Holography, chiral Lagrangian and form factor relations

0.25 MB·English

by  Fen Zuo

BARI-TH-2013-667
Holography, chiral Lagrangian and form factor
relations
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Fen Zuo∗

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IstitutoNazionalediFisicaNucleare,SezionediBari
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J E-mail: [email protected]

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We perform a detailed study of mesonic properties in a class of holographic models of QCD,

] which is described by the Yang-Mills plus Chern-Simons action. By decomposing the 5 di-
h
p mensional gauge field into resonances and integrating out the massive ones, we reproduce the
-
p Chiral Perturbative Theory Lagrangian up to O(p6) and obtain all the relevant low energy con-
e
stants (LECs). The numerical predictions of the LECs show minor model dependence, and
h
[ agree reasonably with the determinations from other approaches. Interestingly, various model-

2 independent relations appear among them. Some of these relations are found to be the large-
v distancelimitsofuniversalrelationsbetweenformfactorsoftheanomalousandeven-paritysec-
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:
v
i
X
r
a
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Holography,chiralLagrangianandformfactorrelations FenZuo
1. Introduction
In recent years the holographic approach towards QCD, based on the gauge/string duality [1,
2,3],hasprovideduswithmanynovelideasandpromisingresults. Ofparticularinterestisaclass
ofmodelsdescribedbytheYang-Mills(YM)andChern-Simons(CS)action[4],whichcomesout
as a large-N generalization of the traditional hidden local symmetry model. In this framework a
c
multiplet of Goldstone bosons is built in at the beginning, and chiral symmetry is realized in the
non-linear pattern. Later a concrete model in this class was constructed by embedding D8 and
D¯8flavorbranesintothebackgroundofD4branes[5]. Chiralsymmetrybreakingisimplemented
geometricallyintheconfigurationwithD8andD¯8branesconnectedsmoothlyintheinfrared.
Since chiral symmetry breaking is quite naturally accommodated in this framework, the low
energypropertiesarenicelyreproduced. Forexample,truncatedtothepionsectoronereproduces
the Chiral Perturbation Theory (χPT) Lagrangian up to O(p4) [5, 6, 7]. In particular, the Chern-
SimonspartimmediatelygivesrisetotheWess-Zumino-Wittenterm[7]. Intheevenparitysector,
the relevant low energy constants (LECs) are quite accurately determined, showing little model
dependence [5, 6, 7]. Recently, the anomaly structure in these models was further studied [8],
and a universal relation for the transverse part of triangle anomalies was found. Studies of such
a relation were carried out in [9, 10, 11]. These progress stimulated us to perform a systematic
investigationwithinthisclassofmodels[12]. HereIsummarizealltheresultsalongtheselines.
2. Reviewoftheholographicframework
2.1 The5Dpictureandcorrelationsfunctions
Theclassofmodelswefocusonaredescribedbytheaction[4,5]
S = S +S (2.1)
YM CS
(cid:90) (cid:20) 1 (cid:21)
S = − d5xtr −f2(z)F2 + F2 , (2.2)
YM zµ 2g2(z) µν
(cid:90) (cid:20) i 1 (cid:21)
S = −κ tr AF2+ A3F− A5 . (2.3)
CS
2 10
HereA(x,z)=A dxM isthe5DU(N )gaugefieldandF =dA −iA ∧A isthefieldstrength.
M f
They are decomposed as A = Aata and F = Fata, with the normalization of the generators
Tr{tatb}=δab/2. The coefficient κ =N /(24π2), with N the number of colors. The functions
C C
f2(z) and g2(z) are invariant under reflection z→−z so that parity can be properly defined in the
model. Thefifthcoordinatezrunsfrom−z toz withz >0. z canbefiniteorinfinitedepending
0 0 0 0
onthebackgrounds.
To calculate the correlation functions, one needs to know the bulk-to-boundary propagators,
which describe the response of the system to external sources. In the present case the most inter-
esting ones are those for the transverse part of the gauge field, which can be further decomposed
intothevectorpartandtheaxialpart. Toobtainthemfirstonederivestheequationofmotionfrom
theaction,whichafterthe4DFouriertransformationbecomes
g2(z)∂ [f2(z)∂ A (q,z)] = −q2A (q,z). (2.4)
z z µ µ
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Holography,chiralLagrangianandformfactorrelations FenZuo
RequiringtheboundaryconditionsV(Q,±z )=1andA(Q,±z )=±1(Q2≡−q2),onethengets
0 0
theexplicitexpressionsofthem.
The original 5D gauge potential can be decomposed as an evolution of the bulk-to-boundary
propagatorwiththeexternalsource. Variouscorrelationsfunctionsarethenobtainedbytakingthe
functional derivative in the 5D action, with respect to the corresponding sources. For example,
fromtheYang-Millstermonefindsthetwo-pointvectorandaxialcurrentcorrelators
1
Π (Q2) = f2(z)V(Q,z)∂ V(Q,z)|z=+z0
V Q2 z z=−z0
1
Π (Q2) = f2(z)A(Q,z)∂ A(Q,z)|z=+z0. (2.5)
A Q2 z z=−z0
FromtheChern-Simonstermoneobtainsthelongitudinalandtransversepartofthetriangleanomaly(in
thekineticlimitwhereonevectorfieldissoft)[8]
w (Q2)= 2Nc, w (Q2)= NC (cid:90) z0 dzA(Q,z)∂ V(Q,z). (2.6)
L Q2 T Q2 z
−z0
Taking into account that V(Q,z) and A(Q,z) are the two independent solutions of eq. (2.4) with
differentboundaryconditions,oneobtainsanovelmodel-independentrelation[8]
N N
w (Q2)= C + C [Π (Q2)−Π (Q2)]. (2.7)
T Q2 f2 V A
π
2.2 The4DpictureandχPT Lagrangian
The equation of motion (2.4) also has normalizable solutions ψ for discrete values q2 =m2,
n n
which correspond to vector and axial meson states depending on the property under the trans-
formation z→−z. The Goldstone bosons are contained in the gauge component A and can be
z
parameterizedthroughthechiralfieldU as
(cid:26) (cid:90) +z0 (cid:27)
U(xµ)=Pexp i A (xµ,z(cid:48))dz(cid:48) . (2.8)
z
−z0
One can further introduce the external sources as the boundary values of the gauge potential A ,
µ
and treat them as if they are dynamical [4, 7]. The propagation of these source fields in the fifth
dimensionisthencontrolledbythezeromodesolutionsof(2.4),namely1=V(0,z)andψ (z)=
0
A(0,z). With all these ingredients included, one finds the on-shell decomposition of the gauge
potentialA ,
µ
∞ ∞
A (x,z)=(cid:96) (x)ψ (z)+r (x)ψ (z)+∑vn(x)ψ (z)+∑an(x)ψ (z), (2.9)
µ µ − µ + µ 2n−1 µ 2n
n=1 n=1
where (cid:96) (x) and r (x) are the source fields at the left and right boundaries, and ψ (z)= 1(1±
µ µ ± 2
ψ (z)). Moreover, one can make the formulas more compact employing the A = 0 gauge, in
0 z
whicheq.(2.8)andeq.(2.9)arecombinedintoasingleexpression
u (x) ∞ ∞
A (x,z)=iΓ (x)+ µ ψ (z)+∑vn(x)ψ (z)+∑an(x)ψ (z), (2.10)
µ µ 2 0 µ 2n−1 µ 2n
n=1 n=1
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Holography,chiralLagrangianandformfactorrelations FenZuo
withthecommonlyusedtensorsΓ (x)andu (x)inχPT.
µ µ
Wecanrewritethe5Dactionina4Dform. Asafirststep,onesetsallthemassivefieldstobe
zero and focuses on the pion sector. Substituting the A decomposition (2.10) in the YM action,
µ
one gets the even-parity χPT Lagrangian up to O(p4). The LECs in the Lagrangian are given by
the5Dintegrals:
(cid:18)(cid:90) z0 dz (cid:19)−1 1 1 1 (cid:90) z0 (1−ψ2)2
f2 = 4 , L = L =− L = 0 dz,
π f2(z) 1 2 2 6 3 32 g2(z)
−z0 −z0
1(cid:90) z0 1−ψ2 1(cid:90) z0 1+ψ2
L = −L = 0 dz, H =− 0 dz. (2.11)
9 10 4 g2(z) 1 8 g2(z)
−z0 −z0
Noticethatatthisorder,therearealreadymodel-independentrelationsamongtheLECs. Further-
more, thesubstitutionoftheA decomposition(2.10)intheChern-Simonsactionreproducesthe
µ
gaugedWess-Zumino-Wittenterm[5,7].
3. χPT LagrangianatO(p6)
Inref.[12]weexplorefurtherthepredictionsalongthesetwolineswithinthisclassofmodels.
First, we notice that the results from the two formalisms are not independent. For example, one
can calculate the quantities on both sides of the relation (2.7) using the resonance decomposition
(2.10). The relation turns into an infinite number of matching conditions among the resonance
parameters. Inparticular,takingtheQ2→0limit,therelationbecomes[9]
N
CW =− C L , (3.1)
22 32π2f2 10
π
whereCW is an O(p6) LEC in the odd-parity sector [13]. While we have shown before that χPT
22
Lagrangian up to O(p4) can be reproduced directly, no derivation for the higher order terms has
been done. Phenomenologically, although the full set of independent operators have been con-
structed about ten years ago [14, 15, 13], accurate determination of the coefficients are still diffi-
cult.
Based on the derivation in the previous section, we know that these higher order terms can
only come from the resonance exchanging diagrams. To reproduce the O(p6) operators, only the
one-resonance interactions are needed since terms with more resonance fields only contribute to
operators of even higher order. Substituting the decomposition (2.10) into the 5D action, one can
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Holography,chiralLagrangianandformfactorrelations FenZuo
extracttheparts
SYM(cid:12)(cid:12)(cid:12)(cid:12) = ∑(cid:90) d4x(cid:104)−12(∇µvnν−∇νvnµ)2+m2vnvnµ2−21(∇µanν−∇νanµ)2+m2ananµ2(cid:105), (3.2)
Kin. n
SYM(cid:12)(cid:12)(cid:12)(cid:12) = ∑(cid:90) d4x(cid:26)−(cid:104) f+2µν(cid:20)(∇µvnν−∇νvnµ)aVvn−2i([uµ,anν]−[uν,anµ])aAan(cid:21)(cid:105) (3.3)
1−res. n
(cid:20)
i i
−(cid:104)4[uµ,uν] (∇µvnν−∇νvnµ)bvnππ − 2([uµ,anν]−[uν,anµ])banπ3](cid:105)
fµν(cid:20) i (cid:21) (cid:27)
+(cid:104) −2 (∇µanν−∇νanµ)aAan − 2([uµ,vnν]−[uν,vnµ])(aVvn−bvnππ) (cid:105) ,
SCS(cid:12)(cid:12)(cid:12)(cid:12) = ∑(cid:90) d4x(cid:26)−3N2πC2cvnεµναβ(cid:104)uµ{vnν,f+αβ}(cid:105)+6N4πC2canεµναβ(cid:104)uµ{anν,f−αβ}(cid:105)
1−res. n
(cid:27)
iN
+16πC2(cvn−dvn)εµναβ(cid:104)vnµuνuαuβ(cid:105) . (3.4)
Here some notations and terms in χPT have been used, and all the couplings are given by the
integral of the corresponding wave functions ψ and ψ over z. In particular, one finds that the
0 n
couplingsfromtheYMpartandCSpartarerelatedduetotheequationofmotion
m2 m2
c = vnb , c = anb . (3.5)
vn 2f2 vnππ an 3f2 anπ3
π π
Itturnsoutthattheserelationsareessentialforthevalidationoftheformfactorrelationsweshow
inthenextsection.
Since we are working at N → ∞, in the above action we have the interactions of infinite
C
number of resonances. One would like to know how it can be approximated with only a few
or even one resonance. From a simple model with the "cosh" metric function [4], one can see
explicitlyhowtheapproximationworks. Themodelisspecifiedby
f2(z)=Λ2cosh2(z)/g2, g2(z)=g2, z =∞, (3.6)
5 5 0
andalltheindependentcouplingsinthismodelread
(cid:115)
1 2(2n+1)
a = a , a =a , a = ,
Vvn 2n−1 Aan 2n n
g n(n+1)
5
g 2g
cvn = √5 δn,1, can = √ 5 δn,1,
3 15
√ √
3g 2 42g
5 5
dvn = δn,1+ δn,2, (3.7)
15 105
Oneseesthattheapproximationwiththefirstoneortworesonancesisaccurateformostcouplings,
the only exception being the couplings to the external sources, a and a . This is because we
Vvn Aan
needaninfinitenumberofresonancestoreproducethelogarithmicbehaviorofthecorresponding
correlatorsatlargeQ2 [16].
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Holography,chiralLagrangianandformfactorrelations FenZuo
The O(p6) χPT operators result from integrating out the resonances from the above La-
grangian, with the coefficients given by the summation of the resonance couplings. For example,
intheodd-paritysector,onefinds
N ∞ a c
CW = C ∑ Vvn vn. (3.8)
22 64π2 m2
n=1 vn
Using the coupling relation (3.5) and the completeness condition for the solutions ψ , this can be
n
furthersimplifiedandfinallybecomes(3.1). Inasimilarway,alltheotherLECsintheoddsector
canbeexpressedthroughtheO(p4)couplingsL ,...,L ,togetherwithanadditionalparameterZ
1 10
(cid:90) +z0 ψ2(1−ψ2)2
Z ≡ 0 0 dz. (3.9)
4g2(z)
−z0
For example, the operator OW [13], which is obtained by replacing the vector source in OW with
23 22
theaxialone,hasthecoefficient
N ∞ a c N
CW = C ∑ Aan an = C (L −8L ). (3.10)
23 128π2 m2 96π2f2 9 1
n=1 an π
Intheeven-paritysector,oneobtainssimilarresonanceexpressionsforalltheLECs. Inparticular,
whenthecontributionsarefromtwooddvertexes,theexpressionscanbefurthersimplifiedasfor
CW. Finallytheseodd-oddtermsarecompletelydeterminedby f andN ,e.g,
22 π C
1 ∞ a b N2
C =− ∑ Vvn vnππ − C . (3.11)
52 8 m2 1920π4f2
n=1 vn π
Many relations among the even couplings at this order exist, extending the previously found rela-
tionsatO(p4). Someofthemostinterestingonesare
3C +C =C +4C , (3.12)
3 4 1 3
2C −4C +C =0, (3.13)
78 87 88
N2
a00=N2, b00= C , (3.14)
2 C 6
a+−=0, b+−=0. (3.15)
2
Herea00, b00anda+−, b+−arecombinationsofLECsrelevantfortheγγ→π0π0andγγ→π+π−
2 2
processes, respectively. Numerically, these relations are satisfied reasonable well [12], see e.g.,
Tab.1.
4. Formfactorrelations
ItturnsoutthatsomeoftheserelationsbetweenLECscanbegeneralizedtorelationsbetween
scattering amplitudes, correlation functions or form factors. As discussed in ref. [6], the large-N
C
assumptionandthefactthatonlyvector/axialresonancesareincludedgivestrictconstraintsonthe
ππ scatteringamplitude. Asaresult,onefindstherelationL = 1L =−1L atO(p4),andfurther
1 2 2 6 3
the relation (3.12) amongC ,...,C at higher order. Similar reasoning for the γγ →ππ processes
1 4
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Holography,chiralLagrangianandformfactorrelations FenZuo
Holo. DSE Reso. Lagr. ENJL
a00 N2 3.79 13±3.3 14.0
2 C
b00 N2/6 1.66 3±1 1.66
C
a+− 0 −0.98 0.75±0.65 6.7
2
b+− 0 −0.23 0.45±0.15 0.38
Table1: Holographicpredictionsoftheparametersa andbrelevantfortheγγ →ππ scatteringprocesses,
2
in comparison with the results from the Dyson-Schwinger Equation approach, the resonance Lagrangian,
andtheextendedNambu-Jona-Lasiniomodel. Moredetailscanbefoundinref.[12].
gives rise to relations (3.14) and (3.15) [12]. The relation L = −L is found to be related to
9 10
the vanishing of the axial form factor in the π →lνγ decay [6], which at higher order results in
(3.13). As for the relation betweenCW and L (3.1), we have already shown that it results from
22 10
therelation(2.7)betweendifferentcorrelationfunctions.
What about the remaining relations of the other LECs? What are the underlying reasons for
them? Orcouldtheyalsobegeneralizedtorelationsvalidinthewholemomentumregion? Fromto
theabovementionedrelations,animmediateobservationisthatCW andL arealsorelatedtoeach
22 9
other. Thus a reasonable guess will be that there could be some relation between the anomalous
πγ∗γ∗ formfactorandtheelectromagneticpionformfactor,whichatlowenergyarerelatedtoCW
22
andL respectively. Theseformfactorscanbecalculatedeitherinthe5Dformalism,orusingthe
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4Dresonancedecomposition. Inthe5Dpicture,theresultsaremorecompactandread
Fπγ∗γ∗(Q21,Q22) = 24NπC2fπ (cid:90)−zz00V(Q1,z)V(Q2,z)∂zψ0(z)dz,
1 (cid:90) z0
F (Q2) = f2(z)V(Q,z)[∂ ψ (z)]2dz. (4.1)
π f2 z 0
π −z0
Employingtheequationofmotionforψ onefinds
0
N
F (Q2,0)= C F (Q2). (4.2)
γ∗γ∗π 12π2f π
π
InFig.1weshowtheresultsforthetwoformfactorscalculatedfromdifferentmodels,andcompare
them with the experimental data. One finds that the “cosh” and hard wall models, in which the
backgroundsareasymptoticanti-deSitter,areabletofitthedatainthelargeQ2 region. However,
more accurate data for F (Q2) are needed to check if the relation (4.2) could be valid or not.
π
Replacingthephotonbytheaxialsource,onegetsarelationas(4.2),whichatlowenergyreduces
to(3.10).
5. Summary
I reviewed our results obtained in ref. [12] for the interactions among the pions, vector/axial
mesonsandexternalgaugesourceswithinaclassofholographicmodels. TheLagrangianwithone
resonancefieldisderivedandfromthisweobtainalltheO(p6)LECs. Variousmodel-independent
relations among these LECs are found. Inspired by these results, we found some interesting rela-
tionsbetweentheformfactorswithdifferentintrinsicparity. Furtherstudyalongtheselinesisstill
inprogress.
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Holography,chiralLagrangianandformfactorrelations FenZuo
Figure1: Anomalousπγ∗γ∗ formfactorandtheelectromagneticpionformfactorcalculatedfromtheflat,
“cosh”,hardwallandSakai-Sugimotomodels,denotedbythedotted,solid,dashedanddash-dottedlines,
respectively. Forthedetailsofdifferentmodelsandtheexperimentaldata,pleaseseeref.[12].
Acknowledgments
I thank Pietro Colangelo and Juan Jose Sanz-Cillero for the collaboration. This work was supported
by the Italian MIUR PRIN 2009 and the National Natural Science Foundation of China under Grant No.
11135011.
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