Near-integrability and confinement for high-energy hadron-hadron collisions

0.14 MB·English

by  Peter Orland

BCCUNY-HEP/08-01
Near-integrability andconfinement forhigh-energy hadron-hadron collisions
Peter Orland
∗
1. TheNielsBohrInstitute,TheNielsBohrInternationalAcademy,Blegdamsvej17,DK-2100,CopenhagenØ,Denmark
2. TheGraduateSchoolandUniversityCenter,TheCityUniversityofNewYork,365FifthAvenue,NewYork,NY10016,U.S.A.and
3. BaruchCollege, TheCityUniversityofNewYork,17LexingtonAvenue, NewYork,NY10010, U.S.A.
(Dated:January20,2008)
WeinvestigateaneffectiveHamiltonianforQCDatlarges,inwhichlongitudinalgaugedegreesoffreedom
are suppressed, but not eliminated. In an axial gauge the effective field theory is a set of coupled (1+1)-
dimensional principal-chiral models, which are completely integrable. The confinement problem issolvable
8 inthiscontext, andwefindthelongitudinalandtransversestringtensionswithtechniquesalreadyusedfora
0 similarHamiltonianin(2+1)-dimensions.WefindsomeaposteriorijustificationfortheeffectiveHamiltonian
0 asaneikonalapproximation. Hadronsinthisapproximationconsistofpartons,whicharequarksandsoliton-
2 likeexcitationsofthesigmamodels.Diffractivehadron-hadronscatteringappearsprimarilyduetoexchangeof
longitudinalfluxbetweenpartons.
n
a
J PACSnumbers:11.10.Jj,11.10Kk,11.15.Ha,11.15.Tk,11.80.Fv,12.38.Aw,12.39.Mk,13.85.Dz

0
2
I. INTRODUCTION Astajt×esAYk)m, ruesstpseacttiisvfeylyGaaunsds’(sAlajw×Ak)a = fbacAbjAck. Physical
]
h
p In the last decade and a half, effective gauge-theory de- (¶ E +¶ 3E3 r )Y =0, (1.3)
- scriptions for QCD at large center-of-massenergy squared s ⊥· ⊥ −
p hasbeenanactiveareaofresearch[1],[2],[3], [4],[5],[6], wherer isthequarkcolor-chargedensity. Asofthiswriting,
e [7], [8]. The approximation of Reference [6] was to elimi- we are not certain that this rescaled theory is definitely de-
h
nate some gauge-theory degrees of freedom by a longitudi- scribingQCD,butthinkitmaybeausefulphenomenological
[
nal rescaling. In this paper we carefully examine the con- approachtosmall-x,large-sscattering.

2 sequences of this rescaling: x0,3 l x0,3, x⊥ x⊥, A0,3 TheHamiltonian(1.2)andtheconstraint(1.3)willberegu-
9v lno−t1eAs0t,h3e,Aco⊥m→poAn⊥en,twshoefrteheAmSU=(N→Aa)m tYa,anag=-M1i,l.l→s..fi,Nel2d−an1d,dt→hee- lwaarirzdesd,xo3nwaisllpaaltsiaolblaetmticaed[e9c]o,knetienpuionugst,ilmeaevcionngtoinnulyouths.eAtrfatnesr--
8
transverseindices1,2aresometimescollectivelydenotedby versecoordinatesdiscrete. Wetakeanaxialgaugecondition,
3
0 ⊥. We normalize Trtatb = d ab and define ifacbtc = [ta,tb]. breakingtheprecedentofconsideringlarge-sscatteringinthe
. Thecenter-of-massenergysquaredchangesass l 2s[6]. light-conegauge. Wedothisbecause,asweshowlater,(1.2)
1 −
0 Sincewewishtoconsiderhighenergies,wetake→l 1. is similar to the anisotropic (2+1)-dimensional Yang-Mills
8 If the scale factor l is small, but not zero, the≪resulting theoryinthisgauge[10],[11],[12],[13],[14].Iffurtherexpe-
0 Hamiltonian has one extremely small coupling and one ex- riencepersuadesustousealight-conelattice[15]instead,so
: tremelylargecoupling.Therescaledactionis beit.Alight-cone-latticeapproachinthisspiritwasdiscussed
v
inReferences[16]. Thiswasthestartingpointofanargument
i
arX S=21g20Z d4xTr l −2F023+j(cid:229)=21F02j−j(cid:229)=21Fj23−l 2F122!,(1.1) jmtuhseatli(ifz2ya+itniog1n)t-hgderiomrueepsnc[sa1iloi7nn]g.alSucisamesdielai[nr13a(1]r,g.1[u1)m4be]ynwtasintwhaoenurietsokptnrreooswpeilncetdreegdnefooorrf-
Reference[17].Weshouldmentionthatalatticegaugetheory
where Fmn =¶ m An ¶ n Am i[Am ,An ]. The Hamiltonian in with differenttransverse and longitudinalcouplingshas also
− −
A0=0gaugeistherefore beenstudiedinReference[18].
The analogous rescaling in (2+1)-dimensions gives the
H= d3x g20E2+ 1 B2 +(g′0)2E2 Hamiltonian,inA0=0gauge1,
Z (cid:18) 2 ⊥ 2g20 ⊥ 2 3 g2 1 (g )2
1 H= d2x 0E2+ B2+ ′0 E2 , (1.4)
+ B2 , (1.2) 2 1 2g2 2 2
2(g )2 3 Z (cid:18) 0 (cid:19)
′0′ (cid:19)
whereg′0=l g0,andtheelectricandmagneticfieldsareEi=
whereg′0=l g0,g′0′=l −1g0 (sothatg′0g′0′=g20),theelectric id /d AiandB=(e jk¶ jAk+Aj Aj),respectively.Gauss’s
andmagneticfieldsare Ei= id /d Ai andBi=e ijk(¶ jAk+ − ×
−
1Wenotethatthecoordinateindices1and2arereversedinReferences[10],
Electronicaddress:[email protected] [11],[12],[13],[14].
∗
2
lawis Afterwediscusstheseconfinementmechanismsforthethe-
oryin(3+1)dimensions,weattemptto justifytheeffective
(¶ 1 E1+¶ 2E2 r )Y =0. (1.5) Hamiltonianasaneikonalapproximation. Thisviewwasad-
· −
vocatedinReference[6],butweinterpretthemeaningofthe
Uponregularization,confinementofquarkscanbereadily
factorl slightlydifferently.
demonstrated. Thusthe hadronizationproblemis essentially
Eachofthetwoincominghadronsinacollisionprocessisa
solved. Thissuggeststhattheelasticpartoftheforwardam-
collectionofFZparticles,joinedbylongitudinal-electricflux
plitude and the soft Pomeron can eventually be understood
lines. Scatteringoftwohadronscaninvolverearrangementof
usingourmethods.
It has been noted before that the l 0 limit of QCD is the flux lines among the FZ particles. It is also possible for
→ anon-AbelianphaserotationofFZparticlestooccurasthey
equivalentto a set of principal-chiralSU(N) SU(N) sigma
× passthrougheachother;thephasefactorisgivenbytheexact
models [6], [16]. Such a sigma model has the Lagrangian
L =1/(2g2)h mn Tr¶ m U†¶ n U, where U SU(N) and m = (1+1)-dimensionalexactS-matrix[19].Thesetwoprocesses
0 ∈ are notentirely distinct, ascan be seen froman examination
0,3. Thisfieldtheoryisintegrable,andtheS-matrix[19]and
ofthe1/NexpansionofthisS-matrix.
evencertainoff-shellinformation(forN =2)[20] isexactly
Inthenextsection,weputtheHamiltonian(1.2)onaspa-
known.Theintegrablenatureoflarge-sscatteringisalsoindi-
tiallattice,thentransformtotheaxialgauge.InSection3,we
catedbytheformofReggeizedamplitudes[21].InReference
takethecontinuumlimitinthex3-directionandshowthatthis
[6]itwaspointedoutthattheeffectivegluon-emissionvertex
Hamiltonian is equivalent to a collection of integrable field
[22] can be derived by considering longitudinal fluctuations
beforetakingthetransverselimitl 0. theorieswhicharecoupledtogether. Wediscussconfinement
→ ofcolorandfindthetransversestringtensioninSection4. We
The particles of the principal-chiral sigma model are la-
findthelongitudinalstringtensioninSection5.Bycomparing
beled by r=1,...,N 1 [19]. The particle with labelr has
− theelectricfieldstrengthsindifferentdirections,wefindsome
anantiparticlewithlabelN r. Themassspectrumis
− justificationoftheeffectiveHamiltonianinSection6. InSec-
sinrp 4p tion7,wedescribethenatureofhadronicstatesandbeginan
mr=m1sinNp , m1=KL (g20N)−1/2e−g20N +... , (1.6) examination of diffractive hadronic scattering, focussing on
N
theexchangeofhadronicflux. We presentsomeconclusions
whereK is a non-universalconstant, L is theultravioletcut- andmentionsomefurtherdirectionsforresearchinSection8.
offandtheellipsesdenotenon-universalcorrections.
WestudythephysicalstatesfortheeffectiveHamiltonian.
They are very similar to those of the (2+1)-dimensional II. REGULARIZATIONANDTHEAXIALGAUGE
model(1.4),(1.5),alreadyinvestigatedindetailinReferences
[10], [11], [12], [13], [14]. Hadronsconsistofquarksjoined Consider a lattice of sites x, whose coordinates are xj,
by electric strings. Longitudinalelectric flux consists essen- j=1,2,3,wherexj/aareintegers,andwhereaisthelattice
tiallyof(1+1)-dimensionalYang-Millsstrings. Incontrast, spacing. Eachlinkisapairx, j,andjoinsthesitextox+jˆa,
transverseelectricfluxisbuiltoutofthemassiveparticlesof where jˆisaunitvectorinthe jthdirection.Wechoosetempo-
thesigmamodel.WecalltheseFaddeev-Zamolodchikov(FZ) ralgaugeA =0. Beforeanyspatialgaugefixing,thedegrees
0
particles,consistentwiththeterminologyoftheoperatoralge- offreedomareelementsofthegroupSU(N)inthefundamen-
braofthe creationandannihilationoperatorsforthese parti- tal(N N)-dimensionalmatrixrepresentationU (x) SU(N)
j
cles. ateach×linkx, j. Inaddition,therearetheelectric-fie∈ldoper-
In(2+1)dimensions,thetransverseandlongitudinalstring atorsateachlinkl (x) ,b=1,...,N2 1. Thecommutation
j b
tensions2 were first foundto leading orderin g′0 [10]. Later, relationsonthelatticeare −
thecorrectionsofhigherordering tothelongitudinalstring
′0
tension[11],andthetransversestringtension[14]werecom- [l (x) ,l (y) ]=id d fd l (x) ,
j b k c xy jk bc j d
puted. Stringtensionsfordifferentrepresentationsofcharges
werefoundanditwasshownthatadjointsourcesarenotcon-
fined[12]. Thelow-lyingmassspectrumwasstudiedinRef- [lj(x)b,Uk(y)]= d xyd jktbUj(x), (2.1)
−
erence[13].
allotherszero.
Both transverse electric flux and longitudinalelectric flux
Thelatticeversionof(1.2)isH=H +H +H ,where
havesomenonzerothickness.Thethicknessoftransverseflux 0 ′ ′′
isduetocolorsmearingoftheFZparticlesinthelongitudinal
direction[14]. Thethicknessoflongitudinalfluxiscausedby H =1(cid:229) g20l (x)2 (cid:229) 1 TrU(cid:3)(x) ,
creationanddestructionofFZparticlepairs,akintovacuum 0 a x " 2 ⊥ −j=1,24g20 j #
polarization[11].
H =(g′0)2(cid:229) l (x)2, H = 1 (cid:229) TrU(cid:3)(x),(2.2)
′ 2a 3 ′′ −4(g )2a 3
x ′0′ x
andwhere,asbefore,g =l g ,g =l 1g and
2InReferences [10],[11],[12],[13],[14],wecalledthetransversestring ′0 0 ′0′ − 0
tensionthe“verticalstringtension”andthelongitudinalstringtensionthe
“horizontalstringtension”. Ui(cid:3)(x)=e ijkUj(x)Uk(x+jˆa)Uj(x+kˆa)†Uk(x)†. (2.3)
3
We need the adjoint representation of the SU(N) gauge x1,2 =0,a,2a,...,L1,2 a, andx3 =0,a,2a,...,L3. Gauss’s
field Rj(x), defined by Rj(x)bctc =Uj(x)tbUj(x)†. This has lawisstillgivenby(2.5−),provided(2.6)ismodifiedto
the properties Rj(x) SU(N)/ZN, Rj(x)TRj(x) = 1, and
detRj(x)=1. Notice∈that D3l3(x)=d x36=L3l3(x)−d x36=0R3(x⊥,x3−a)l3(x⊥,x3−a),
[Rj(x)bclj(x)c,Uk(y)]=−d xyd jkUj(x)tb. DD21ll21((xx))==ll21((xx))−RR21((xx11,−x2a,xa2,,xx33))ll21((xx11,−x2a,xa2,,xx33))., (2.7)
− − −
Thus l (x) generates infinitesimal SU(N) transformationson
j
theleftofUj(x)andRj(x)lj(x)generatesinfinitesimalSU(N) Tofixthelinksinthe3-direction,weneedtouse(2.5)and
transformations on the right of Uj(x). The squares of these (2.7)tosolveforl3:
operatorsarethesame
x3
[lj(x)]2=[Rj(x)lj(x)]2, l3(x)= (cid:229) q(x⊥,y3) (D l )(x⊥,y3) . (2.8)
− ⊥· ⊥
byvirtueoftheorthogonalityoftheadjointrepresentation. y3=0h i
Colorchargeoperators,denotedbyq(x) ,satisfy
b Somenon-Abeliangaugeinvarianceremains,namely
[q(x) ,q(y) ]=ifdd q(x) . (2.4)
b c bc xy d L3
Schro¨dingerwave functionsare complex-valuedfunctions G (x⊥)Y = (cid:229) [(D⊥·l⊥)(x)−q(x)]Y =0. (2.9)
x3=0
ofallthelinkdegreesoffreedomU (x). Physicalwavefunc-
j
tionsY ( U )satisfyGauss’law
Thisremaininggaugeinvariancemeansthat(2.8)isequivalent
{ }
to
[(D l)(x)b q(x)b] Y ( U )=0. (2.5)
· − { }
L3
where(withnosummationof j) l3(x)= (cid:229) q(x⊥,z3) (D l )(x⊥,z3) . (2.10)
− − ⊥· ⊥
[Djlj(x)]b=lj(x)b−Rj(x−jˆa)bclj(x−jˆa)c. (2.6) Thenewexpz3r=esxs3+ioanhsforl3,namely(2.8)and(2.i10),allow
us to completely eliminate all degrees of freedom but U .
NextweimposetheaxialgaugeconditionU =1. Wetake 2,3
3 We now can write the term in the Hamiltonian (2.2) which
the lattice to be open at x3 =0,L3, which means we do not
dependsonthelongitudinalelectricfieldas
fix any non-contractible Wilson loops. The open boundary
condition means, however, that a relic of Gauss’s law must
stilWlbeecihmopoosseedsp.ace to be a lattice “cylinder” of size L1 H′=−(g2′0a)2(cid:229) (cid:229)x3 (cid:229)L3 q(x⊥,y3)−(D⊥·l⊥)(x⊥,y3)
L2 L3, with periodic boundary conditions in the 1- and 2×- x⊥y3=0z3=x3+ah i
dire×ctions only. This means that for any function of po- q(x⊥,z3) (D l )(x⊥,z3) ,
sition f(x), we have f(x1+mL1,x2+nL2,x3) = f(x), for × − ⊥· ⊥
h i
any m,n Z. We take components of x to have the values or
∈
H′=−(4ga′0)22(cid:229) (cid:229)L3 |y3−z3| q(x⊥,y3)−(D⊥·l⊥)(x⊥,y3) q(x⊥,z3)−(D⊥·l⊥)(x⊥,z3) , (2.11)
x⊥ y3,z3=0 h ih i
Intheaxialgauge,thelongitudinal-electric-field-squaredterm nature of the vacuumstate is subtle. Mandelstam, who con-
H ishighlynon-localinthe3-direction. Thisnon-localityis sideredtheanalogouscontinuumHamiltonian,arguedthatthe
′
astandardfeatureofphysicalgauges. Thisfacthasasimple vacuumstate canonlyhavefinite energydensityif magnetic
physical interpretation in the case of (2.11). The longitudi- condensationtakesplace[24]. Hisreasoningwasthatthisis
nal electric field has been eliminated in favor of the trans- thewayinwhichthevacuumcorrelator
verse degrees of freedom. Now electric flux between two
charged transverse plates must be proportional to the longi- 0 [q(x⊥,y3) (D l )(x⊥,y3)]
h | − ⊥· ⊥
tudinal separation of these plates. This is accounted for by [q(x⊥,z3) (D l )(x⊥,z3)] 0 ,
the linear factor in (2.11). On the other hand, the vacuum × − ⊥· ⊥ | i
expectation value of H must be proportional to the spacial canfalloffsufficientlyquicklyin y3 z3 .
′
volume L1L2L3. The non-locality of (2.11) means that the What of the remainder of the H| am−ilto|nian H +H after
0 ′′
4
gaugefixing?SettingU1(x)=1,wefind respectively,wherem =0,3.TheoperatorJ0L(x⊥,x3,j)pro-
ducesanSU(N)rotationattheedgeoftheband(x ,j)atx .
⊥ ⊥
H0=(cid:229) H0(x⊥,2)+H0(x⊥,3) , (2.12) TheoperatorJ0R(x⊥,x3,j)producesanSU(N)rotationatthe
x⊥h i otheredgeoftheband(x⊥,j)atx⊥+jˆa.
where
H
H0(x⊥,j)= (cid:229)L3 g20lj(x)2 (cid:10)H(cid:10)(cid:10)H(cid:10)(cid:10)HH(cid:10)HH
Ocohfnathnthegeegdoa,uthsgieenrcfihe−ealindtLxd(cid:229)33.d=−,e0taxph32e=egn01l20do2asnaogRnietluyTdroinUnajlt(hmxe)aUtgrajn(nexst⊥vice,xrtse3er+mcoaHm)†′p′.,o(in2se.u1nn3ts-) x63 (cid:10)(cid:10)(cid:30)x2 H(cid:10)(cid:10)HH(cid:10)(cid:10)HHH(cid:10)(cid:10)(cid:10)(cid:10)HHHH(cid:10)(cid:10)(cid:10)HHHH(cid:10)(cid:10)(cid:10)(cid:10)HHHH(cid:10)(cid:10)(cid:10)(cid:10)HHHHH(cid:10)(cid:10)H(cid:10)(cid:10)HHHH(cid:10)(cid:10)(cid:10)H(cid:10)(cid:10)HHH(cid:10)(cid:10)H(cid:10)(cid:10)(cid:10)(cid:10)HH(cid:10)H(cid:10)HH(cid:10)(cid:10)(cid:10)H(cid:10)HH(cid:10)(cid:10)
NowthattheHamiltonianhasbeenrecastinanaxialgauge, (cid:10)
wemay,atleastinprinciple,takethethermodynamiclimit,in (cid:10)
H
whichL1,L2 andL3 allbecomeinfinity. H
H
Hj
x1
III. COUPLED(1+1)-DIMENSIONALFIELDTHEORIES
FIG.1: Spacewithx discreteandx3 continuousresemblesthein-
⊥
Next let us examine each of the terms of the Hamilto- teriorofawinetransitbox.
nian more closely. We will take a continuumlimit in the 3-
direction. Thus only the transverse coordinateswill be latti-
cized. Theresultingstructureresemblesatransitboxforbot-
tlesofwineasshowninFig. 1. EachtermH0(x⊥,j),defined WewriteH0as
in(2.13), isalattice (1+1)principal-chiralnonlinearsigma
model, with couplingconstant g . We will regard H as the
Hasaminitletroancitaionnosf. tNheotuicnepethrtautrtbheedc0toheefofircyieanntds torfeaetacHh′o0afndtheHs′e′ H0=x(cid:229) ,jZ dx321g20[J0L(x⊥,x3,j)2+J3L(x⊥,x3,j)2].(3.1)
terms is small, by virtue of l 1. Thus one coupling, g ⊥
0
≪
issmall(totakethecontinuumlimit),anothercouplingg is
′0
muchsmallerstillandthethirdcouplingg iscomparatively The connection between (2.12), (2.13) and (3.1) is made
′0′
extremelylarge.Weshallsaymoreaboutthesizesofthecou- through the Heisenberg equation of motion for Uj. This
plingsinthenextsection. gives lj(x) ≈ ag−02J0L(x⊥,x3,j), for small a. Similarly,
priWnceipwali-lclhaisrsaulmsiegmthaemlaottdiceelssapsacnineagristhsemiratlhlearnmdodtryenaatmthiec Risnj(oxw)ljJ(x)L≈R.ag−02J0R(x⊥,x3,j). Thetransverseelectricfield
0
andcontinuumlimits. Eachsigmamodellivesinaband,that
isaatwo-dimensionalstripoflengthL3 ¥ andwidtha,in The residualgauge-invarianceconditionandthelongitudi-
the j-3 plane, where j=1,2 (we referre→dto bandsas layers nal-electric-field-squaredtermarewrittenbythesamesubsti-
inthepaperson(2+1)-dimensionalgaugetheories. Inthree tution into (2.9) and (2.11), respectively. Residual gaugein-
spatial dimensions, a different name seems appropriate). In varianceisnowtheconditiononphysicalstatesY
our wine-transit-boxanalogy, thereare fourbandssurround-
ing each wine bottle, and two wine bottles adjacent to each
abtanxd⊥.+Wjˆeadbeyno(txe⊥t,hje).baTnhdebleetfwt-heeanndthedelainndearitgxh⊥t-haannddtehdecliunre- Z dx3j=(cid:229) 1,2hJ0L(x⊥,x3,j)−J0R(x⊥−jˆa,x3,j)
rentsofsigmamodelintheband(x⊥,j)are r (x ,x3) Y =0, (3.2)
⊥
−
JmL(x⊥,x3,j)b = iTrtb¶ m Uj(x⊥,x3)Uj(x⊥,x3)†, i
JmR(x⊥,x3,j)b = iTrtbUj(x⊥,x3)†¶ m Uj(x⊥,x3), andthelongitudinal-electric-field-squaredtermis
5
H′= − 4(gg4′0a)22(cid:229) (cid:229) dx3 dy3|x3−y3| J0L(x⊥,x3,j)−J0R(x⊥−jˆa,x3,j)−r (x⊥,x3)
0 x⊥ j=1,2Z Z h i
× J0L(x⊥,y3,j)−J0R(x⊥−jˆa,y3,j)−r (x⊥,y3) (3.3)
h i
wherer (x ,x3)isalinearchargedensity,satisfyingthealge- We can regard a line in the 3-direction as a choice of x .
⊥ ⊥
bra Consider now the four bands which meet at this line x ,
⊥
namely(x ,1), (x ,2), (x 1ˆa,1)and(x 2ˆa,2),shown
⊥ ⊥ ⊥ ⊥
[r (x⊥,x3)b,r (y⊥,y3)c] = id x⊥y⊥d (x3−y3) inFig. 2. Therearefivecolo−rchargesatthelin−ex⊥,onefrom
fdr (x ,x3) . (3.4) eachbandandonefromthecolorsource dx3r (x⊥,x3)onthe
× bc ⊥ d line. Thenontrivialconstraint(3.2) meansthat sum of these
R
ThelasttermintheHamiltonianisthelongitudinal-magnetic- fivecolorchargesiszero.
field-squaredterm
H
H = 1 (cid:229) dx3ReTrU(cid:3)(x ,x3), (3.5) (cid:10)(cid:10) HH
′′ −4(g′0′)2a2 x⊥Z ⊥ (cid:10)(cid:10) HHH
(cid:10) (cid:10)(cid:10)
whereU(cid:3)(x⊥,x3)isgivenby(2.3),asbefore. (cid:10) (cid:10)(cid:10)(cid:10)HH (cid:10)
(cid:10) (cid:10)(cid:10) HHYHH(cid:10)
H
H(cid:10)(cid:10)(cid:10)(cid:29) (cid:10)
H
H (cid:10) (cid:10)(cid:10)
x3 H
H(cid:10) (cid:10)
6 (cid:10)
(cid:10)(cid:10) HH
x2 HH (cid:10)(cid:10)(cid:10)(cid:30)
H (cid:10) HHj
HH(cid:10) (cid:10)(cid:30) HH (cid:10)(cid:10)
H (cid:10) H(cid:10)(cid:10)
(cid:10) H
H (cid:10)
(cid:10)
(cid:10)
x3 (cid:10) HH
H
6 Hj
x1
x2
(cid:10)(cid:30)
(cid:10)
(cid:10) FIG. 3: The Wilson loop U(cid:3)(x ,x3), directed counter-clockwise
H ⊥
(cid:10) aroundfourbands.
H
H H
H (cid:10)(cid:10) H
Hj H
x1 (cid:10) The interaction H′, as discussed before, is simply the E12
(cid:10) termoftheHamiltonianinaxialgauge.
FInally, the interaction H is a discrete version of the in-
′′
FIG.2:Thefourbandsmeetingatthelineoffixedx . tegral of the square of the longitudinal magnetic flux. The
⊥ quantityTrU(cid:3)(x ,x3)istheWilsonlooparoundfourbands(a
⊥
winebottle standsin the middleofthese fourbands),shown
ThetermsintheHamiltonian(3.1),(3.3),(3.5)andthecon-
inFig. 3.
straint(3.2) may seem a bitcomplicated, butthey each have
a straightforwardgeometricalinterpretation. A sigma model
lives in each band (x ,j), whose Hamiltonian is H (x ,j).
⊥ 0 ⊥
TheexcitationsofH aretheFZparticles,whichbehavelike IV. TRANSVERSECONFINEMENT
0
solitons, though they are not quantized versions of classical
solutions. Withouttheinteractionterms,theseparticlesmove Nextweexplainhowtransverseconfinementworksforour
onlyin the 3-direction,scattering with a nontrivialS-matrix. effectivegaugetheory. The mechanismdoesnotrely onour
Thescatteringisintegrable,whichimpliesthatthereisnopar- takingthe continuumlimit ofthe coordinatex3; it holdsjust
ticle creationordestruction. The elementaryr=1FZ parti- aswellonthethree-dimensionallattice. Thekeyingredients
clesareadjointgluon-likeparticles(seeequation(1.6)).They arethemassgapofH0 andresidualgaugeinvariance(2.9)or
can be thought of a color dipoles with a right fundamental (3.2). Wewillshowthatourexplanationmakessensefor
color charge (anti-charge)at x and a left fundamentalanti-
⊥
ochuatrogfeth(cehsaerrge=)a1t“xd⊥if+frajˆcat.ivAellgoluthoenrs”F.Zparticlescanbebuilt (g′0)2= (gg′0′40)2 ≪ g10e−4p /(g20N). (4.1)
6
OurargumentsareadaptedfromReference[10].
Letusfirstconsidertheextremecaseofl =0.Inthiscase,
g =0 and g =¥ , so that H =H . Let us place a quark
′0 ′0′ 0
at u ,u3 and an antiquark at v ,v3. We now ask what the
⊥ ⊥
ground-state energy is, with these sources introduced. This
|
energyisaneigenvalueofH . SinceH isthesumofnonlin-
0 0 | |
earsigmamodels,wetrytoputasmanyaspossibleofthese q¯
| | | | |t
sigmamodelsintheirgroundstates. Inotherwords,wewant |
as many bands of our wine transit box as possible to be un-
occupiedbyFZ particles. Considerthe effectoftheresidual x2 |
conditionon states, (3.2). This tells us that we cannotmake 6
|
allfourofthebandsmeetingatu inacolor-singletstate,due
⊥ | || |
to the presence of the quark at u . Now we want the sigma
⊥ |
modelineachofthesebands(u ,1),(u ,2),(u 1ˆa,1)and
⊥ ⊥ ⊥ t
(u 2ˆa,2)tobeinaneigenstate(sinceweseek−thelowest- q
⊥
−
energyeigenstateofH ). Nowatleastoneofthesefoursigma
0
models cannot be in a color-singlet state. The Hohenberg- -x1
Mermin-Wagnertheoremtellsusthatthereisnospontaneous
symmetrybreakingforthe vacuumof a (1+1)-dimensional FIG. 4: The transverse projection of a transverse string, built out
field theory. Hence thevacuummustbe a singlet. Anynon- ofFZparticles(representedbylargebullets),joiningaquarktoan
singlet eigenstate must therefore have energy of at least the antiquark(representedbysmallbullets).Longitudinalfluxlinesjoin
massgap,m . Soatleastoneofthefoursigmamodelsmust theparticlestogether.Thesefluxlinesareparalleltothelineofsight,
1
containanFZparticle,say(u ,1). Ifthereisnocolorsource hencearenot visible. Theneighboring FZparticlesandsourcesin
⊥
in the line u +1ˆa, then by the constraint(4.1), at least one thisfigurearenotadjacent,ingeneral,sincetheymaybelocatedat
oftheothert⊥hreesigmamodelsinbandsadjacenttothisline, differentvaluesofx3.
namely(u +1ˆa,1),(u +1ˆa,2)and(u +1ˆa 2ˆa,2),must
⊥ ⊥ ⊥
−
also be excited, by the same reasoning. Continuing in the
Eachofthecorrelationfunctionsinthisproductdecaysexpo-
this way, there must be a connected two-dimensional union
nentiallyas
ofbandsinwhichallthesigmamodelsareexcited,terminat-
ing at u⊥. For the energy to be finite, this two-dimensional 0 U1(x⊥,x3)U1(x⊥+1ˆM1,x3)† 0 e−m1M1 ,
unionofbandsmustalsoterminateatv ,wheretheantiquark h | | i∼
⊥
ispresent. where 0 is the vacuum of the sigma model. Therefore the
| i
The physical picture of transverse confinement is the fol- Wilsonloopbehavesas
lowing. Imagine we look at the quark-antiquarkpair from a
A exp s M1M3. (4.3)
great distance along the 3-direction. We see a string of FZ
∼ − ⊥
particlesinthetransverseplane,joiningthequarktotheanti- Loops oriented in the x2-x3-plane behave exactly the same
quark,asinFig. 4. Thepotentialisthesumofthemassesof
way.
alltheseFZparticles. Asithappens,thepotentialisnotrota- Let us now suppose we increase l a bit, so that it is no
tion invariant, even in the transverseplane. If the quarkand
longer zero, but so that (4.1) is satisfied. Then the coeffi-
antiquarkareseparatedalongthe1-or2-directions,however,
cients of H and H are small compared to m /a. We have
′ ′′ 1
the potential between these sources is linear with transverse three mass scales in the problem, namely m , (g )2/a and
stringtension 1 ′0
1/[(g )2a]. If the first of these mass scales is much greater
′0′
s = m1 . (4.2) thantheothertwo,transverseconfinementwillstillhold. To
⊥ a determine the correction to the potential, as g′0 is increased
andg is decreasedwill requirethe applicationof exactma-
Thelackofrotationalinvariancearoundthex3-axisseemsun- ′0′
trixelements;currentformfactorsforH [20]andfieldform
realistic, butwewillargueattheendofthissectionthatthis ′
factors for H [23]. The inclusion of these terms will mean
invariancecomesaboutwhenH andH areincluded. ′′
′ ′′ that FZ particles interactbetween adjacentbands. There are
WecanreadilyseethatcertainspacelikeWilsonloopshave
alsobeprocesseswherebyFZparticlescanbecreatedorde-
an area law. Consider a rectangular loop in the x1-x3-plane.
stroyed,duetotheseinteractions.Theseeffectsmeanthatthe
ByvirtueofthegaugeconditionU =1,thisloopbreaksapart
3 theoryisnolongerintegrable,thoughitisnearlyso,provided
intoaproductofsigma-modelcorrelationfunctions.Suppose
(4.1)holds.
the dimensions of the loop are M1 and M3 in the 1- and 3-
The correction to the transverse string tension for (2+
directions, respectively. Thenthe Wilson loopisroughly(in
1)-dimensional SU(2) gauge theory comes from the lower-
thatwearebeingsloppywithcontractionsofgroupindices)
dimensional analogue of H (there is no term analogous to
′
y3+M3 H′′)[14]. Thiscorrectionisduetofluctuationsofpositionsof
A (cid:213) 0 U (x ,x3)U (x +1ˆM1,x3)† 0 . FZparticles,boundbylongitudinalstrings. Thelongitudinal-
1 ⊥ 1 ⊥
∼ x3=y3 h | | i string potential(which we discuss in the nextsection) is not
7
linearatshortdistances,butquadratic;thisisduetofactthat cle,boostthemtothelabframe,andconsiderthewavefunc-
thecolorofanFZparticleissmearedinthelongitudinaldirec- tionofthesecondparticleinthepresenceofthosefields(af-
tion. TheactualdistributionofcolorisapproximatelyGaus- terwards one can improve the result by iteratively imposing
sian, which can be foundusing a form factor for the current unitarity).Wepointoutinthissectionthattherescalingofthe
operatorofthesigmamodel[20]. couplingconstantin(1.1),(1.2)and(2.2)produces“boosted”
We expect that corrections coming from the inclusion of electricfields.
H makethepotentialinvariantwithrespecttorotationsabout Thevalueofg mustbechosensothattheratioofthelongi-
′′ ′0
the3-axis. ThisisbecauseH actingonstringstatesdeforms tudinalstringtensiontothetransversestringtensionissmall.
′′
contourofthestringinthetransverseplane. Perturbationthe- Thisratio,from(4.2)and(5.1)is
ory in this term should therefore cause the string to fluctu-
ateenoughtomakearotation-invariantpotential(thisiswhat s L = (g′0)2 = l 2g20
happens in Hamiltonian-strong-couplingperturbation theory s m a m a
1 1
[25]). ⊥
Wecanturnthisexpressionaroundtofind
m a s
V. LONGITUDINALCONFINEMENT l 2= g12 s L . (6.1)
0
⊥
Next we will show how longitudinal confinement takes Supposenowweconsidertwoincominghadrons,travelingin
place. Again, the reasoning is essentially that of Reference the3-direction,withvelocity v. Iftheelectricandmagnetic
±
[10]. fieldsofahadroninitsrestframeareE andB ,respectively,
k′ k′
Letusnextconsideraquarkandantiquarkseparatedinthe theninthelabframe
3-direction only, with coordinates u ,u3 and u ,v3, respec-
⊥ ⊥ E vB E vB
tthiveelsyo.urIcfegs,′0s=inc0e, tlhoenrgeituisdijnuastl ealeccotnrisctaflnutxpoctoesntstianlobeentweregeyn. E1= √1′1± v22′, E2= √2′1∓ v21′, E3=E3′. (6.2)
− −
Ifweassumeg =0and(4.1)instead,thenelectricfluxdoes
′06 If we assume the magnetic fields inside the hadron are ran-
costsomeenergy. Furthermorethisfluxwillbeconcentrated
dom,thentheaverageoftheratioofthelongitudinalelectric
alongthelineu . Thereasonforthisconcentrationisthatif
⊥ fieldtothetransverseelectricfieldis√1 v2. We substitute
apnliyesfltuhxatleoanveeosfththiselfionuer,breasniddsua(luga,1u)g,e(iunv,a2ri)a,n(cue(4.11ˆ)a,i1m)-, thisfors L/s in(6.1)toobtain −
⊥ ⊥ ⊥ ⊥
(u 2ˆa,2)cannotbeinasingletstate. Henceatlea−stoneof m a
⊥− l 2= 1 1 v2. (6.3)
thesefourbandsisexcitedwithanenergyatleastthesigma- g2 −
0
modelgapm1. Themassgaptendstopreventfluxspreading p
transversely. Note that (6.3) means that large velocity v implies small l .
We can now estimate the longitudinalstring tension. It is Wehavetherebyinterpretedtherescalingofthecouplingcon-
simplythestringtensionofaYang-Millstheoryinonespace stantsin(1.1),(1.2)and(2.2)astheresultofhadronsmoving
andonetimedimension,withcouplingg′0/a: at high velocities. To make such an interpretation sensible,
however,thevelocityin thetransformedcoordinates(thatis,
s = (g′0)2C , (5.1) the coordinates defined after the rescaling) should be small.
L a2 N Otherwise, the expression (6.1) cannotbe used. This means
thattoconsiderlarge-sprocesses,l shouldbetakensmall,but
whereCN is the Casimir of SU(N). Notice thatthe physical velocitiesinournewcoordinatesshouldbenonrelativistic.
mechanismoflongitudinalconfinementisadualMeissneref- Ourjustificationoftherescalingasaneikonalapproxima-
fect, though no assumptions of magnetic condensation have tion is rather heuristic. It seems strange, because it contra-
beenmade. dicts the fact that under a simple rescaling, the longitudinal
Notethatifwesimplytookl =0[6],wewouldhavetrans- componentofvelocitydoesnotchange. We are arguingthat
verseconfinement,butnolongitudinalconfinement.Theterm there really is an anomalous transformation of the velocity.
H′ isessentialforthelongitudinalstringtension. Beforerescalingthevelocityisclosetozero,butafterwards,
Correctionsofhigherorderin(g′0)to(5.1)comefromvir- itisgivenby(6.3).Itseemsworthmakingtheargumentmore
tualpairsofFZparticles.Theanalogouscalculationin(2+1) rigorous. This could conceivably be done with anisotropic
dimensionshasalreadybeendone[11]. renormalization-groupmethods[16],[17],[13],[14].
VI. THEEFFECTIVEHAMILTONIANASANEIKONAL VII. HADRONICSTATESANDDIFFRACTIVE
APPROXIMATIONFORQCD SCATTERING
We nowarguethatthe effectiveHamiltonianis aneikonal The structure of hadrons for our effective action is rather
approximationforQCD.Aconventionalapproachtothisap- similar to that of the strong-couplingpicture [9], despite the
proximation[26]istotakethefieldsfromoneincomingparti- factthatonlyonecouplingg ,isstrong,allothersbeingweak
′0′
8
(the same is true in the (2+1)-dimensional case where all
couplingsareweak).Hadronsarebuiltoutofstringsconnect-
ing quarks. Strings terminate at quarks and N of them can
meet in junctions. Thus we have a string-parton picture, in
whichthepartonsarequarksandFZparticles. Xj Xj Xj
jr jr jr jr jr jr
Xj Xj Xj
x2
eqq q
rj X⊗j 6
jr Xj
q rjA jr rj rjeqq
X⊗j rj Xj
rj
x2 Xj rj Xj
6 Xj Bj Xj Xj Xj Xj q - x1
Xj Xj jr ⊗
Xj
FIG.6:Anexampleoflongitudinalfluxexchange(LFE)isresonance
jr
betweentheconfigurationsshownhere,resembling(A)inFigure5.
eqrq
Longitudinalfluxlinesarepresent,ingeneral,inthesecondandthird
diagrams.TheparticlesshowncanbeeitherquarksorFZparticles.
-x1
VIII. CONCLUSIONS
FIG.5:Longitudinallyapproachingbaryons.Largecirclesrepresent
FZparticlesandsmallcirclesrepresentquarks.Thesymbol means
⊙
thattheconstituentparticle(FZparticleorquark)iscomingoutofthe Inthispaper,we haveconsideredthe anisotropiceffective
pageandthesymbol meansthatitisgoingintothepage. Strings theory of Reference [6], which is a longitudinally-rescaled
⊗
of FZ particles can intersect at lines of fixed x⊥ (A) or overlap at Yang-Mills theory, with rescaling parameter l . We have
transversebands(B). shownthatthistheoryconfinesforsufficientlysmalll anda
naturalstring-partonpictureemerges.Thepartonsarevalence
The transverse projection of a state of two baryons ap- quarksandthesoliton-likeFZparticlesoftheprincipal-chiral
proachingeach other longitudinallyis shownin Fig. 5. The non-linear sigma model. The interpretation of this effective
hadronscancollideatcoincidentlinesor bands. Letussup- actionasaneikonalapproximationisrathersubtle,sinceelec-
pose we neglect H +H . In the case of overlapping sites tricfieldsare muchstrongerin thetransversethanthe longi-
′ ′′
(butwithoutoverlappingat adjacentlinks), longitudinalflux tudinal direction, even for slow-moving hadrons. We have
exchange(LFE) atlinesoffixedx canhappen,butnothing argued that this interpretation is correct, provided colliding
⊥
else occurs. TheLFE amplitudecan be thoughtofas due to hadrons are moving anomalously slowly in the rescaled co-
resonancebetweendifferentfluxarrangementsandisoforder ordinates. Animportantprocessinhadronicscatteringinthe
1/N,asweexplaininthenextparagraph. forwarddirectionistheexchangeoflongitudinalflux,evenif
TheLFEprocessisanalogoustothecaseofmesonscatter- FZparticlescollide.
ingin(1+1)dimensions. WecanthinkofthetwoFZparti- Thereareessentiallytwoareaswhichcometomindforfur-
clesadjacenttothelinex (inonehadron)asbeingsimilarto therinvestigation.
⊥
two sources of fundamentalcolor in that line (recall that FZ We would like to improve our understanding of confine-
particles are dipoles, and each of them has one fundamental mentfortheanisotropicgaugetheoryconsideredhere.Thisis
sourceadjacenttotheline).ThusLFEisessentiallysimilarto notastrong-couplinggaugetheory,butratherahybridgauge
flux exchangein (1+1)-dimensionalQCD. This non-planar theorywhereonecoupling,namelyg , isstrongandtherest
′0′
processisoforder1/N[27]. AnexampleofLFEisshownin areweak.Ademonstrationthatconfinementstillholdsifg is
′0′
Fig. 6. weakwouldberealprogressontheQCD confinementprob-
We notethatworldsheetoflongitudinalelectricfluxlines lem. We havebeeninvestigatingexpansionsin (g ) 2 using
′0′ −
during LFE has the same topology as that of Pomeron ex- fieldformfactorsofthesigmamodel[23],butasyethaveno
changeinstringmodels. simpleargumentthatconfinementholdsforsmallg . Inany
′0′
In the case of two hadrons intersecting the same band, case, it seems possible to generalize our calculations of the
i.e. withcollidingFZparticles,theS-matrixoftheprincipal- corrections to the longitudinal [11] and string tensions [14],
chiralsigmamodelcomesintoplay.Wenotethatthis(1+1)- andtothemassspectrum[13]to(3+1)dimensions,withg′0′
dimensional S-matrix is unity in the large-N limit, N ¥ , large,butnotinfinite.
g2N fixed. Thus in this limit, no interaction occurs. I→n the Theotherproblemofimportanceisthe forwardscattering
0
1/NexpansionofthisS-matrix,however,LFEoccurs[19]. amplitude of hadron-hadronscattering. We hope that its so-
9
lutionwillyieldaquantitativeunderstandingofthePomeron. Acknowledgments
ThesolutionwillrequiresomecontrolofLFEprocesses. We
believethatthisproblemistractable. WehavearguedinSec-
tion6 thatin thenew coordinates,momentashouldbetaken
as small as possible; in this way the electric fields are lon- We thankGregoryKorchemskyforabrief,butveryinfor-
gitudinally boosted as they should be. Thus, LFE processes mative discussion, Jamal Jalilian-Marian for inspiration and
canbestudiedinanonrelativisticcontext. We hopetomake PoulHenrikDamgaardforencouragement.
progressonthisproblemsoon. Someassumptionsmayneed
tobemadeforthedistributionsofpartonswithinahadron.A
good starting point should be the solution of hadron-hadron ThisresearchwassupportedinpartbytheNationalScience
scattering in (2+1) dimensions, where the parton distribu- Foundation, under Grant No. PHY0653435 and by a grant
tionsaresimpler. fromthePSC-CUNY.
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