Regge-like quark-antiquark excitations in the effective-action formalism

0.13 MB·English

by  D. Antonov

Regge-like quark-antiquark excitations in the
2 effective-action formalism
1
0
2

n
a
J

4 Dmitri Antonov
∗
DepartamentodeFísicaandCentrodeFísicadasInteracçõesFundamentais,

]
h InstitutoSuperiorTécnico,UTLisboa,
p Av. RoviscoPais,1049-001Lisboa,Portugal
-
p E-mail:[email protected]
e
h JoséEmílio F.T.Ribeiro
[ DepartamentodeFísicaandCentrodeFísicadasInteracçõesFundamentais,

2 InstitutoSuperiorTécnico,UTLisboa,
v Av. RoviscoPais,1049-001Lisboa,Portugal
4
E-mail:[email protected]
8
9
5
Radialexcitationsofthequark-antiquarkstringsweepingtheWilson-loopareaareconsideredin
.
3
theframeworkoftheeffective-actionformalism. Identifyingtheseexcitationswiththedaughter
0
1 Reggetrajectories,wefindcorrectionswhichtheyproducetotheconstituentquarkmass. Theen-
1
ergyofthequark-antiquarkpairturnsouttobemostlysaturatedbytheconstituentquarkmasses,
:
v
rather than by the elongation of the quark-antiquark string. Specifically, while the constituent
i
X
quarkmassturnsouttoincreaseasthesquarerootoftheradial-excitationquantumnumber,the
r
a energyofthestringincreasesonlyasthefourthrootofthatnumber.
InternationalWorkshoponQCDGreen’sFunctions,ConfinementandPhenomenology
5-9September2011
Trento,Italy
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Regge-likequark-antiquarkexcitationsintheeffective-actionformalism DmitriAntonov
1. Introduction
Low-energyQCDcanbecharacterized byfournonperturbative quantities, ofwhichthegluon
condensate (gFmna )2 and the vacuum correlation length l (that is, the distance atwhich the two-
h i
point, gauge-invariant, correlation function of gluonic field strengths exponentially falls off) are
relatedtoconfinement(sothatthestringtensioncorresponding tothetwostaticsourcesinthefun-
damental representation is s (cid:181) l 2 (gFmna )2 [1]),whilethequarkcondensate yy¯ ,together with
h i h i
the constituent quark mass m, characterizes spontaneous breaking of chiral symmetry. The quark
condensate is expressible in terms of gluonic degrees of freedom, yy¯ = ¶ G [Aam ,m] /¶ m,
h i − h i
where G [Aam ,m] ,theone-loopeffectiveaction,canberepresentedasaworld-lineintegraloverthe
h i
closedquark’s trajectories zm (t ) andtheiranticommuting counterparts y m (t ) [2],
{ } { }
¥ ds
G [Aam ,m] = 2Nf e−m2s Dzm Dy m e− 0sdt(14z˙2m +12y m y˙m )
h i − Z0 s ZP ZA R ×
s
trPexp ig dt Ta Aam z˙m y m y n Fmna Nc . (1.1)
× 0 − −
(cid:26)(cid:28) (cid:20) Z (cid:21)(cid:29) (cid:27)
In (1.1), N is the number of light-quark flavors(cid:0), s is the Schwinge(cid:1)r proper time “needed” for the
f
quark to orbit its Euclidean trajectory, Fmna =¶ m Ana ¶ n Aam +gfabcAbm Anc is the Yang–Mills field-
−
strengthtensor,andTa’sarethegenerators ofthegroupSU(N )inthefundamentalrepresentation,
c
obeying thecommutation relation [Ta,Tb]=ifabcTc. Notice that, since thequark condensation is
generally argued to occur due to the gauge fields, the free part of the effective action in Eq. (1.1)
hasbeensubtracted, sothat G [0,m] =0. Furthermore, PandAstandtherefortheperiodic (
h i P ≡
) and the antiperiodic ( ) boundary conditions, which are imposed,
zm (s)=zm (0) A ≡ y m (s)= y m (0) R
rRespectively, on the trajectories zm R(t ) anRd their−Grassmannian counterparts y m (t ) describing g -
matricesorderedalongthetrajectory. Thetrajectories obeythecondition sdt zm (t )=0,meaning
0
that the center of each trajectory is the origin. That is, the factor of volume associated with the
R
translation of a trajectory as a whole is divided out, and the vector-function zm (t ) describes only
the shape of a closed trajectory, not its position in space. Finally, throughout this talk, we mean
bythequark masstheminimalvalueofthemassparameter m,entering Eq.(1.1), whichrenders a
finite yy¯ —seeRef.[3].
h i
Fromthemathematical viewpoint, theeffective action(1.1)represents anintegral overclosed
quarktrajectories, withtheminimalsurfacesboundedbythosetrajectories appearingasarguments
oftheWilsonloops. Thatis,
¥ ds
G [Aam ,m] = 2Nf e−m2s Dzm Dy m e− 0sdt(41z˙2m +21y m y˙m )
− Z0 s ZP ZA R ×
(cid:10) (cid:11)
s d
× exp −2 0 dty m y n d smn (z(t )) W[zm ] −Nc , (1.2)
(cid:26) (cid:20) Z (cid:21) (cid:27)
(cid:10) (cid:11)
withtheWilsonloopgivenby
s
W[zm ] = trP exp ig dt TaAam z˙m (1.3)
0
(cid:28) (cid:18) Z (cid:19)(cid:29)
(cid:10) (cid:11)
andd /d smn beingtheareaderivativeoperator,whichallowsustorecoverthespinterm y m y n Fmna
∼
inEq.(1.1). Equation(1.2)allowsustoreducethegauge-fielddependence ofEq.(1.1)tothatofa
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Regge-likequark-antiquarkexcitationsintheeffective-actionformalism DmitriAntonov
Wilson loop. TheWilson loop isunambiguously defined bythe minimal-area surface bounded by
thecontour zm (t ).
World-line integrals of this type were first calculated by imposing for the minimal surface a
specific parametrization, which in 3D corresponds to a rotating rod of a variable length [4]. We
notice that similar parametrizations of minimal surfaces spanned by straight-line strings (which
interconnect quarks or gluons in the one-loop diagrams) are widely used in the existing litera-
ture [5]. With the 4D parametrization of this kind, we were able to obtain, within the effective-
actionformalism,arealisticlowerboundfortheconstituentquarkmass: m=460MeVfor yy¯
h i≃
(250MeV)3 (cf. Ref.[3]). Forbookkeeping purposes, anexplicit derivation ofthisparametriza-
−
tionfromtheNambu–Gotostringactionwillbeprovided inSectionIIofthistalk.
It happens that for sufficiently large m’s, the mean size of the trajectory is smaller than the
vacuum correlation length l , so that the nonperturbative Yang–Mills fields inside the trajectory
can be treated as constant, leading to the area-squared law for the Wilson loop [6]. By using
the world-line representation (1.1), the area-squared law can be shown (see Ref. [3] for details)
to yield the known heavy-quark condensate of QCD sum rules, yy¯ heavy (cid:181) (gFmna )2 /m, in
h i −h i
whichcasembecomestheconstituent massofaheavy quark. Forlighterquarks, themeansizeof
their Euclidean trajectories exceeds l , so that one can expect the area-squared law tobe morphed
into an area law. However, unlike static color sources transforming according to the fundamental
representation ofSU(N ),whichyieldanarealawwithaconstantstringtensions ,lightquarksare
c
subject to azigzag-type motion. This type ofmotion can only be reconciled with the area law for
someeffectivescale-dependent stringtensions˜(s)(cid:181) 1/s,soastoobtainanonvanishing yy¯ [3].
h i
Wenoticethatthefractalizationofquarktrajectories,whichtakesplaceuponthedeviationfromthe
heavy-quark limit,hasbeenstudied inRef.[7]intermsofthevelocity-dependent quark-antiquark
potentials.
It is then clear from the above discussion that the minimal value for the mass parameter m
and the mean size of the quark trajectory are not independent from each other so that, in general,
G [Aam ,m] is a functional of the minimal area. To this we have to add that the quark condensate
characterizes the QCD vacuum, and, therefore, should remain invariant under the variations of
(cid:10) (cid:11)
the minimal area. The final ingredient should come from energy conservation: the total energy
of a given excited quark-antiquark system should be equal to the sum of the quark masses and
the energy stored in the quark-antiquark string. Then, the purpose of our analysis is to study the
effective action as a functional of the minimal area for quark-antiquark excited systems, that is,
to evaluate the corresponding variation d m such that yy¯ remains constant for a given series of
h i
quark-antiquark radialexcitations. Atthispoint,weneed aphysicalinputonhowtoquantifythese
excitations. AnaturalchoicewillbetolinktheseexcitationstothedaughterReggetrajectories[8].
Inthiswork,weadoptthispointofview.
The talk is organized as follows. In the next Section, using as an input the excitation energy
corresponding tothe n-th daughter Reggetrajectory, weintroduce anAnsatz forthescaling factor
Z , which describes an increase in the area of the excited-string world sheet. In this way, we
n
consider only the radial excitation modes of the quark-antiquark pair, which correspond to the
“breathing modes” of the string world sheet. Consideration of angular excitations, which would
correspond to the disclinations of the world sheet, lies outside the scope of our present analysis.
Then, we proceed to a selfconsistent determination of the “critical index” g , which defines that
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Regge-likequark-antiquarkexcitationsintheeffective-actionformalism DmitriAntonov
Ansatz. With this knowledge at hand, we calculate the constituent quark-mass correction d m as
n
afunction ofthe radial-excitation quantum number n. InSection III,using thelarge-n asymptotes
of the formulae obtained for Z and d m , we show that the primary contribution to the excitation
n n
energyofthequark-antiquark pairstemsfromtheconstituentquarkmassm ,andnotfromthearea
n
increase. Wealso obtain the lowest (that is, n=1)correction tothe constituent quark mass, d m ,
1
whichturnsouttobeabout26MeV.AlsoinSectionIII,wepresentsomeconcluding remarksand
anoutlook.
2. A correction to the constituent quark mass
InthisSection,wederiveageneralexpressionforthecorrectiontotheconstituentquarkmass,
coming from radial excitations of the quark-antiquark string sweeping the surface of the Wilson
loop. Tothisend,weusefortheeigenenergies ofradialexcitations ofthequark-antiquark pairthe
Reggeformula[8]:
E = ps (4n+3). (2.1)
n
Here, n is the quantum number of a radial epxcitation, and s (440MeV)2 stands for the string
≃
tension inthefundamental representation ofthegroupSU(3).
The energy gap E E can be filled in by both deformations of the quark-antiquark string
n 0
−
and/orbyincreasing inthequarkconstituent masses. Thatis,
E E =s (L L )+2(m m ), (2.2)
n 0 n 0 n 0
− · − −
whereL ’saretheeigenvalues ofthelengthofthestring. Noticethatm turnsouttobeanimplicit
n n
function of L . Wefurthermore denote by 2R the diameter of the semiclassical Euclidean trajec-
n n
toryperformedbythequarkinthe n-thexcitedstateofthesystem. Thevalueof2R cannotexceed
n
some 2R , at which string-breaking occurs. The string elongation is given by the ratio L /L ,
n,max n 0
whereinthe(n=0)statewemusthave2R =L .
0 0
Since wewill be calculating the constituent quark masses m in the units of √s , it is conve-
n
nienttointroduce adimensionless function
f m /√ps . (2.3)
n n
≡
Thesoughtcorrection totheconstituent quarkmasscanbeobtainedfromtheeffectiveaction(1.1)
if one uses there for the surface, entering the Wilson loop, the world sheet of the excited quark-
antiquark string. To this end, let us start by defining the scaling factor Z which describes an
n
enlargement oftheworld-sheet areaoveritsvalueinthe (n=0)state. Wehave
L p /s
n =1+S , where S √4n+3 √3 2 d f , (2.4)
n n n
L ≡ 2R − − ·
0 p 0
h i
and d f f f . Accordingly, the scaling factor, which describes an increase of the area of the
n n 0
≡ −
stringworldsheetinthe n-thexcitedstate,reads
2
L
Z = n =(1+S )2. (2.5)
n n
L
0
(cid:18) (cid:19)
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Regge-likequark-antiquarkexcitationsintheeffective-actionformalism DmitriAntonov
Wefurther notice that the role of aglobal characteristic of the quark trajectory can be played
either by its semiclassical radius R , orby the proper time sduring which the trajectory is orbited
n
bythequark. Therefore, thesetwoquantities arerelatedto eachotherthroughascalingrelationof
theform
g
s
R =R . (2.6)
n n,max
· s
max
(cid:18) (cid:19)
Heres isthepropertimenecessary forthequarktoorbitatrajectory ofthemaximumdiameter
max
2R equal to the string-breaking distance. The actual value of the “critical index” g will be
n,max
selfconsistentlydeterminedbelow. Thus,wegetforthecorrectingtermS inEq.(2.5)thefollowing
n
expression:
x p /s sg
S =S (s)= n, where x max √4n+3 √3 2 d f . (2.7)
n n sg n≡ 2R − − · n
p 0,max
h i
Weproceednowtothediscussionofparametrizations forthe minimalareaofthestringworld
sheetandfortheWilsonloop. Fortheminimalareaoftheunexcited-string worldsheetweusethe
followingparametrization:
1 s
S4D = dt e mnlr zl z˙r . (2.8)
2√2 0 | |
Z
It represents a four-dimensional generalization of S = 1 sdt z z˙ (cf. Ref. [4]), which is the
3D 2 0 | × |
area-functional of a surface swept out by a rotating rod of a variable length. We notice that S
R 4D
stemsdirectlyfromtheusualformulaforthearea(correspondingtotheNambu–Gotostringaction)
upon the parametrization of the surface by the vector-function wm (z 1,z 2)=z 2 zm (z 1/s ), where
z =st andz [0,1]. Indeed, theusualformulafortheareareads A = s sdz· 1dz √detg ,
1 2∈ 0 1 0 2 ab
where gab =¶ awm ¶ bwm is the induced-metric tensor, and each of the inRdices aRand b takes the
·
values 1 and 2. Using the above parametrization for wm (z 1,z 2), one can then readily prove the
followingequality:
1 s
S4D =A = dt z2m z˙n2 (zm z˙m )2.
2 0 −
Z
q
Next,wefollowRef.[3]forwhatconcernstheparametrization oftheWilsonloop(1.3),whichcan
bewrittenintheform
N
hW[zm ]i= 2a 1Gc(a )·(s˜|S mn |)a ·Ka (s˜|S mn |). (2.9)
−
Here, S mn =e mnlr 0sdt zl z˙r istheintegrated surface element, whoseabsolute valueisimpliedin
1/2
thesensethat S mn R= 2 (cid:229) S 2mn . Furthermore,G (x)andKa (x)inEq.(2.9)stand,respectively,
| | m <n
for Gamma- and Mac(cid:16)Donald fun(cid:17)ctions, a & 1 is some parameter, and s˜ = s˜(s) stands for the
effective string tension of dynamical quarks. As it was shown in Ref. [3], Eq. (2.9) provides an
interpolation between the area law for large loops and the area-squared law [6] for small loops.
This statement is illustrated byFigs. 1and 2, which compare the combined parametrization (2.9),
for a =1.9, with the area and the area-squared laws. For this illustration, we choose a circular
contour with the minimal area S, set Nc = 3, and use the relation [6] h(gFmna )2i= 7p2 ls2, where
l = 1.72GeV 1 [9] for the case of QCD with dynamical quarks considered here. The chosen
−
value of a = 1.9 has been shown in Ref. [3] to provide the best analytic approximation of the
5
Regge-likequark-antiquarkexcitationsintheeffective-actionformalism DmitriAntonov
1
’Combined’ parametrization
Area law
0.9 ’Area-squared’ law
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.5 1 1.5 2 2.5 3 3.5 4
s S
Figure1: 2a −11G (a )·(s S)a ·Ka (s S)ata =1.9, e−s S, e−h(g4F8mnaNc)2iS2. Thevalues S=4correspondstothe
radiusofthecontourequalto0.51fm.
9
’Combined’ parametrization
Area law
’Area-squared’ law
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6
Figure2: −ln 2a −11G (a )·(s S)a ·Ka (s S) ata =1.9, s S, h(g4F8mnaNc)2iS2. Thevalues S=5.5corresponds
to the radius ofhthe contourequal to 0.6 fim. The gap between ln[ ] and s S, being ln(s S) at large
− ··· ∼
distances,isirrelevantforthestaticpotential.
area-squared lawbythecombined parametrization (2.9)atsmalldistances. Figure1illustrates the
efficiencyofthisapproximation.
Wearenowinapositiontocalculate G [Aam ,mn] ,whichisdefinedbyEq.(1.1)withmreplaced
h i
bymn. Forthispurpose, wemultiplytheinfinitesimalsurfaceelementdt zl z˙r inS mn abovebythe
scaling factorZ . FollowingRef.[3],wearriveattheexpression
n
hG [Aam ,mn]i=−2NfNcZ0¥ dsse−m2ns·a (a (+2p1s)˜(2a)3+2) m(cid:213) <n Z−+¥ ¥ dBmn ! 1I(+Zn4B,s2mFn˜2mn a )+3, (2.10)
(cid:16) (cid:17)
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Regge-likequark-antiquarkexcitationsintheeffective-actionformalism DmitriAntonov
where
I(Zn,Fmn )= Dzm Dy m e− 0sdt(14z˙2m +21y m y˙m +2iZnFmn zm z˙n −iZnFmn y m y n ) (2.11)
ZP ZA R
andFmn e mnlr Blr .
≡
The world-line integral (2.11) describes an infinite sum of one-loop quark diagrams, each
having its own number of external lines of the auxiliary gauge field Bmn . Similarly to what was
done in Ref. [3], we retain in this diagrammatic expansion only the two leading terms – the one
corresponding to a free quark, which cancels out in Eq. (1.1), and the one corresponding to a
diagram withtwoexternallinesofthegaugefield. Thelatter yieldsthefollowingexpression:
1 4
I(Zn,Fmn )= (4p s)2 3(sBZn)2+O (sBZn)4 , (2.12)
(cid:20) (cid:21)
(cid:0) (cid:1)
1/2
whereB= (cid:229) B2mn .
m <n
We sta(cid:16)rt our an(cid:17)alysis with the limit n 1, where one can approximate the factor Z in
n
≫
Eq. (2.12) by S2. In the same large-n limit, it is legitimate to disregard the quartic and the higher
n
termsintheexpansion (2.12), providedtheamplitude Bisbounded fromaboveas
1 s2g 1
−
B< = . (2.13)
sS2 x 2
n n
In the last equality, we have used the explicit parametrization (2.7) of S in terms of s. Now, as
n
it was already mentioned in Introduction, the quark condensate, hyy¯ i=−¶ ¶mnhG [Aam ,mn]i, being
the quantity which characterizes the QCD vacuum, should remain n-independent. For this to be
possible,theenergyofstringexcitationsshouldbelargelyabsorbedbytheconstituentquarkmasses
m ,anditwillbedemonstrated belowthatthisiswhatindeedhappens.
n
Anexpression forthequarkcondensate followingfromEqs.(2.10)-(2.13) reads
yy¯ = a (a +1)(a +2)Nf m ¥ dse−m2nsS4 s2g−1/xn2dB B7 , (2.14)
h i − 8p 2 · nZ0 s˜6 nZ0 1+ B2 a +3
2s˜2
(cid:16) (cid:17)
wherefromnowonwesetN =3. TheB-integrationinthisformulacanbeperformedanalytically,
c
toyield
3N ¥ f˜[A (s),a ]
hyy¯ i=−4p 2f ·mnZ0 dse−m2ns· 2s2nAn(s) , (2.15)
where
1 s2g 1 2
−
A (s) , (2.16)
n ≡ 2 sx˜ 2
(cid:18) n (cid:19)
whileasomewhatcomplicated function f˜[A,a ]wasintroduced inRef.[3]. Animportantproperty
ofthisfunctionisthat,fora &1ofinterest,theratio f˜[A,a ]/A,asafunctionofA,hasamaximum
atA 1,whichsharpensandreaches afinitevalue( 1.18),withthefurtherincrease of a .
≪ ≃
In order forthe quark condensate to stay finite in the small-mass limit, one should be able to
represent f˜[An(s),a ] intheform(cf. Refs.[10,3])
2s2An(s)
f˜[A (s),a ] s 3/2
n = 0 , (2.17)
2s2A (s) √s
n
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Regge-likequark-antiquarkexcitationsintheeffective-actionformalism DmitriAntonov
where s is some parameter of dimensionality (mass)2 unambiguously related tothe phenomeno-
0
logical value of yy¯ . Moreover, this representation should remain valid up to the values of the
h i
propertimesuchthat
m2s &1. (2.18)
n max
Equation(2.17)canequivalently bewrittenas
f˜[A ,a ]
n =x, where x 2(s s)3/2. (2.19)
0
A ≡
n
Owing to the above-mentioned form of the function f˜[A,a ]/A, a solution to Eq. (2.19), which
provides a physical decrease of s˜ with s, reads A xe , where e 0 for a &1 of interest, and
n
≃ →
x.1 (cf. Ref. [3]). Therefore, to a very good approximation, one can set in Eq. (2.16) A 1.
n
≃
This allows us to obtain the actual value of the power g in the initial Ansatz (2.6). To this end,
we notice that, due to the energy conservation in the quark-antiquark system, a variation of the
semiclassical radius R of the trajectory leads to the variation d m = s˜d R of the constituent
n n n
quark mass. Therefore, a difference between the values of the radius R in the n-th and the 0-th
n
states, d R R R ,reads
n n 0
≡ − d m √2x 2
d R = n =d m n , (2.20)
n s˜ n· s2g 1
−
whereEq.(2.16)hasbeenusedatthefinalstep. Now,inorderforEqs.(2.6)and(2.20)tohavethe
sames-dependence, g =1/3.
Withthisvalueofg athand,wecannowcalculatethecorrection d m totheconstituent quark
n
mass, which is produced by the radial excitations of the quark-antiquark string. To this end, we
first insert the value of g =1/3 into Eq. (2.20), that leads to the following relation: (d R ) =
n max
√2d m x 2 s1/3. Furthermore, for x in this formula we use its expression provided by
n· n,max max n,max
Eq.(2.7)withg =1/3. Thatyields
2
d m s 3 3
(d R ) =p √2 n max n+ (d f ) . (2.21)
n max · s R2 4− 4− n min
0,max "r r #
Next, weuse theapproximation d m s 1/(d m ) , whichreflects thefact thatthe trajectory
n max n min
≃
of a maximum size (and therefore requiring the maximum proper time to be orbited) is reached
when the value of the constituent quark mass is minimal. [Notice that this approximation paral-
lels condition (2.18).] Substituting this approximation into Eq. (2.21), we arrive at the following
equation:
2
p √2 3 3
(d m ) n+ (d f ) . (2.22)
n min≃ s R2 (d R ) 4− 4− n min
0,max n max"r r #
In order to solve this equation, we represent it entirely in terms of (d f ) . That can be done by
n min
virtueoftherelation(d m ) =√ps (d f ) ,whichstemsfromEq.(2.3). Asaresult,weobtain
n min n min
thefollowingquadratic equation:
3 3 s 3/4R (d R )1/2
(d f ) +b (d f )1/2 n+ =0, where b 0,max n max. (2.23)
n min · n min− 4− 4 ≡ (2p )1/4
r r !
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Regge-likequark-antiquarkexcitationsintheeffective-actionformalism DmitriAntonov
Asolution tothisequation yieldsthesoughtcorrection totheconstituent quarkmass:
2
√ps 3 3
(d m ) =√ps (d f ) = b2+4 n+ b . (2.24)
n min n min
· 4 v 4− 4 − 
u r r !
u
t 
Thelimitsofthisformulaatlargeandsmalln’swillbeanalyzed inthenextSection.
3. The limitingcases oflargeand smallexcitations. Concluding remarks
Ignoring for a moment the effect of string-breaking, we see that the obtained (d m ) , Eq.
n min
(2.24),vanishesinthelimitofb ¥ ,whichcorrespondstothequark-antiquarkstringofaninfinite
→
length [cf. the definition of b in Eq. (2.23)]. In reality, however, the string-breaking phenomenon
imposes an upper limit on the possible values of b. Indeed, the upper limit for both R and
0,max
(d R ) is given by d /2, where d is the string-breaking distance. Lattice simulations and
n max s.b. s.b.
analytic studies [11] suggest for this distance the value of d 1.5fm. Using also the phe-
s.b.
≃
nomenological valueofs (440MeV)2,wegetb.1.37. Therefore, wefindfromEq.(2.24)the
≃
realistic asymptotic behavioroftheconstituent quarkmasstobe
(d m ) √ps n for n 1. (3.1)
n min
→ ≫
Comparison of this result with the initial Eqs. (2.1) and (2.2) shows that, in the large-n limit,
the leading contribution to the excitation energy E of the quark-antiquark pair stems from the
n
constituent quark masses. Indeed, one can perform the large-n expansion of (d f ) given by
n min
Eq. (2.24), which yields (d f ) =√n 1 bn 1/4+O(n 1/2) . Then, inserting this expansion
n min − −
−
intotheformulaforS ,Eq.(2.7),weobtaintheleadinglarge-n behavior
n (cid:2) (cid:3)
p /s
S bn1/4, (3.2)
n s=smax → pR0,max ·
(cid:12)
whichissubdominant comparedtoEq(cid:12).(3.1). Recalling Eq.(2.4),weconclude that
m L2.
n,min n
∼
Thus, the constituent quark mass appears as a primary ingredient of the excitation energy of the
quark-antiquarkpairinthelarge-nlimit,whereastheelongationofthestringplaysonlyasecondary
role. Still, we observe an increase of S with n, which, for sufficiently large n’s, validates the
n
approximation Z S2 usedafterEq.(2.12).
n n
≃
Let us now evaluate a correction to the constituent quark mass, which is associated with the
(n=1)excitation. Wenotethatthisexcitationisdeveloped justontopofthemaximally-stretched
unexcited-string configuration. For this reason, one can use for such an evaluation the above-
adopted maximum values of b=1.37 and R =0.75fm, and also set s=s . Extrapolating
0,max max
thenEq.(3.2)downton=1,wegetS S =1.45. Thefactthatthisextrapolationton=1
1≡ 1|s=smax
of the initial parametrically large result of Eq. (3.2) leads to S 1, signals the need to introduce
1
∼
some correcting numerical factor k. It can be defined through the relation (kS )2 = Z , where
1 1
again Z =(1+S )2. Thisequation yields thevalueofk=1.69. Next,according toEq.(3.2), the
1 1
9
Regge-likequark-antiquarkexcitationsintheeffective-actionformalism DmitriAntonov
multiplication of S by a factor of k is equivalent to the multiplication of b by such a factor. This
1
observation yields the corrected value of b=2.32. Inserting it into Eq. (2.24), we get the sought
estimateforthe(n=1)correction totheconstituent quarkmass:
(d m ) =26.0MeV.
1 min
This value looks like a reasonable additive correction to the leading result, m=460MeV, quoted
intheIntroduction.
In conclusion, we notice that excitations of the quark-antiquark pairs can in general lead to
an increase of the constituent quark mass and to an elongation of the quark-antiquark string. In
this talk, we have shown that, for large radial excitations n, the constituent quark mass grows
as O(n1/2), while the length of the string grows only as O(n1/4). This result clearly means that
the excitation energy of a quark-antiquark pair stems mostly from the increase of the constituent
quark masses and not from string elongation. Thus, atleast within the effective-action formalism,
excited quark-antiquark bound states tend to have essentially the same size, irrespective of their
radial excitations. This is true even if we had energy dependence for bound states different from
that of Eq. (2.1), because that will be just another prescription for the area scaling and hence, the
final qualitative result would not depend on actual details on how this scaling is obtained, but just
from the fact that for a given scaling up of the area, the mass would go like the square of the
string elongation, whereas the string energy would dimensionally go with s L . Finally, such size
n
stiffnesswillprecludedecaychannelstobecomelargewithstringelongations,becausetheychiefly
measure the size of the parent hadron and that does not change appreciably. We also emphasize
thattheadoptedcalculational methoddoesnotrelyonanyspecificclassofthestringdeformations
(such as, e.g., the normal modes). Finally, it looks natural to apply the present approach to the
description of an interesting lattice result [12] that, for quarks in the fundamental representation,
thedeconfinement andthechiral-symmetry-restoration temperatures arenearly thesame,whereas
for quarks in the adjoint representation, the chiral-symmetry-restoration temperature exceeds the
deconfinement onebyafactorof8. Workinthisdirection iscurrently inprogress.
Acknowledgments
D.A. thanks the organizers of the QCD-TNT-II International Workshop for an opportunity to
present these results in a very nice and stimulating atmosphere. The work of D.A.was supported
by the Portuguese Foundation for Science and Technology (FCT, program Ciência-2008) and by
the Center for Physics of Fundamental Interactions (CFIF) at Instituto Superior Técnico (IST),
Lisbon.
References
[1] Forreviews,see: A.DiGiacomo,H.G.Dosch,V.I.ShevchenkoandYu.A.Simonov,Phys.Rept.
372,319(2002);D.Antonov,SurveysHighEnerg.Phys.14,265(2000).
[2] Z.BernandD.A.Kosower,Phys.Rev.Lett.66,1669(1991);Nucl.Phys.B379,451(1992);
M.J.Strassler,Nucl.Phys.B385,145(1992);M.G.SchmidtandC.Schubert,Phys.Lett.B318,
438(1993);ibid.B331,69(1994);forreviews,see: M.Reuter,M.G.SchmidtandC.Schubert,
AnnalsPhys.259,313(1997);C.Schubert,Phys.Rept.355,73(2001).
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