (cid:3)
e e e e
1 4πv
[nn;nn] , = √s √N +1 , (35)
T00 ≤ 2 ⇒ ≤ √3π
which is, apparently, delayed relative to the customary unitary limit for ππ scattering by a factor
of (N +1)(2/3).
However, the deconstructed theory has many “KK” levels of πa, with n = 1,2, N, and we
p n ···
must consider coupled channels as well. Consider a normalized state, consisting of Goldstone boson
pairs with “KK” levels up to N0, e
1 N0
Ψab = πaπb , (36)
√N ℓ ℓ
0
(cid:12) E Xℓ=1(cid:12) E
4Attheleadingorderofthe1/N-expansi(cid:12)(cid:12)onthemassspectrum(cid:12)(cid:12)eofegaugefields{Aanµ}alsobecomesidenticaltothat
of the thecompactified 5D Yang-Mills theory,i.e., Mn =(n/R)[1+O(n2/N2)], as shown in eqn.(11).
5 Here we do not include the contribution of the leading order Lagrangian LLGOBeto the scattering amplitude since
it behavesas constant and does not grow with theenergy, as computed in Ref.[18].
8
from which we deduce the (1/N) scattering amplitude, at high energies √s 2M ,
O ≫ N0
N0 1
Ψab Ψcd = πaπb πaπb
T → N T ℓ ℓ → k k
0
(cid:2)(cid:12) (cid:11) (cid:12) (cid:11)(cid:3) ℓX,k=1 h i
=(cid:12) (N 1(cid:12)) πaπb πa πb e e + e πeaπb πaπb (37)
0− T n n → m m n6=m T n n → n n
h i(cid:12) h i
(cid:12) (N + 1)
= N + 1 eπaeπb eπaeπb (cid:12)= 0 e2 e sXaeb,cde+tXac,bd+uXad,bc .
0 2 T n n → m m (N +1)v2
h i
(cid:0) (cid:1) (cid:2) (cid:3)
Thus, we see that whenethee numebereof invoked “KK” levels reaches N N, we recover the
0
∼
customary unitarity limit, for SU(m) = SU(2),6
2 4g 2 4(1+N−1)
√s . √8πv = = , (38)
π g2 π ga
r 5 r
which is of the order v g/g2 1/(ga), and is neither enhanced by extra √N +1 [cf. the
∼ 5 ∼
single channel analysis in eqn.(35)] nor further suppressed by 1/√N +1 [cf. the naive estimate in
eqn.(22)].
In summary, we have systematically analyzed the gauge and Goldstone interaction Lagrangians
in four-dimensional deconstructed Yang-Mills theory. For the low “KK” levels, the gauge sector
differs from the compactified 5D theory only in the mass-spectrum, but the Goldstone sector ex-
plicitly differs in its interaction Lagrangian at order (1/N). We have analyzed the relationship
O
between the scale of unitarity violation in longitudinal vector-boson scattering and the scale of the
underlying chiral symmetry breaking dynamics responsible for spontaneously breaking the repli-
cated gauge groups. As in compactified 5D gauge theory, the low-energy unitarity of longitudinal
vector-boson scattering is ensured through an interlacing cancellation among contributions from
various “KK” levels. We have shown that the behavior of these amplitudes can be also understood
in the deconstructed theory by analyzing would-be Goldstone boson scattering via the equivalence
theorem. Taking into account the non-cancelled E2-contributions at the order 1/N, we find that
unitarity violation in the deconstructed theory is delayed to the intrinsic ultraviolet scale 1/g2 or
5
1/a, and is above the customary Dicus-Mathur/Lee-Quigg-Thacker limit. We have also demon-
strated explicitly the correspondence between the Higgs mechanism in the 4D deconstructed theory
and the “geometric Higgs mechanism” in the compactified 5D gauge theory.
Acknowledgments
We thank Duane Dicus, Howard Georgi and Chris Hill for discussions. This work was supported
by the Department of Energy under grants DE-FG02-91ER40676 and DE-FG03-93ER40757.
References
[1] N. Arkani-Hamed, A. G. Cohen, and H. Georgi, Phys. Rev. Lett. 86, 4757 (2001), hep-
th/0104005.
6The limit for thearbitrary SU(m) varies only by a factor of O(m/2)=O(1), with no conceptual difference.
9