Effective Field Theory for Long Strings

0.15 MB·English

by  M. Baker

Effective Field Theory for Long Strings
3
1
0
2

n M. Baker
a
DepartmentofPhysics,UniversityofWashington,Seattle
J
Box351560,SeattleWA98195,USA
8
1 E-mail:[email protected]

]
h InpreviousworkweusedmagneticSU(N)gaugetheorywithadjointrepresentationHiggsscalars
p
to describe the long distance quark-antiquark interaction in pure Yang-Mills theory, and later
-
p to obtain an effectivestring theory. The empiricallydeterminedparametersof the non-Abelian
e
effectivetheoryyieldedZ fluxtubesresemblingthoseoftheAbelianHiggsmodelwithLandau-
h N
[ Ginzburg parameter equal to 1/√2, corresponding to a superconductoron the border between

typeIandtypeII.However,thephysicalsignificanceofthedifferencesbetweentheAbelianand
1
v the ZN vortices was not elucidated and no principle was found to fix the value of the ’Landau-
7 Ginzburgparameter’k ofthenon-AbeliantheorydeterminingthestructureoftheZ vortices.
N
6
4 Here we reexaminethis pointof view. We propose a consistency conditionon ZN vortices un-
4 derlying a confining string. This fixes the value of k . The transverse distribution of pressure
.
1 p(r)intheresultingZ fluxtubesprovidesaphysicalpictureofthesevorticeswhichdifferses-
N
0
sentially from that of the vortices of the Abelian Higgs model. We speculate that this general
3
1 picture is valid independentof the details of the effective magnetic gaugetheoryfromwhich it
:
v wasobtained. LongwavelengthfluctuationsoftheaxisoftheZ vorticesleadfromaneffective
N
i
X fieldtheorytoaneffectivestringtheorywiththeNambu-Gotoaction.Thiseffectivestringtheory
r dependsona single parameter,thestringtension s . Incontrast, the effectivefield theoryhasa
a
secondparameter,theintrinsicwidth1/Mofthefluxtube,andisapplicableatintermediatedis-
tancesinarangebetween0.2fmand1fm,wherethecontributionoftheintrinsicwidthincreases
thefluxtubewidthoverthatpredictedbyeffectivestringtheory.
XthQuarkConfinementandtheHadronSpectrum
8–12October2012
TUMCampusGarching,Munich,Germany
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
EffectiveFieldTheoryforLongStrings
1. Introduction
TheprincipalgoalofthistalkistoreexaminemagneticSU(N)gaugetheorywhichwe
have used [1] as an effective field theoryof the long distance quark-antiquarkinteraction,
andto elucidate thepropertiesofthe Z flux tubesfoundin thetheory,
N
In Sections 2 to 4 we write downthe Lagrangianofthe effective SU(N)gaugetheory.
We obtain a relation, applicable for any configuration of the Higgs fields, between the
string tension s and the transverse distribution of pressure p(r) in the resulting Z flux
N
¥
tubes. In section 5, using this relation, we impose a constraint on p(r), rp(r)dr =0,
0
which we speculateis a necessaryconditionfor a fluxtubeto behaveasa string.
R
In sections 6 and 7 we consider SU(3), where we have found explicit classical Z flux
3
tubesolutions,andwecomparethesesolutionstothosefoundintheAbelianHiggsmodel.
Weplotthepressuredistributions p(r)intheZ fluxtube,anddescribethephysicalpicture
3
thatemerges. Thepressureispositiveneartheaxisandatlargerdistancesitisnegative.
It is natural to associate the boundary between the outside and inside of the string with
the pointat whichthe pressure vanishes.
In sections 8 and 9 we show that long wavelength fluctuations of the flux tube axis
lead from an effective field theory to an effective string theory with a single parameter, s .
The contribution of these fluctuations to the flux tube width [2] fixes the value of the short
distancecutoff1/L oftheeffectivefieldtheoryatavaluelessthantheintrinsicwidth1/M.
Thusthetheorycan resolve distancescales onthe orderof1/M. Finally,weexaminethe
impactof string fluctuations onthe domainofapplicabilityofthe effective field theory.
2. Effective FieldTheory
The Lagrangian Leff couples magnetic SU(N) gauge potentials, Cm to three adjoint
representationscalar fieldsf . Thegaugecouplingconstantis g .
i m
1 1
Leff(Cm ,f i)=2tr( Gmn Gmn + (Dm f i)2) V(f i), (2.1)
−4 2 −
Gmn =¶ m Cn ¶ n Cm igm[Cm ,Cn ], Dm f i=¶ m f i igm[Cm ,f i]. (2.2)
− − −
Thecomponentsofthe field tensorGmn definecolorelectric andmagneticfields~E and~B.
1
Ek= e Glm,Bk =Gk0. (2.3)
klmn
2
TheHiggspotentialV(f )is generatedfrom oneloopgraphsofSU(N)gaugetheory:[1]
i
V(f )= m 2N (cid:229) 22trf 2+4Nl tr((cid:229) f 2f 2)+ 1(tr((cid:229) f 2))2+ 2 (cid:229) (trf f )2 , (2.4)
i 4 i 3 i j N i N i j
i ij i ij !
where the parameter m 2 has dimensions of mass squared and l is dimensionless. The
SU(N)gaugesymmetry ofL reflects thatoftheoriginalSU(N)Yang-Millstheory.
eff
In the confining vacuum the magnetic SU(N) /Z gauge symmetry is completely bro-
N
ken by a Higgs condensate<f >=f =f J, wherethe three matrices J are the gener-
i i0 0 i i
ators of N-dimensionalirreducible representationof the three-dimensionalrotation group.
TheHiggspotentialhasanabsoluteminimumatf =f : f 2= 9m 2 .
i i0 0 −8(N2 1)l
−
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EffectiveFieldTheoryforLongStrings
3. Z Electric FluxTubes
N
Atlargedistancesrfromthefluxtubeaxisf i andCm areagaugetransformationW (q )
of thevacuumf i =f i0,Cm =0, whichwe can chooseto beAbelian; W (q )=exp(iq Y).
i
f i W −1(q )f i0W (q ), Cm W −1(q )¶ m W (q ). (3.1)
→ → g
m
Therequirementthatf besingle valued exp(i2p Y)is an elementofZ .
i N
→
Asr ¥
→
1 k
C~ eˆq Y, exp(igm C~ d~l) exp(2p i ), k=1,2,N 1. (3.2)
→ g ,r · → N −
m I
Assuming thegaugepotentialC~ =C(r)eˆq Y everywhereimpliesthatthe electric field
1 d(rC(r))
~E = (cid:209) C~(~x)Y = eˆ Y. (3.3)
z
− × −r dr
Thefiniteness oftheflux tubeenergy f =0on theflux tubeaxis.
i
→
4. RelationBetween String Tensionand Stress Tensor inSU(N) FluxTubes
Usingthe Abelianansatz(3.3)andtheresulting classical static equation
(cid:209) ~E = ~j =ig [f ,D~f ] (4.1)
m i i
× −
to evaluate L gives the following general relation between the string tension s , the
eff
stress tensor component Tqq , and ~E(r=0), the color electric field on the axis of the flux
tube:
¥ 2p rTqq (r)dr= 2tr(2p Yeˆ ~E(r=0)) s . (4.2)
Z0 r2 − gm · −
validforanyconfigurationoftheHiggsfieldsf . Thequantity 2tr(2p Yeˆ ~E(r=0))R=W,
i − gm z·
the work necessaryto separatea qq¯ pairlying on thez-axis bya distance R.
IfTqq >0thegaugerepulsionexceedstheHiggsattractionproducedbythecirculating
magnetic currents generated by the Higgs condensate, and (4.2) implies that W > s R.
Thatis,whenthereisnetrepulsion,theworkW neededtoseparatetheqq¯pairadistance
R inthe fixed finalvortexfield~E(r=0) isgreaterthans R,whichitself is equalto thework
donein a field~E thatis beingbuiltupasthe qq¯ pairis separated.
If there is compensationbetween the net attractive and repulsive contributions to the
pressure p(r)=Tqq /r2 averagedoverthe widthofthe flux tube;thatis, if
¥ 2p rTqq dr= ¥ 2p rp(r)dr=0, (4.3)
r2
0 0
Z Z
then thestring tension s =W/R,determinedby thefield ~E(r=0) on theflux tubeaxis.
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EffectiveFieldTheoryforLongStrings
5. Speculation onEffective FieldTheories Describing Long Strings
Consider now a flux tube connecting a qq¯ pair located at z = R/2 having energy
±
V (R), the heavy quark potential. The force acting on the quarks is determined by the
0
color field at the positions of the quarks and is equal to dV /dR. If the long distance
0
potentialV (R)=s Rpersiststo distancesR,thenthisfieldisfixed bythestring tensions .
0
If condition (4.3) is met, the field ~E on the z axis near the midpoint of the flux tube is
also fixed by the value of s . In this situation, it is consistent to assume the field has the
same value, proportionalto s , at all points on the z axis between the qq¯ pair ; that is, the
flux tube behaves like a string, consistent with the assumption that the long distance qq¯
interactionpersiststoshortdistances. Thisargumentfailsifcondition(4.3)isnotsatisfied.
We assume that (4.3) must be satisfied for any effective field theory describing the
confining string in SU(N)Yang-Mills theory, andwe impose this condition to constrain the
parametersin L .
eff
6. ClassicalStatic SU(3)Flux Tube Solutions
ForSU(3)we havefoundexplicit classical static solutions[1]with
l
J = l , J = l , J =l , Y = 8 , (6.1)
x 7 y 5 z 2
− √3
(l il ) (l +il )
f 1 = f 1(~x) 7−2 6 +f 1∗(~x) 7 2 6 ,
( l il ) ( l +il )
f 2 = f 2(~x) − 52− 4 +f 2∗(~x) − 52 4 ,
f = f (~x)l ,
3 3 2
C~ = C(r)eˆq Y, f 1(~x)=f (r)exp( iq ), f 2(~x)=f (r)exp(iq ), f 3(~x)=f 3(r).
−
Thecommutationrelations
[Y,l il ]=l il , [Y, l il ]= ( l il ), [Y,l ]=0 (6.2)
7 6 7 6 5 4 5 4 2
− − − − − − −
showthattheHiggsfieldsf f ,f carryY charge 1,1,0respectively,andthattheansatz
1 2 3
−
(6.1) is consistentwith equation(4.1).
We rescale the fields choosing the flux tube radius 1 as the scale of length, making
M
the replacementr r/M,C MC, f f f , f f f , with M=√6g f . Inserting (6.1)
→ → gm → 0 3 → o 3 m 0
into the effective Lagrangian (2.1) yields the energy density T and the stress tensor
00
componentTqq :
4M2 1 1d(rC) 1 1 1 df 1 df
T = ( )2+ (C )2f 2+ ( )2+ ( 3)2+V(f ,f ) , (6.3)
00 3 g2 2 r dr 2 − r 2 dr 4 dr 3
m (cid:20) (cid:21)
Tqq = 4M4 1(1d(rC))2+1(C 1)2f 2 1(df )2 1(df 3)2 V(f ,f ) . (6.4)
r2 3 g2 2 r dr 2 − r −2 dr −4 dr − 3
m (cid:20) (cid:21)
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EffectiveFieldTheoryforLongStrings
where
(f 2 1)2 (f 2 1)2 (f 2 1)(1 f 2) 25 l
V(f ,f )=k 2 − +9 3 − 7 3 − − , k 2 . (6.5)
3 4 100 − 50 ≡ 9 g2
(cid:18) (cid:19) m
Separatingthegaugecontribution andthe Higgscontribution to T00 andTqq gives
¥ 4M2 4M2
2p rT dr=s = s (k ) = (s (k )+s (k )), (6.6)
00 3 g2 3 g2 g h
Z0 m m
¥ 2p rTqq dr = 4M2(s (k ) s (k )), (6.7)
Z0 r2 3 g2m g − h
where
¥ 1 1d(rC) 1 1
s (k ) 2p rdr ( )2+ (C )2f 2 , (6.8)
g
≡ 0 2 r dr 2 − r
Z (cid:18) (cid:19)
and
¥ 1 df 1 df
s (k ) 2p rdr ( )2+ ( 3)2+V(f ,f ) . (6.9)
h 3
≡ 0 2 dr 4 dr
Z (cid:18) (cid:19)
7. Results forSU(3)String Tensionand Stress Tensor
Figure1:Tqq /rvsr. Red,longdashed,k 2=0.5; blue,thick,k 2=0.59; green,shortdashed,k 2=0.8.
Note that if f (r) has its vacuum value f =1, V(f ,f ) reduces to the Higgs potential
3 3 3
of the Abelian Higgs model with Landau Ginzburg parameter k . Furthermore, numerical
solutionoftheclassicalequationsshowsthatf <1andf >1everywhere;hencetheterm
3
coupling f and f in (6.5) is attractive. This additional attraction in V(f ,f ) reduces the
3 3
energy of the Z vortex below that of the Abrikosov-Nielsen-Olesen vortex of the Abelian
3
Higgs model. The ANO vortex can then be viewed as an unstable configuration of the
non-Abeliantheorythatsubsequentlydecays to the stationaryclassical solution f (r).
3
Condition (4.3) along with (6.7) yield s (k )= s (k ), which determines the physical
g h
valueofk ;k 2 0.6. Thestring tensions (k 2 0.6) 3.1, approximatelyequaltoits value
≈ ≈ ≈
in the Abelian Higgs model at k 2 =1/2. Figure 1 shows Tqq /r evaluated at the classical
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EffectiveFieldTheoryforLongStrings
solution as a function of r for three values of k 2. For k 2 0.6, where condition (4.3) is
≈
satisfied,Tqq =0atr r 1.7/M; thereisrepulsionatr r andattractionatr>r . Itis
≡ ∗∼ ≤ ∗ ∗
naturaltoidentifyr asa boundary,separatingthe insideofthe fluxtubefrom its exterior.
∗
In contrast, in the Abelian Higgs modelk = 1 is a BPS state [3], and condition (4.3)
√2
is satisfied exactly because the componentsTqq andTrr ofthe stress tensor vanish for all
r [4], andthustheprofile ofTqq (r) doesnotrevealtheboundaryof theflux tube.
We speculate that the difference between the properties of the stress tensor inside
(positive netpressure) andoutside the flux tube (negative netpressure) is a fundamental
physical property of flux tubes giving rise to a confining string. The difference between
the Abelian and non-Abelian theories is caused by the additional attractive interaction
between the scalar particles, which breaks the supersymmetry [5] giving rise to the BPS
AbelianHiggsvortex, andstabilizes thenon-Abelianflux tube. Indeed,aswe haveseen,
the additional attraction in the Higgs potential of the non-Abelian theory is approximately
balancedatk 2 0.6 bythe additionalrepulsionassociated with the fact thatk 2>1/2.
≈
8. From Effective FieldTheory to Effective String Theory
The Higgs fields f vanish on the axis L of the static flux tube. Long wavelength
fluctuations of the axis L of a flux tube connecting a quark-antiquark pair sweep out a
spacetimesurfacex˜m (z )onwhichf vanishes. TheWilsonloopW(G )ofYang-Millstheory
is the path integral over all field configurations for which the Higgs fields vanish on some
surface x˜m (z ) whoseboundaryis the loopG .
W(G )= DCm Df exp(iS(Cm ,f ), S(Cm ,f )= dxLeff(Cm ,f ). (8.1)
Z Z
We transformW(G ) to apath integraloverthevortex sheetsx˜m (z ) in two stages:
1. Wefixthelocationx˜m (z )ofthevortexandintegrateoverfieldconfigurationsCm (x), f (x)
for which f (x)x=x˜(z ) =0. The integration (8.1) over these configurations Seff(x˜),
| →
the action ofthe effective string theory.
2. We then integrate over all surfaces x˜m (z ). This integration putsW(G ) into the form
of apartition functionofan effective string theory: [6]
W(G )= Dx˜m ...exp[iS (x˜m )]. (8.2)
eff
Z
Thepathintegral(8.2)goesoverthe two transverse fluctuationsoftheworld sheetx˜m (z ).
m
ThefieldmodescontributingtoS [x˜ ]havemasses>M. Fluctuationsofwavelength
eff
> 1/M are string fluctuations accounted for by (8.2). We can then replace integrations
(8.1) over field configurationsCm ,f by the classical configuration minimizing S(Cm ,f ) for
fixed position xm (z ) ofthe vortex.
exp(iS (x˜m (z ))) exp(iSclass(x˜m (z ))).
eff
≈
Whencondition(4.3)issatisfied,thelinearpotentialpersistswhenastraightfluxtube
isshortened. Likewise,bendingthefluxtubeslightlygivesachangeinenergyproportional
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EffectiveFieldTheoryforLongStrings
to the change D R in length: D E =s D R. The action of the effective of the effective field
theorybecomesthe Nambu-Gotoactionproportionalto the areaofthe vortexsheet.
S (x˜m )=s d2x g(x ) S (x˜m ). (8.3)
eff NG
− ≡
Z
p
9. HeavyQuarkPotentials andFlux Tube Shape
To obtain the heavy quark potential V (R) and transverse energy profiles between
0
static quarks separated by distance R we couple the vector potentialC~ to a Dirac string,
writing
2p
~E = (cid:209) C~ d (x)d (y)(q (z+R/2) q (z R/2))eˆ Y (9.1)
z
− × − g − −
m
in the Lagrangian L , and solving the resulting static equations [7]. We compared the
eff
results [8] with lattice data for heavy quark potentials [9], and found that g 3.91; i. e.,
m
≈
M 1.9√s . Furthermore,thesecalculationswereconsistentwithSU(2)latticesimulations
≈
[10]fortransverseenergyprofilesforarangeofinterquarkspacings0.25/√s R<2/√s .
≤
Theabovecalculationsdidnotexplicitly includethe contributionofstring fluctuations.
However,stringfluctuationsrenormalizetheintrinsicwidthandthereforetheyaretosome
extent accounted for in the empirically determined value of M. For distances larger than
1/√s , string fluctuations become dominant, leading to the logarithmic increase of the
∼
meansquarewidth oftheflux tubeatits midpoint[11];
d 2 R
w2(R/2)= − log . (9.2)
2ps r
0
Recent lattice simulations of (2+1)dSU(2) Yang-Mills theory [2] extending to dis-
tances R=36/√s gave excellent agreement with the predictions of effective string the-
ory for distances R>1.5/√s , and determined the value of r =0.364/√s . (Interpreting
0
1/r =L asthecutoffoftheeffectivefieldtheorygivesL 2.75√s 1.41M.) However,for
0
≈ ≈
distances1.5/√s >R>0.2/√s thelattice simulationsofw2(R/2)lie abovetheprediction
(9.2),indicatingthattheintrinsicwidthofthefluxtubemustbetakenintoaccountatthese
qq¯ separations. It is in this intermediate range, shown schematically in Figure 2, that we
can test the physical picture oftheconfining string given bythe effective field theory.
Figure2: SchematicshowingapproximatedomainsofapplicabilityofEffectiveFieldTheory(EFT)(solid
blueline)andEffectiveStringTheory(EST)(reddashedline).
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EffectiveFieldTheoryforLongStrings
With use of analytic regularization [12], string fluctuations do not renormalize the
string tension s , andhence its physical interpretationas the energyperunit lengthof the
classical flux tube is preserved. The leading large distance correction to the heavy quark
potential is the Lüscher term p (d 2)/24R [13], which can be regarded as a renormal-
− −
ization of the intrinsic width at the intermediate distances shown as the region of overlap
in Figure2.
10. Summary and Future Work
We have presented a physical picture of the Z flux tubes giving rise to a confining
N
string. In this picture net positive pressure in the interior of the Z vortices balances
N
net negative pressure outside. (Perhaps at the deconfinement temperature the flux tube
’bursts’! ) Wespeculatethatthisgeneraldescriptionisvalid,independentofthedetailsof
the effective magneticgaugetheoryfrom whichit wasobtained.
Comparisonwithlatticesimulationsatintermediatedistanceswouldtestourhypothe-
sisthatthereisaneffectivefieldtheoryunderlyingtheconfiningstring. SinceM 3 times
∼
T , the SU(3) deconfinementtemperature, the theory should be applicable for a range of
C
temperatures in the deconfined phase, where it was used [14] in a preliminary study of
spatialWilson loopsandwhereweexpect somemanifestationofthe Higgsfield.
Acknowledgment:
I would like to thank the organizersfor the opportunityto participate in this very stim-
ulating conference.
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