On the Couplings of the Rho Meson in AdS/QCD

0.17 MB·English

by  Sungho Hong

UW/PT 05-03, hep-th/0501197
On the Couplings of the Rho Meson in AdS/QCD
Sungho Hong, Sukjin Yoon, and Matthew J. Strassler
Department of Physics, P.O Box 351560, University of Washington, Seattle, WA 98195
(Dated: January 20, 2005)
We argue that in generic AdS/QCD models (confining gauge theories dual to string theory on a
weakly-curvedbackground),thecouplingsgρHH ofanyρmesontoanyhadronHarequasi-universal,
lyingwithinanarrowbandnearm2ρ/fρ. Theargumentreliesuponthefactthattheρisthelowest-
lying state created by a conserved current, and the detailed form of the integrals which determine
the couplings in AdS/QCD. Quasi-universality holds even when rho-dominance is violated. The
argument fails for all other hadrons, except for the lowest-lying spin-two hadron created by the
energy-momentumtensor. Explicit examples are discussed.
PACSnumbers: 11.25.Tq,12.40.Vv,14.40.Cs
5
0
0 The couplings of the ρ meson to pions and to nucle- subscript-0todenotequantitiesinvolvingtheρ.) Charge
2
ons are remarkably similar, g g [1]. Is this conservation normalizes the form factor exactly: we fac-
ρππ ∼ ρNN
n accidental or profound? Most other couplings of the ρ toroutthetotalchargeofthehadronH inourdefinition
a cannot be easily measured; even the coupling g is un- of F (q2), so that
J ρρρ H
known. Lattice gauge theory at large number of colors

4 N, where hadrons are more stable and the quenched ap- F (0)=1= fngnHH . (2)
2 proximation is valid, could potentially be used to obtain H Xn m2n
additionalinformation,buttherehavebeenfewifanyef-
3
A classic argument in favor of ρ–coupling universality
v fortsinthis direction. Inthe absenceofconstraintsfrom
rests upon an assumption, ρ–dominance, which is sup-
7 data or from numerical simulation, the issue of whether
9 the ρ hasuniversalcouplingstoallhadronshas beenleft ported to some degree by QCD data. There is some
1 to theoretical speculation. ambiguity in the terminology, but by our definition, “ρ–
1 dominance” means that the ρ gives by far the largest
In this letter we will reexamine the long-standing “ρ–
0 contribution to the form factor at small q2:
coupling universality” conjecture [1]. We view the con-
5
0 jecture as having two parts: (a) the ρ has universal cou- f g f g
h/ Tplhinegres,iasnvder(yb)litthteleurneiavseornsaltocoeuxppleincgt itshiesqucaolnjteoctmu2ρre/ftρo. q20+0HmH20 ≫ q2n+nHmH2n (|q2|.m20 , n>0) (3)
p
- be exact in QCD, but even attempts to explain its ap- Combinedwith(2),thisconditionimplies,subjecttocer-
p proximatevalidityhaverelieduponparticulararguments tain convergence criteria, that
e which themselves are open to question. We will reex-
h ∞
maine this web of arguments in AdS/QCD. AdS/QCD f g f g f g
: 1= 0 0HH + n nHH 0 0HH (4)
v offers us the opportunity to compute all the couplings of m2 m2 ≈ m2
i an infinite number of hadronsin a four-dimensionalcon- 0 nX=1 n 0
X
fining gauge theory. This makes it an ideal setting for whichinturnprovesg m2/f forallH —inshort,
ar testing theoretical arguments concerning the properties ρ–coupling universalit0yH. H ≈ 0 0
of hadrons. We will examine the AdS/QCD calculation However,the convergenceconditionsforthesumsover
of the ρ’s couplings, and make estimates that show they n are not necessarily met. Convergence cannot, of
lieinanarrowbandnearm2ρ/fρ. Wewillthencheckthat course,becheckedusingdata. Meanwhile,ρ–dominance,
this is actually true in explicit models. though a sufficient condition for ρ–coupling universality,
We will consider the form factor FH(q2) for a hadron isclearlynotnecessary. Theindividualtermsinthe sum
H with respect to a conserved spin-one current. The in(4)couldbelargeandofalternatingsign,andstillper-
| i
same current, applied to the vacuum, creates a set of mit ρ–coupling universality. We will see later that this
spin-one mesons n , where n=0,1,...; we will refer to does happen in explicit AdS/QCD examples.
| i
the n=0 state as the “ρ”. At large N, a form factor for A second and qualitatively different argument treats
a hadron H can be written as a sum over vector meson the ρmesonasagaugeboson[1],withthe hope thatthe
poles, broken four-dimensional gauge symmetry might assure
ρ–coupling universality. Hidden local symmetry [2] is a
f g
F (q2)= n nHH, (1) consistent formulation of this idea, but does not have
H Xn q2+m2n exact ρ–coupling universality as a consequence (except,
possibly, in a limit when the ρ becomes masslessrelative
where m and f are the mass and decay constants of to all other vector mesons, as proposed in [3].) Instead,
n n
the vector meson n , and g is its coupling to the it is related [4] to a deconstructed version of AdS/QCD,
nHH
| i
hadronH. (Henceforth we will use both subscript-ρ and and is subject to the arguments given below.
2
InAdS/QCD,where gaugetheorieswith ’tHooftcou- have some model-dependent structure in the remaining
plingλ=g2N (gtheYang-Millscoupling,N thenumber fivedimensions,butthis structurealwaysfactorsoutbe-
ofcolors)arerelated[5]tostringtheoriesonspaceswith cause the gauge boson arises from a symmetry — for
curvature 1/√λ, these issues take on a new light. Al- examples see [6, 9].) The form factor is obtained by in-
∼
though the ρ cannot be treated as a four-dimensional tegrating this mode against the current built from the
gauge boson, a ρ meson in AdS/QCD is the lowest cav- incomingandoutgoinghadronH. WetakeH tobespin-
ity mode of a five-dimensional gauge boson. In the limit zero;thegeneralizationtohigherspinisstraightforward.
λ , the ρ meson, along with the entire tower of vec- Fromthe ten-dimensional wavefunction ofthe incoming
to→r m∞esons — the remaining cavity modes of the five- hadron of momentum p, ΦH(x,z,Ω) = eip·xφ(z,Ω), and
dimensional gauge boson — becomes massless. More the corresponding mode for the outgoing hadron with
precisely, the vector mesons become parametrically light four-momentum p′, we construct the current JH(x,z):
compared to the inverse Regge slope of the theory, and
thus to all higher-spin mesons. However, the universal Jµ = i R5d5Ω gˆ Φ∗ ←→∂ µΦ′ . (5)
propertiesofthefive-dimensionalgaugebosondonotim- H − Z p ⊥ n H Ho
ply ρ–coupling universality; they simply ensure that the
Here gˆ is the metric of W. Equivalently, Jµ(z) = (p+
corresponding global symmetry charge is conserved and H
that F(q2 0)=1. p′)µei(p−p′)·xσH(z) where
→
Nevertheless, the AdS/QCD context offers a new and
logically distinct argument for generic and approximate σ (z)= R5d5Ω gˆ φ∗ (z,Ω)φ (z,Ω) 0 .
H Z ⊥ H H ≥
ρ–coupling universality, as suggested by [6]. This argu- p
mentdoesnotrelyupontheexistenceofalimitinwhich The hadron wave functions are normalized to unity:
ρ–coupling universality is exact, and indeed there is no
suchlimit. EvenatinfiniteN and/orinfinite λ, the cou-
dz µ (z) σ (z)=1 . (6)
plings of the ρ always remain quasi-universal. Z H H
We will consider the string-theoretic dual descrip-
tions of four-dimensional confining gauge theories which whereµH =e2A(z)R/z(foraspin-zerohadron.) Interms
are asymptotically scale-invariant in the ultraviolet. of σH, the form factor then reduces to
For large λ and N the dual description reduces to
ten-dimensional supergravity, on a space with four- F (q2)=g dz µ (z) Ψ(q2,z)σ (z) . (7)
dimensional Minkowski coordinates xµ, a “radial” co- H 5Z H H
ordinate z, and five compact coordinates Ω. The co-
As q2 0, the non-normalizable mode Ψ(q2,z) goes to
ordinates can be chosen to put the metric in the form →
a constant, namely 1/g ; then Eq. (7) becomes equal to
ds2 =e2A(z)η dxµdxν+R2dz2/z2+R2dsˆ2;hereR2dsˆ2 5
µν ⊥ ⊥ Eq. (6), enforcing the condition F (0)=1.
isthemetriconthefivecompactdirections,andR λ1/4 H
∼ Meanwhile, the ρ, as the lowest-mass state created
is the typical curvature radius of the space. The co-
by the conserved current, appears in AdS/QCD as the
ordinate z corresponds to 1/Energy in the gauge the-
lowest normalizable four-dimensionalcavity mode of the
ory. The ultraviolet of the gauge theory corresponds to
same five-dimensional gauge boson. The coupling g
z 0; the gauge theory is nearly scale-invariant in this 0HH
→ is computed in almost the same way, by integrating the
region, and the metric correspondingly is that of AdS
5 ρ’s ten-dimensional wave function ϕ (which is trivial in
(with e2A(z) = R2/z2) times a five-dimensional compact 0
the five compact dimensions) against the current of the
space W. The infrared, where confinement occurs, is
ten-dimensional wave function Φ :
more model-dependent, but generally the radial coordi- H
nate terminates, at z = z 1/Λ [7, 8]. Importantly,
max
∼ g =g dz µ (z)ϕ (z)σ (z) . (8)
the metric deviates strongly from AdS5 — scale invari- 0HH 5Z H 0 H
ance is stronglybrokenin the gaugetheory — only for z
on the order of zmax. Couplings gnHH for the nth vector meson are computed
The form factor of a hadron H associated to a con- by replacing ϕ (z) with the nth mode function ϕ (z).
0 n
served current Jµ can be computed easily in AdS/CFT. Thefunctionϕ (z),asthelowestnormalizablesolution
0
Aconservedcurrentinthegaugetheorycorrespondstoa ofa second-orderdifferentialequation, is typically struc-
five-dimensionalgaugeboson,whichmayariseasagauge turelessandhasnonodes. Itcanbechosentobepositive
bosonintendimensions(orasubspacethereof)orasthe definite. On general grounds it grows as z2 at small z
dimensional reduction of a mode of a ten-dimensional (near the boundary) until scale-invariance is badly bro-
graviton. This gauge field satisfies Maxwell’s equa- ken, in the region z z . Also, because the ρ is a
max
∼
tion (or the linearized Yang-Mills equation) in the five- mode of a conserved current, it must satisfy Neumann
dimensionalspace,subjecttoNeumannboundarycondi- boundary conditions (so that Ψ(0,z) = 1/g is an al-
5
tions at z z . For each four-momentum qµ there is lowed solution.) Generically it will not vanish at z .
max max
→
acorrespondingfive-dimensionalnon-normalizablemode Wethereforeexpectthatintheregionz z thewave
max
Ψ(q2,z)eiq·x of the gauge boson. (The mode will also function ϕ (z) has a finite typical size ϕˆ∼.
0 0
3
We will now use these properties to make some es-
timates of the above integrals. The normalizable and
non-normalizable modes are related by
f ϕ (z)
Ψ(q2,z)= n n . (9)
q2+m2
Xn n
Meanwhile, the ρ meson is normalized:
FIG. 1: The combination f0g0∆aa/m20 in the hardwall model,
plotted as a function of a for all ∆<40.
1= dz µ (z)ϕ (z)2 µˆ ϕˆ2δz , (10)
Z 0 0 ∼ 0 0
where µ (z) = R/z is a slowly-varying measure factor,
0
µˆ is the typical size of µ in the region z z , and
0 0 max
∼
δz z /2 is the region over which ϕ (z) ϕˆ . Since
max 0 0
∼ ∼
Ψ(0,z) = 1/g , applying Eq. (10) to Eq. (9) and using
5
orthogonality of the modes ϕ implies
n
1 1
f0/m20 = g5 Z dz µ0(z)ϕ0(z)∼ g5µˆ0ϕˆ0δz . (11) FIG. 2: The combination fngn∆aa/m2n in the hardwall model,
plotted as a function of n for all a and ∆<40.
For a scalar hadron H created by an operator of dimen-
sion ∆ 1, the integrand of Eq. (8), µ ϕ σ z2∆−1,
H 0 H
≥ ∼
is small at small z. Thus the integral is dominated by thescalarstates ∆,a ofthe hard-wallmodel,asafunc-
| i
large z z , where the slowly-varying function ϕ is tionoftheexcitationlevela,forall∆<40. Asexpected,
max 0
∼
oforderϕˆ . Therefore,usingEqs.(6),(8),(10)and(11), the couplings lie in a narrow range near 1. That this is
0
true only of the ρ is indicated in Fig. 2, where the cou-
plings of all hadrons to the nth vector meson are shown.
f g f g ϕˆ
0 0HH 0 5 0 dz µ (z)σ (z) 1 . (12) Only the ρ’s couplings lie in a narrow range near 1 and
m20 ∼ m20 Z H H ∼ arealwayspositive. These propertiesarealsotrue inthe
D3/D7 model, shown in the next two figures (with the
ThisapproximatebutgenericrelationappliestoallH;
additional feature [6] that the large-∆ limit and large-a
it can fail for individual hadrons or in extreme models,
limit of g∆ are equal.)
butwillgenerallyhold. Thisexplainstheobservationsin 0aa
[6]. Innolimit,accordingtothisargument,isρ–coupling
universality exact across the entire theory. Instead, our
estimatesshowthatapproximateρ–couplinguniversality
is a general property to be expected of all ρ mesons in
all AdS/QCD models. Strictly we have only shown this
for scalar hadrons, but the argument is easily extended
to higher spin hadrons.
These estimates are invalid when the ρ meson mode
is replaced with the mode for any other hadron, unless
FIG. 3: Asin Fig. 1, for theD3/D7 model.
that hadron is also the lowest mode (structureless and
positive-definite) created by a conserved current (with
As is clear from the figures, both models include
model-independent normalization and boundary condi-
hadrons for which ρ–coupling universality is approxi-
tion at z = z .) The only other hadrons of this type
max
mately true but ρ–dominance fails badly. For instance,
are the lowest spin-two glueball created by the energy-
momentum tensor and (if present) the lightest spin-3/2 for the |2,9i state in the D3/D7 model, f0g0(2,9),9/m20 ∼
hadron created by the supersymmetry current. 1, but the terms in Eq. (4) fall off slowly: for n =
Now let us see that this argumentapplies in the hard- 0,1,2,3,4...
wallmodelandD3/D7model[4,10],reviewedintheap-
pendices of [6]. (These should be viewed as toy models, f g(2) /m2 =1.49, 1.21, 1.23, 1.12, 1.08,... (13)
n n,9,9 n − −
as neither is precisely dual to a confining gauge theory.
Calculations in specific gauge theories, such as the dual- Yet a naive test (Fig. 5) of Eq. (3) moderately supports
ity cascade [8], are often straightforward but cannot be ρ–dominance. Thus ρ–coupling universality can lead to
done analytically.) Our statements about the ρ meson apparentρ–dominanceevenifρ–dominanceisfalse. This
modes in these models can be checked using [6], while mightberelevantforthesuccessoftheρ–dominancecon-
thoseregardingthe couplingsg areillustratedinthe jecture in QCD, which cannot be directly checked with-
0HH
figures below. Figure 1 shows the ρ’s couplings g∆ to out measuring the higher g .
naa nHH
4
The pattern of couplings thus depends on the pattern
of masses; both ρ–dominance and ρ–coupling universal-
itycanfail,andρ–couplinguniversalitycanbe true even
when ρ–dominance is false. If, for large k, m kp
k
∼
with p 1/2, then ρ–coupling universality fails at large
≤
∆. A flat-space stringy spectrum m √k is a border-
k
∼
linecase,whilethelowsupergravitymodesofAdS/QCD
FIG. 4: As in Fig. 2, for theD3/D7 model. have p = 1. Meanwhile examples in which ρ–dominance
breaks down are easily found. If m (k+1)m , then
k 0
f g /m2 is largest for n = 0 but∼remains large for
| n nHH n|
moderate n. If m (k+2)(k+1)/2 m , as in the
k 0
D3/D7 model, then,∼forplarge∆, f g /m2 is largest
| n nHH n|
forn>0andρ–dominanceisbadlyviolated. Somespin-
onemodesintheD3/D7modelaremaximallytruncated,
including the ρ, and for them ρ–dominance indeed fails.
In sum, neither ρ–dominance, nor the hypothesis that
FIG.5: F(q2)forthe|2,9istateoftheD3/D7model;hereq2 the ρ is somehow a four-dimensional gauge boson, are
hasasmall constantimaginary part. Afit(thickline)witha logically necessary for ρ–coupling universality, exact or
single pole misleadingly supports ρ–dominance at small |q2|. approximate. An independent AdS/QCD argument, ap-
plicableonlytothe lightestmesonscreatedbyconserved
currents, ensures the ρ’s couplings are quasi-universalat
The exploration of a toy scenario provides some ad- large λ. Could this argument be generalized to QCD?
ditional, though limited, perspective; it emphasizes the With suitable definition of hadronic wave-functions, us-
role of the mass spectrum of the vector mesons. For a ing operator matrix elements, it might be possible to di-
scalar hadron created by an operator of dimension ∆, rectlyextendthislineofthinkingtotheoriesatarbitrary
the condition of ultraviolet conformal invariance implies λ, including QCD. Alternatively, the applicability of our
F(q2) q−2(∆−1). This requires that at least ∆ 1 of arguments at smaller λ could be tested using numerical
→ −
thetermsinthesumbenon-vanishing. Supposethesum simulations of large-N (quenched) QCD-like theories.
is “maximally truncated”: only the first ∆ 1 couplings
−
g , n = 0,...,∆ 2, are non-vanishing. Then the
nHH
−
condition that F(0)=1 uniquely fixes the coefficients in
We thank D.T. Son and L.G. Yaffe for useful conver-
the sum:
sations. This work was supported by U.S. Department
f g ∆−2 m2 of Energy grants DE-FG02-96ER40956and DOE-FG02-
n nHH = k (14) 95ER40893,andbyanAlfredP.SloanFoundationaward.
m2 m2 m2
n kY6=n k− n
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