The A5 and the pion field

0.13 MB·English

by  Johannes Hirn

The A and the pion field
5 ∗
6
0 Johannes Hirna and Ver´onica Sanzb
†
0
2 aIFIC, Departament de F´ısica Teo`rica, CSIC - Universitat de Val`encia
n Edifici d’Instituts de Paterna, Apt. Correus 22085,46071 Val`encia, Spain
a [email protected]
J
3 bCAFPE, Departamento de F´ısica Teo´rica y del Cosmos, Universidad de Granada
1 Campus de Fuentenueva, 18071 Granada, Spain
[email protected]

2
v
Inthistalk,anSU(N )×SU(N )Yang-Millsmodelwithacompactextra-dimensionisusedtodescribethespin-
3 f f
2 1 mesons and pions of massless QCD in the large-Nc. The right 4D symmetry and symmetry-breaking pattern
0 is produced by imposing appropriate boundary conditions. The Goldstone boson (GB) fields are constructed
0 using a Wilson line. We derive the low-energy limit (chiral lagrangian), discuss ρ-meson dominance, sum rules
1 betweenresonancecouplingsandtherelationwiththeQCDhigh-energybehavior. Finally,weprovideananalytic
5 expression for thetwo-point function of vector and axial currents.
0
/
h
p
- 1. Introduction Section 4. Sum rules for their couplings are ob-
p
e tained,relyingon5Dgaugestructure: theyimply
Based on [1], we present a model for a sector
h asofthigh-energybehavior. Thisallowsapartial
v: of large-Nc QCD in the chiral limit: that of pi- matching to QCD. Up to this point, the geome-
ons and the infinite tower of spin-1 mesons. In
i try (5Dmetric)influences noneofthe qualitative
X view of a leading 1/N approximation, we limit
c features of the model. We discuss ρ-meson dom-
r ourselvesto tree-level. The model never refers to inance in Section 5. In Section 6, we focus on
a
quarks, but rather to mesons. These will be de-
the AdS metric and give analytic results for the
5
scribedby Kaluza-Klein(KK)excitationsoffive-
vector and axial two-point functions.
dimensional (5D) fields.
Consider SU(N ) SU(N ) Yang-Mills fields
f × f 2. The model
in a compact extra dimension. From the four-
dimensional (4D) point of view, these are seen Using conformally flat coordinates ds2 =
as two infinite towers of resonances. Imposing w(z)2 η dxµdxν dz2 ,weconsideracompact
µν
−
appropriate boundary conditions (BCs) on one extra (cid:0)dimension extend(cid:1)ing from z = l0 (UV
boundary will induce spontaneous breaking of brane) to z = l (IR brane). The standard
1
the 4D SU(Nf)×SU(Nf) symmetry acting on SU(Nf)×SU(Nf)Yang-Millsactionin5Dreads
the otherboundary (wherethe standard4Delec- M =(µ,5)
troweak interactions are located).
We define the model in Section 2. We con- 1 l1
= d4x dzw(z)ηMNηRS
centrate in Section 3 on the predictions for low- SYM −4g2 Z Z
5 0
energy physics, i.e. pion interactions. At higher
L L +R R . (1)
MR NS MR NS
energies, resonances play the leading role: we × h i
consider the constraints on their interactions in Havingalarger(i.e. 5Dinsteadof4D)SU(N )
f
×
SU(N ) symmetry will have important conse-
f
∗TalkgivenatQCD05,Montpellier,France,July2005 quencesonthehigh-energybehaviorofthemodel,
†Speaker. see Section 4. As for now, we discuss the fate of
1
2
theelementsofthegaugegroupactingonthetwo 4. Resonances
boundaries, i.e. the BCs.
Having extracted the massless scalar, we per-
UV brane: The fields at z = l are taken
0
• form redefinitions to eliminate the fifth compo-
to be classical sources: this defines a generating
nents. Also, due to the BCs (2-3), the actionwill
functional. The4DSU(N ) SU(N )localsym-
f f
× bediagonalintermsofvectorandaxialcombina-
metry at z = l (a subgroup of the 5D gauge in-
0
tionsoftheoriginalgaugefields. Onethenfollows
variance) then guarantees the Ward identities of
the standard KK procedure: a 5D field V (x,z)
a 4D SU(N ) SU(N ) global symmetry. µ
f f
IR brane:×Of the SU(Nf) SU(Nf) trans- is decomposed as a sum of 4D modes Vµ(n)(x),
for•mations acting at z = l1, w×e allow only the each possessing a profile ϕVn (z) in the fifth di-
vector subgroupas a symmetry of the whole the- mension, i.e. Vµ(x,z) = ∞n=1Vµ(n)(x)ϕVn (z).
ory. This is done by imposing the BCs The wave-functionsϕ(z) arPe entirely determined
bythemetric,assolutionsof ∂ (w(z)∂ )ϕX =
Rµ(x,z =l1)−Lµ(x,z =l1) = 0, (2) M2 w(z)ϕX, with BCs ded−ucezd from tzhosne of
Xn n
R (x,z =l )+L (x,z =l ) = 0. (3) Section 2. The BCs are found to be, for the vec-
5µ 1 5µ 1
torcaseϕV (z) =∂ ϕV (z) =0,andfor
n z=l0 z n z=l1
3. Goldstone bosons the axial one ϕ(cid:12)A(z) =ϕA((cid:12)z) =0.
(cid:12)n z=l0 n (cid:12) z=l1
This yields an alte(cid:12)rnating tower(cid:12)of vector and
With the above BCs, the spectrum contains a (cid:12) (cid:12)
axial resonances. The masses of the heavy reso-
massless 4D spin-0 mode, related to the (axial
nances behave as M n as a consequence
combinationof)thefifthcomponentofthe gauge Vn,An ∼
of the 5D Lorentz invariance broken by the finite
fields3. Thecorrespondingcovariantobjectisob-
size of the extra-dimension. This conflicts with
tained by constructing the following Wilson line
the expectation M √n from the quasi-
U(x) P ei ll10dzR5(x,z) classical hadronic stVrni,nAgn[5∼].
≡ n R o Couplings of resonances are expressed as bulk
P ei ll01dzL5(x,z) . (4) integrals over the fifth coordinate. One derives
× n R o sum rules using the completeness relationfor the
U depends on the first four coordinates only. KK wave-functions 4
Moreover, of the whole (5D) group, U is only
affected by those transformations acting on the ∞
f g = 2L , (7)
UV brane, in a manner that is precisely appro- Vn Vn 9
nX=1
priateforGBs ofthe 4Dchiralsymmetrydefined
∞
in z =l0: the breaking is spontaneous. fVngVnMV2n = fπ2, (8)
Theinteractionsofthemasslessmodes(thepi- nX=1
ons) at low energy can then be obtained. Paying
∞
attention to preserve the 4D chiral symmetries, g2 = 8L , (9)
Vn 1
oneperformstheidentificationwiththe chiralla- nX=1
grangian operator by operator. The pion decay f2
constant can be read off as 1/f2 = g2 l1 dz . gV2nMV2n = 3π, (10)
π 5 l0 w(z) Xn
The (p4) low-energy constants of the RχPT la-
O ∞
grangian can be computed. In addition, the fol- f2 f2 = 4L . (11)
Vn − An − 10
lowing relations hold independently of the metric nX=1(cid:0) (cid:1)
(the first two are familiar from [4])
The first three relations are reminiscent of
L = 2L = 1/3L , (5) those considered in [4], but generalized to infi-
2 1 3
−
L = L . (6) nite sums. They ensure that the vector form fac-
10 9
− tor (VFF) andGB forwardelastic scatteringam-
3This is different from the 5D model of [2,3], where new
bulkscalarsmixwiththeA5 toyieldazeromode. 4Fordefinitions andderivationssee[1],andalso[6].
3
plitude respectively satisfy an unsubtracted and
once-subtracted dispersion relation, as expected w(z)=1 w(z)= l0 Experiment
z
in QCD. The sum rule (10) shows that the natu- 103L1 0.5 0.5 0.4 0.3
±
ral value for the KSFR II ratio is 3 here, rather 103L2 1.0 1.1 1.35 0.3
±
than 2 5. 103L 3.1 3.1 3.5 1.1
3
− − − ±
103L 5.2 6.8 6.9 0.7
9
±
103L 5.2 6.8 5.5 0.7
10
5. ρ-meson dominance Γ −4.4 −8.1 7−.02 ±0.11
ρ ee
→ ±
Γ 135 135 150.3 1.6
Before extracting numerical values, we con- Mρ→ππ 1.6 103 1.2 103 1230±40
sider the question of ρ-meson dominance. As A1 × × ±
M 2.3 103 1.8 103 1465 25
can be expected in any 5D model, only the light V2 × × ±
M 3.9 103 2.8 103 1688.1 2.1
resonances will have non-negligible overlap inte- V3 × × ±
grals with the pion field, since the GBs are non-
local objects in the fifth dimension [7]. As a
Table2: Numericaloutputs. Experimentalvalues
consequence, the sums (7-10) are dominated by
fortheL ’scorrespondtoarenormalizationscale
i
the first resonance, the ρ. In Table 1, we fo-
µ=M [8]. Dimensionful quantities are in MeV,
ρ
cus on the VFF, which can be put in the form
except for Γ in KeV.
aFre(cid:0)qc2o(cid:1)ns=idePre∞nd=,1flfaVtnsgpVfnπa2MceV2nanMdMV2nAV2−ndqS2.: two metrics 6. Two-pointρ→fueenctions
w(z)=1 w(z)= l0 Weparticularizetoanexactlyconformalmetric
fV1gV1MfπV221 0.81 1.11 z own(lzy)b=ylt0h/ezp:rceosnefnocremoafltihnevaIrRiabnrcaenies.thTehnebvreockteonr
fVngVnMfVπ22n n∼1 n12 n∼1 (−√1n)n annudmbaexriaolftwpool-epso,inantdfucnacntiobnesecxopnrteasisnedanfoirntfiinmitee-
≫ ≫
like momenta as 6
Table1: BehaviorofthequantityfVngVnMV2n/fπ2: Π (q2) = l0 log q2 +λ 2πY0,1(ql1()12),
compare with the results quoted in [5]. V,A −g2 (cid:18) µ2 ∓ J (ql )(cid:19)
5 0,1 1
The situation is then the following: indepen- where λ=log(µ2l2/4)+2γ .
0 E
dently of the metric, the 5D gauge structure en- The first term in this expression reproduces
sures two properties: soft HE behavior through the partonic logarithm, as already emphasized in
the sum rules (7-9) and approximate saturation [7,2,3]. Thelastpieceinthecorrelator(12)comes
of these sums by the first term (ρ-meson domi- from the breaking of conformality produced by
nance). Approximating the sum rules by keep- (different) BCs on the IR brane. Correspond-
ing only the first term, we would recover exactly ingly, in the euclidean Q2 = q2 > 0, it is sup-
−
the same relations as in [4]. As in this reference, pressed by an exponential factor e 2Ql1 at high
−
wearethereforeassuredthat, once the twoinput energies. Therefore, (12) does not reproduce any
parameters are matched to Mρ 776MeV and ofthe condensatesthatappearin the QCDoper-
≃
fπ 87MeV, the predictions for the low-energy ator product expansion (OPE) [9].
≃
constants and decays of the ρ will provide good The absence of these power corrections (in the
estimates, see Table 2. This result does not de- differenceofvectorandaxialtwo-pointfunctions)
pend stronglyon the chosen metric. Ontheother translates as an infinite number of generalized
hand, the masses of the resonances above the a1 Weinberg sum rules [10]: this result will hold as
grow too fast with n, as mentioned before.
6The limit l0 → 0 with l0/g52 fixed is implied here. The
divergence inλisabsorbedbyasourcecontact term,see
5Asimilarresultwasnotedin[7,3]. [1].
4
long as the symmetry-breaking effects are local- timepreservingchiralsymmetry. Thisensures
ized at a finite distance from the UV brane. that the lagrangian obtained is that of Chiral
Expanding now at low energies ql 1, the Perturbation Theory.
1
≪
logarithmic contribution is canceled by part of
2. We have extracted resonance interactions and
the last term in (12) to end up with an analytic
demonstrated the ensuing soft HE behavior.
function in powers of ql
1
Thisresultisindependent of the 5D metric. It
g2 1 5 allowstomatchthebehaviorofQCDforsome
5Π (q2) c+ l2q2+ l4q4+ q6 (.13)
l V ≃ 2 1 64 1 O amplitudes.
0 (cid:0) (cid:1)
The constant term is the expected one from the
3. ρ-mesondominanceholdsduetothe5Dstruc-
chiral expansion: in RS1, we can verify that c =
ture. Itensuresthatlow-energyquantitiesgive
2log(l1/l0)= 4L10 8H1. good estimates of the QCD ones, and are not
− −
We haveseenthatthe result(12)lackedpower very sensitive to the geometry. However, it
corrections to reproduce QCD results. Also, our
should be noted that the way ρ-meson domi-
matching at low-energy—to f and M — corre-
π ρ nance is realized depends on the precise form
sponds to N 4.3 (identifying the factor l /g2
c ≃ 0 5 of the metric.
in (12)). To correct these flaws, one can modify
the metric 7. A metric that behavesnear the UV 4. Wehavederivedananalyticexpressionforthe
brane as zw(z) 1 (z l )2d will gener- vectorandaxialtwo-pointfunctionsatallmo-
ate a 1/Ql20d corre−ctionz→∼tlo0 (12−). T0his introduces menta. This is done for AdS5 space, which
enables a further matching with the partonic
newparameters,allowingtoimposeN =3while
c logarithmsofQCD[7]. Wealsoproposemodi-
preserving numerical agreements for low-energy
ficationsofthemetricasameansofimproving
quantities. The case d=2 will represent a gluon
the matching with the QCD OPE.
condensate. Modifications with d = 3, if felt dif-
ferently by the vector and axial fields, will de-
Acknowledgments
scribe the qq 2/Q6 term.
h i
We thank Misha Shifman for stimulating ex-
7. Conclusions changes. JH is supported by the European net-
work EURIDICE (HPRN-CT-2002-00311) and
An SU(N ) SU(N ) Yang-Mills model with
f × f by the Generalitat Valenciana (GV05/015).
a compact extra-dimension implements some es-
sentialpropertiesofthe pionsandvectorandax-
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