On the Analytic Structure of Scalar Glueball Operators

1.4 MB·English

by  Andreas Windisch

On the Analytic Structure of Scalar Glueball
Operators
3
1
0
2

n
a AndreasWindisch
∗
J
InstitutfürPhysik,Karl-FranzensUniversitätGraz,Universitätsplatz5,8010Graz,Austria

5
E-mail:[email protected]
1

MarkusQ.Huber
]
h InstitutfürKernphysik,TechnischeUniversitätDarmstadt,Schlossgartenstrasse2,64289
p
Darmstadt,Germany
-
p E-mail:[email protected]
e
h ReinhardAlkofer
[
InstitutfürPhysik,Karl-FranzensUniversitätGraz,Universitätsplatz5,8010Graz,Austria

1 E-mail:[email protected]
v
5
2 ThecorrelatorofthesquareoftheYang-Millsfield-strengthtensorcorrespondstoascalarglue-
5
ball, i. e., to a bound-state formed by gluonic ingredients only. It has quantum numbers 0++
3
. and its mass, as predicted by differenttheoreticalapproaches, is expected to lie between 1 and
1
0 2 GeV. Here we restrictourconsiderationsto the Bornlevel, thatis, we considerthe correlator
3
tozerothorderinthecoupling. Gluonicself-interactionistakenintoaccountindirectlybyusing
1
: non-perturbativegluon propagators. The employed closed expressions are motivated by lattice
v
i and Dyson-Schwingerstudies. The analyticcontinuationof the integralsthemselvesis compli-
X
catedbyadditionalobstructivestructureslikebranchcutsandpolesthatareinducedbytheinner
r
a integralinthecomplexplaneoftheouterintegrationvariable. Wedealwiththisproblembyde-
formingtheouterintegrationcontouraccordingly.Fordifferentinputgluonpropagatorswefinda
positiveglueballspectraldensitywhichisrequiredforphysicalstates. Polesare,however,absent
whichismostlikelyanartifactofworkingatBornlevel.
XthQuarkConfinementandtheHadronSpectrum,
October8-12,2012
TUMCampusGarching,Munich,Germany
Speaker.
∗
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
OntheAnalyticStructureofScalarGlueballOperators AndreasWindisch
1. Introduction
Ascolor-carrying statesgluonsdonotappearasasymptotic physicalstates. Theyareconfined
to observable color singlet objects by some mechanism, see, e. g., [1] for a short review. In pure
Yang-Millstheorytheonlypossibility togeneratecolorneutral –andthusobservable –statesisto
combineseveralgluonstoformaboundstate,aglueball. Experimentally,theyareveryhardtofind
due to mixing with mesonic states. On the theoretical side, several approaches for gluonic bound
statesareavailable, see,forinstance, [2]forarecentreview.
Inthefollowingwecalculatethecorrelatorofa0++ glueballcandidateanddetermineitsana-
lyticproperties[3]fromwhichwecanextractthespectraldensity. Foraphysicallyobservablestate
the spectral density must be positive in order to allow a probabilistic interpretation [4, 5]. Thus,
the detection of positivity violations indicates that a certain state is expelled form the asymptotic
state space (and in this sense confined), while the converse does not hold necessarily. Glueballs
mustthereforepossessapositivespectraldensity,althoughtheirconstituentsmaynot. Indeed,pos-
itivity violations of gluons are established in the Landau gauge from lattice [6, 7] and functional
calculations [8,9].
AsafirstapproximationwetakeintoaccountonlytheAbelianpartofthefieldstrengthtensor.
This simplifies the calculations in several respects. For example, renormalization would become
more complicated but the required machinery is available [10]. However, the input we use was
obtained from fullYang-Mills theory and therefore contains interactions. Inthefollowing wewill
usetwodifferent fitsforthegluonpropagator andcalculate theglueball correlator numerically. In
simplecasesthisisalsopossible analytically, see,e.g. [11,12]. Forfutureapplications anumeric
procedureiscertainlyadvantageous asitallows,forinstance,tousealsonumericaldataasbecame
available onlyrecently[13].
Thecalculation ofthecorrelator boilsdowntoatwodimensional integral. Sinceweconsider
complex external momenta, an additional subtlety arises: The inner integral leads to non-analytic
structures, likebranchcuts,inthecomplexplaneoftheouterintegrationvariablewhichhavetobe
taken into account properly. We do this here by deforming the integration contour, i. e., also the
radialintegration variablebecomescomplex. Thismethodisalreadyknown,see,forexample,[8],
butinthepresentcasethestructure ofthearisingnon-analyticities isespecially tediousasdetailed
below.
Thetwogluonpropagatorfitsweemployhereareofthedecoupling[14,15,16,9,17]andscal-
ingtype[18]. FortheformerweuseafittolatticedatamotivatedbytherefinedGribov-Zwanziger
scenario (RGZ) [19]. Forthe chosen parameter values it possesses two complex conjugate poles.
Notethatotherfits,forinstance,in[20],weresuggestedaswell,butmostofthemarespecialcases
oftheusedone. Forthescaling typepropagator weuseafittothesolution ofaDyson-Schwinger
study[8]. Ithasabranchcutonthenegativerealaxis.
2. SomePrerequisites
Weconsiderthecorrelatorofacandidate forascalarglueball withquantum numbers0++,
F2(x)F2(0) d = Fmna (x)Fmna (x)Frsb (0)Frsb (0) d, (2.1)
h i h i
2
OntheAnalyticStructureofScalarGlueballOperators AndreasWindisch
where d isthe space-time dimension and Fmna (x) is, aspart of ourapproximation, just theAbelian
partoftheYang-Millsfield-strength tensorgivenby
Fmna =¶ m Aan ¶ n Aam . (2.2)
−
Weareinterested inthemomentumspacerepresentation ofthiscorrelator,
ddp
F2(x)F2(0) = eipxO (p2). (2.3)
h id Z (2p )d · d
Thedesiredexpression, O (p2),reads[12]
d
ddk
O (p2)=8(N2 1) G((p k)2)G(k2)(k2(p k)2+(d 2)(k (p k))2) . (2.4)
d C− Z (2p )d − − − · −
(cid:0) (cid:1)
Foratransverse gluonpropagatorwehave
pm pn
Dmn (p2)= d mn G(p2), (2.5)
(cid:18) − p2 (cid:19)
where only the scalar part G(p2) enters the expression (2.4). A further complication we have not
addressed so far is the fact that in 4 Euclidean space time dimensions, the integral as given in
eq. (2.4) diverges like p4. To render the integral finite we employ the BPHZ renormalization,
∼
i.e.,weTaylorsubtract thedivergentterms:
¶ 2 ¶ 4
Or(p2)=O (p2) O (0) p2 O (p2) p4 O (p2) . (2.6)
d d − d − ¶ p2 d (cid:12)p=0− ¶ p4 d (cid:12)p=0
(cid:12) (cid:12)
(cid:12) (cid:12)
Theoddderivativesvanishbecauseoftheanti-symmetry oftheangularintegral. Inordertoobtain
theanalyticstructureofthescalarglueballcorrelator,wehavetosolveeq.(2.6)forcomplexvalues
of the square of the external momentum. Thespectral density isthen accessible by evaluating the
discontinuity of the branch cut along thenegative real axis. Forthe two-point function D (p2)of a
givenspinzerooperator F ,thespectral densityreads
1
r (p2)= lim [D ( p2 ie ) D ( p2+ie )] (2.7)
2p ie 0+ − − − −
→
andthespectral representation ofthetwo-pointfunction is
ddp ¥ r (t )
D (p2)= eipx F (x)F (0) = dt , (2.8)
Z (2p )d · h i Zt t +z
0
ifnopolesorcutsexceptfortime-likemomentaexist. t isthemulti-particle threshold.
0
Eq.(2.4) holds forarbitrary dimensions. Hereweconsider only d =4. Thetwo-dimensional
case, which of course has a trivial glueball spectrum, served as a test-case forthe development of
thenumericsandispresented togetherwiththefour-dimensional resultsin[3].
3
OntheAnalyticStructureofScalarGlueballOperators AndreasWindisch
3. The Method
The algorithm we use here is described in detail in [21], where as an example the analytical
results from[12]werereproduced. LetusconsiderthecaseoftheRGZpropagatorfitof[19],
p2+s
G(p2)=C . (3.1)
p4+u2p2+t2
Thefit-parametersare s=2.508GeV2,t =0.72GeV2,u=0.768GeVandC=0.784[19]. In[21]
thefollowingstepsaregiveninordertoevaluatetheintegral(2.4):
STEP1: Express(2.4)inhyper-spherical coordinates
•
8C2 ¥ 1 x+y 2√x√yz+s
O (x)= dy y dz 1 z2 −
4 p 3 Z0 Z 1 p − (x+y 2√x√yz)2+u2(x+y 2√x√yz)+t2
− − −
y
(x+y 2√x√yz)y+2(√y√xz y)2 , (3.2)
× y2+u2y+t2 − −
(cid:2) (cid:3)
wherex= p2,y=k2 and p k=√x√yz.
·
STEP2: Renormalization
•
Theintegral(3.2)divergesquadratically inx. Therenormalizedexpression isgivenby(2.6).
STEP3: Analyticcontinuation
•
Forthepresent casethisstepcanbeperformed eitheranalytically ornumerically. Forx C
∈
the inner integral of eq. (3.2) can produce an integrable singularity together withthe rest of
theintegrand. Whenzrunsthroughitsintegrationinterval [ 1,1],itpicksupawholelineof
−
these singular points resulting in abranch cut in the complex plane of the radial integration
variable y. Thus the contour of the radial integral has to be deformed in order to avoid the
cut. Foreq.(3.2)wefindtwobranch cutsaswellasapairofcomplexconjugate poles. The
branch cuts, parametrized by z, in the y-plane can be determined analytically by finding the
zeroes of the integrand of eq. (3.2) for a given x C. We compared these results with a
∈
numerical integration. Forx= 2+2ibothareshowninFig.1.
−
ItisclearlyvisibleinFig.1thatthedeformation ofthecontourofthey-integration, required
to connect y=0 to y=x 2 where x is a UV cutoff, can be quite tricky. In general the open
piece between the branch cuts always points in the direction of Arg(x). Thus, if x is on the
positive real axis, the integration is straightforward since y can be kept real as well. Now
let us consider a complex x=(r,f ) by keeping r fixed while 0<f <p /2. There are no
poles in the first quadrant, and the opening of the branch cuts always point in the direction
of Arg(x), thus the contour can be deformed continuously in that case. The same is true
for the fourth quadrant. However, the complex conjugate poles of the integrand located in
the quadrants II and III require more care. Obviously the contour cannot be deformed as
easily for Arg(p )>Arg(x) >Arg(p ), where p and p are the pole locations in the
III II II III
second andthirdquadrants, respectively. Forsomevaluesofxthebranch cutendpoints are
narrowing down the area for a possible contour, see Fig. 2. When an endpoint of a branch
4
OntheAnalyticStructureofScalarGlueballOperators AndreasWindisch
6
4
2
HLmy 0
I
-2
-4
-6
-6 -4 -2 0 2 4 6
ReHyL
Figure1: Left:Analyticresultsforthebranchcutsandpolesinthecomplexy-planeforx= 2+2i.Right:
−
Numericalverificationoftheanalyticresult.
6 6
4 4
2 2
HLmy 0 HLmy 0
I I
-2 -2
-4 -4
-6 -6
-6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6
ReHyL ReHyL
Figure2:Left: Theregionforpossiblecontoursnarrowsdown,x= 2.Right:x= 1.18+2.7i.Thesame
− −
situationoccursforx= 1.18 2.7i,withthecutsflippedaroundtherealaxis.
− −
cutcoincideswithoneofthepoles,thecontourcannotbedeformedcontinuously andanon-
analyticity arisesintheintegral. In[3]weconfirmedthatforallpointswherethishappens a
branch pointisalsopredicted fromtheCutkoskyrules[22].
There are two further steps which we omit here as they are purely technical. What is relevant
here is that the complex conjugate poles together with the two branch cuts severely restrict the
possibilities for the contour deformation. It is hard to obtain stable results for complex values of
x when the argument of x coincides with the argument of one of the pole locations. As discussed
in the next section, we find three branch cuts for the RGZ case, one along the negative real axis,
and two along the directions Arg(p ) and Arg(p ). The numerical determination of the branch
II III
points in this case is very troublesome, because for x-values close to the cuts in the x-plane the
contour necessarily always comes very close to the cuts in the y-plane what leads to numerical
artifacts. Eventhoughthescalingpropagatorof[8]hasabranchcutandtheintegrandinducestwo
more cuts in the y-plane, the absence of poles allows a continuous contour deformation to values
very close to thenegative real axis. Theresults forthe scaling propagator arethus not plagued by
numerical issues.
5
OntheAnalyticStructureofScalarGlueballOperators AndreasWindisch
Figure3: Left: TheimaginarypartofthescalarglueballcorrelatorwithRGZgluonsasinput. Right: The
realpartofthecorrelator.
4. Results
4.1 Decoupling
IntheprevioussectionwealreadydiscussedseveralaspectsoftheRGZpropagatorasgluonic
input. Mostimportantly, weconfirmedthelocationofthebranchpointsknownfromtheCutkosky
rules. Fig.3showstheimaginaryandrealpartsofthecorrelator. Thethreebranch cutsareclearly
visible. Thetwo’unphysical‘ onesopenveryslowly. Theextracted discontinuity ofthe’physical’
branch cutis depicted infig. 4. Itbecomes negative forsmall values of p2 and rises earlier than
−
expect from theCutkosky analysis. Frominvestigating thecomplexplane oftheradial integration
variable we know that these phenomena are numerical artifacts which we expect to vanish if the
contour deformation isbettertuned; see[3]foramoredetailed discussion. Thusweconclude that
thespectral densityispositive.
100
60
80
2p)] 40 2p)] 60
O( O( 40
c[ 20 c[
dis dis 20
0
0
-20 -20
0 1 2 3 4 5 0 1 2 3 4 5
2 2 2 2
-p [GeV ] -p [GeV ]
Figure4: Thediscontinuityofthephysicalbranchcuts. Left: Decouplinggluons.Right:Scalinggluons.
6
OntheAnalyticStructureofScalarGlueballOperators AndreasWindisch
Figure5: Left: Theimaginarypartofthescalarglueballcorrelatorwithscalinggluonsasinput.Right:The
realpartofthecorrelator.
4.2 Scaling
TheIRpartofthescalinggluonfitisgivenby[8],
1 p2 2k
G(p2)=w , (4.1)
p2(cid:18)p2+L 2(cid:19)
withk =0.595353. Theexponentleadstoabranchcutofthepropagatorforcomplexmomenta. We
neglected the UV part of the propagator fit, which involves a logarithm, as weare only interested
inIRrelevantpartsofthepropagator. Theotherparameters arew=2.5andL =0.51GeV.
Fig. 5 shows the imaginary and the real parts of the correlator. Since there are no non-
analyticities besides the branch cut on the negative real axis, a Källén-Lehmann representation
is possible. Thecorresponding positive spectral density is depicted infig.4. Wealso observe that
in this case the evaluation of the correlator in the complex plane is not plagued by numerical ar-
tifacts. Strictly speaking the Cutkosky analysis is in this case not applicable, since the employed
propagator does not have the required form. However, a naive application leads to a threshold in
precise agreementwithournumericresult.
5. Summary
InthisworkwestudiedtheanalyticpropertiesofascalarglueballcorrelatoratBorn-level. The
self-interaction of gluons entered via using non-perturbative gluon propagator fits. These exhibit
positivity violations and describe thus confined gluons. The resulting glueball correlators have a
branch cut for time-like momenta and no poles. The extracted spectral densities are positive as
required for aphysical state. Forthe fitof the decoupling propagator wealso findtwo unphysical
cuts which are due to the analytic structure of the fit. Possible continuations include the addition
of higher order terms and the use of numerical results for the propagator in the complex plane.
Theemployed techniques fortheevaluation oftheintegrals maybeusefulforotherstudies, where
complexmomentaareinvolved, aswell.
7
OntheAnalyticStructureofScalarGlueballOperators AndreasWindisch
Acknowledgments
MQHacknowledges support bytheAlexandervonHumboldtfoundation. AWwassupported
bytheDoctoralprogram”Hadrons inVacuum,NucleiandStars“,fundedbytheAustrianScience
FundFWF,contract W1203-N16.
References
[1] R.AlkoferandJ.Greensite,J.Phys.G34(2007)S3,hep-ph/0610365.
[2] V.Mathieu,N.KochelevandV.Vento,Int.J.Mod.Phys.E18(2009)1,arXiv:0810.4453[hep-ph].
[3] A.Windisch,M.Q.HuberandR.Alkofer,arXiv:1212.2175[hep-ph].
[4] G.Kallen,Helv.Phys.Acta25(1952)417.
[5] H.Lehmann,NuovoCim.11(1954)342.
[6] K.Langfeld,H.ReinhardtandJ.Gattnar,Nucl.Phys.B621(2002)131,hep-ph/0107141.
[7] P.O.Bowman,U.M.Heller,D.B.Leinweber,M.B.Parappilly,A.Sternbeck,L.vonSmekal,
A.G.WilliamsandJ.-b.Zhang,Phys.Rev.D76(2007)094505,hep-lat/0703022.
[8] R.Alkofer,W.Detmold,C.S.FischerandP.Maris,Phys.Rev.D70(2004)014014,hep-ph/0309077.
[9] C.S.Fischer,A.MaasandJ.M.Pawlowski,AnnalsPhys.324(2009)2408,arXiv:0810.1987
[hep-ph].
[10] D.Dudal,S.P.Sorella,N.VandersickelandH.Verschelde,Eur.Phys.J.C64(2009)147,
arXiv:0812.2401[hep-th].
[11] D.Zwanziger,Nucl.Phys.B323(1989)513.
[12] L.Baulieu,D.Dudal,M.S.Guimaraes,M.Q.Huber,S.P.Sorella,N.Vandersickeland
D.Zwanziger,Phys.Rev.D82(2010)025021,arXiv:0912.5153[hep-th].
[13] S.Strauss,C.S.FischerandC.Kellermann,arXiv:1208.6239[hep-ph].
[14] D.Dudal,S.P.Sorella,N.VandersickelandH.Verschelde,Phys.Rev.D77(2008)071501,
arXiv:0711.4496[hep-th].
[15] P.Boucaud,J-P.Leroy,A.L.Yaouanc,J.Micheli,O.PeneandJ.Rodriguez-Quintero,JHEP0806
(2008)012,arXiv:0801.2721[hep-ph].
[16] A.C.Aguilar,D.BinosiandJ.Papavassiliou,Phys.Rev.D78(2008)025010,arXiv:0802.1870
[hep-ph].
[17] R.Alkofer,M.Q.HuberandK.Schwenzer,Phys.Rev.D81(2010)105010,arXiv:0801.2762
[hep-th].
[18] L.vonSmekal,R.AlkoferandA.Hauck,Phys.Rev.Lett.79(1997)3591,hep-ph/9705242;
L.vonSmekal,A.HauckandR.Alkofer,AnnalsPhys.267(1998)1,hep-ph/9707327.
[19] A.Cucchieri,D.Dudal,T.MendesandN.Vandersickel,Phys.Rev.D85(2012)094513,
arXiv:1111.2327[hep-lat].
[20] A.C.AguilarandJ.Papavassiliou,Eur.Phys.J.A35,189(2008),arXiv:0708.4320[hep-ph].
[21] A.Windisch,R.Alkofer,G.HaaseandM.Liebmann,Comput.Phys.Commun.184(2013)109,
arXiv:1205.0752[hep-ph].
[22] D.DudalandM.S.Guimaraes,Phys.Rev.D83(2011)045013,arXiv:1012.1440[hep-th].
8