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ADP-98-67/T334UK/TP-98-16hep-ph/9810422Hidden Symmetry and Georgi’s “Vector Limit”H.B.O’Connell a and A.G.Williams b a Department of Physics and Astronomy,University of Kentucky,Lexington,KY 40506-0055USA hoc@pa.uky.edu b Department of Physics and Mathematical Physics and Special Rearch Centre for the Subatomic Structure of Matter,University of Adelaide 5005,Australia awilliam@physics.adelaide.edu.au Abstract We study the generation of vector meson mass in the Hidden Local Symmetry (HLS)model of low energy QCD.After demoting the HLS to a hidden global symmetry (HGS),the ρis explicitly shown to be massless for any value of the HS parameter a and the chiral partner of the pion appears as the Goldstone boson associated with the spontaneous breaking of the HGS,cloly rembling the predictions of Georgi’s vector limit.Keywords:Chiral symmetries,Spontaneous symmetry breaking,Chiral La-grangians,Vector-meson dominance.PACS numbers:11.30.Rd,11.30.Qc,12.39.Fe,12.40.Vv
Typet using REVT E X
Recently there has been interest in the behaviour ofρmass under certain conditions[1,2] and attention has turned to Georgi’s“vector limit”in which the scalar and pudoscalar fields are on equal footing and theρmass is presumed to vanish[3].Our purpo here is to examine Georgi’s assumption that theρmass vanishes in the vector limit and discuss the more general issue of the role of theρas a“dynamically generated”gauge boson[4].
We shall begin with a very brief outline of the Hidden Local Symmetry(HLS)model, introduced in Ref.[4]and reviewed in Ref.[5].The HLS model has the vector mesons as the gauge bosons of a hidden local symmetry,as oppod to,say,a localid chiral sym-metry(the so-called“massive Yang-Mills model”e Ref.[5]).The vector mesons acquire mass through the Higgs-Kibble(HK)mechanism[6].In this manner,the HLS Lagrangian provides an accurate description of low energy QCD through its reproduction of the phe-nomenologically successful vector meson dominance(VMD)model(for a review e Ref.[7]). Let us consider the chiral Lagrangian[8],
L chiral=1
U(x)≡ξ†L(x)ξR(x),ξL,R(x)=exp[iS(x)/f S]exp[∓iP(x)/f P](6) where
ξL,R(x)→h(x)ξL,R(x)g†L,R.(7) One now eks to incorporate HLS into the low energy Lagrangian in
a non-trivial way, thereby introducing the lightest vector meson states[4,5].The procedure is tofirst rewrite L chiral explicitly in terms of theξcomponents
L chiral=−f2
谷歌翻译朗读Tr (DµξLξ†L−DµξRξ†R) 2,L V=−f2P
4
∂µS′.(14)
gf S
With this condition L V produces the Lagrangian mass term,M L,for the vector mesons,
M2L=af2P g2.(15) The interaction terms of theρ,photon and pions are given by
L int=i[(a/2)gρµ+(1−a/2)eAµ](π−∂µπ+−π+∂µπ−)−eaf2P g2ρµAµ.(16) The choice a=2[4]gives the usual formulation of VMD,fixing g=gρππand removing the
direct photon-pion contact term(the photon can only couple to the pudoscalars through the vector mesons).Eq.(15)then reproduces the KSRF relation[14].
Let us now discuss exactly how the vectors acquire mass,and the role of the scalar
field in this in order to understand Georgi’s massless vector limit and associated claim of a vanishing coupling constant through
mρ/f P→0⇒g→0(17) as might be inferred from Eq.(15).We shall continue to work in the HLS model and restrict
our specific attention to the neutralρ.The relevant Lagrangian term is given by
nac
L SV=af2P Tr[gVµ−∂µS/f S]2.(18) The gauge symmetry,H local,is spontaneously broken by the vacuum expectation value of the compensatorfield e iS(x)/f S,and thefield S is the“would be”Goldstone boson associated
with this.The gaugefixing in Eq.(14)removes the Sfields and reduces Eq.(18)to simply
the Lagrangian mass term of the vectorfield.The vector limit cannot be realid,becau the scalarfield is unphysical,and so cannot appear as the chiral partner of the pion.
nibaHowever,as Georgi points out[3],there is no physics in the hidden local symmetry.
Indeed,while global symmetries are associated with obrvable currents,gauge symmetries merely reprent a redundancy in the description of a physical system[15].Let us now
consider demoting the HLS H local to a hidden global symmetry(HGS),H global,but keep the vectorfields which transform now as Vµ→hVµh†(h∈H global).Naturally,L A is unaffected by this:we still have G broken to SU(3)V by the vacuum expectation value of the F(x)
fields.Let us concentrate on L V.In this ca the vacuum expectation value of thefield e iS(x),transforming as e iS(x)→he iS(x),spontaneously breaks the H global symmetry(the e iP(x)are si
nglets under H global just as e iS(x)is a singlet under G)and the S(x)are now legitimate Goldstone bosons.
We shall now consider the effect of this on the vector meson mass,in particular that of
the neutralρ,which is of interest as Georgi describes a vanishingρmass in his vector limit [3].Introducing the chiral partner of the pion,σ(an isospin triplet),which is an element of S the wayπis an element of P,Eq.(18)gives
ρµ∂µσ.(19) Lρσ=12af2P f
S
contractingThefirst term on the RHS of Eq.(19)is simply the Lagrangian mass term of theρgenerated by the vacuum expectation value of thefield e iS(x).The cond term is the kinetic term for theσmeson.The third term couples the masslessσto theρ.We now e that Eq.(19)
suggests a slightly more general limit than Georgi’s a=1.So that theσappears as a physical particle with a normalid kinetic energy term we have,
f2S=af2P.(20)
We shall refer to this as the BKY limit(following Sect.6of Ref.[5]).However,f S which has appeared here so far is merely a parameter.Let us relate it to the physical scalar decay constant of theσ,which defines Georgi’s vector limit through[3]
0|Aµ|π =ifπpµ, 0|Vµ|σ =ifσpµ,fπ=fσ,(21) The extraction of the pudoscalar decay constants from the HLS Lagrangian is discusd in Section IV.A of Ref.[13].Following this wefind,
fσ=af2Pbecome的意思
f2S
f S,fπ=f P(22)
and wefind the BKY limit(Eq.(20))yields fσ=f S.Setting a=1,would then give Georgi’s
footmanfσ=f P[3],though this value for a ems disfavoured by data[16].
We are now in a position to determine the physical mass of theρ.Firstly,let us state
that we shall ignore the effects of pudoscalar loops in the following analysis.The terms
contribute to the vector pole through the polarisation function,Π(s),which we define as being generated solely through meson loops.The physical pole,s=p V is then given by
p V−m2−Π(p V)=0.The polarisation functions develop an imaginary part above the pudoscalar production threshold and move the pole of the vector meson propagator from
慷慨大方the real axis to the complex plane,thus generating the physically important meson widths
[17,18].However,they have little effect on the real part of the mass term:numerical studies find the real part of theρmass to be typically altered only by a few percent,and even less for theω[19].Furthermore,as we are interested in examining Georgi’s statement that the vector meson mass vanish,our interest lies in the vector propagators around s=0. In a model such as the HLS model,the vectors couple to conrved currents,as can be en form the interaction Lagrangian given,for example,in the Appendix of Ref.[13].The vector currents take the form Jµ=a∂µb−b∂µa,where a and b are pudoscalars and thus ∂µJµ=0from the pudoscalar equations of motion∂2a=∂2b=0.AsΠ(s)is generated from couplings to conrved currents we haveΠ(s=0)=0[20].
蓝调
Therefore,the tree level is entirely adequate for our purpos.The physical mass is then
determined by two things.Thefirst,naturally,is M L,the cond is the dressing of theρpropagator by theσ(a similar effect is discusd for the weak bosons by Farhi and Susskind [21]and Peyranere[22]and for theπ−a1system by Kaloshin[23]).We have
(Dρµν)−1=(M2L−s+M2σρ)gµν+qµqν.(23) The contribution to theρdressing from theσis given by
juexiang
iM2ρσ= −ag f2P q2 ag f2P f2S.(24)
The mass contribution Mρσis en to be imaginary,due to the relative minus sign between theσmomentum,qµ,going in to eachρσvertex in Eq.(24).This is much like what Sakurai
discovered for the vector meson contribution to the vacuum polarisation of the photon[24]. Sakurai realid that this could be cancelled by a Lagrangian mass term for the photon(e also Ref.[17]).Such a term is provided automatically in the HLS model,M L.The physical mass for theρ,
)2=M2L+M2ρσ=ag2f2P−ag2f2P af2P
(m phys
ρ
af P due to the normalisation of the scalar kinetic term,and Georgi’s vector limit constraint a=1rves only to t f S=f P.
Acknowledgements
We would like to thank T.D.Cohen,K.F.Liu,J.McCarthy,J.Sloan and W.Wilcox for helpful correspondence and discussions.This work is supported by the US Department of Energy under grant DE–FG02–96ER40989(HOC)and the Australian Rearch Council (AGW).
