From Decay to Complete Breaking: Pulling the Strings in SU(2) Yang-Mills Theory

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by  M. Pepe

From Decay to Complete Breaking: Pulling the Strings in SU(2) Yang-Mills Theory
M. Pepea and U.-J. Wieseb,c
a INFN, Istituto Nazionale di Fisica Nucleare, Sezione di Milano-Bicocca
Edificio U2, Piazza della Scienza 3 - 20126 Milano, Italy
b Center for Research and Education in Fundamental Physics,
Institute for Theoretical Physics, Bern University, Sidlerstr. 5, 3012 Bern, Switzerland
c Institute for Theoretical Physics, ETH Zu¨rich, Schafmattstr. 32, CH-8093 Zu¨rich, Switzerland
Westudy{2Q+1}-stringsconnectingtwostaticchargesQin(2+1)-dSU(2)Yang-Millstheory.
Whilethefundamental{2}-stringbetweentwochargesQ= 1 isunbreakable,theadjoint{3}-string
2
connectingtwochargesQ=1canbreak. Whena{4}-stringisstretchedbeyondacriticallength,it
decays into a {2}-string by gluon pair creation. When a {5}-string is stretched, it first decays into
a{3}-string,whicheventuallybreakscompletely. Theenergyofthescreenedchargesattheendsof
9 a string is well described bya phenomenological constituent gluon model.
0
0 PACSnumbers: 11.15.Ha,12.38.Aw,12.38.Gc.
2

n
a Studies of the strings connecting two static color be characterized by its N-ality k = 0,1,...,N 1 .
{ − }
J chargesprovidevaluable insightsinto the physicsofcon- Strings connecting external charges with N-ality k = 0
6
6 finement in SU(N) Yang-Mills theories. The properties are known as k-strings. These strings are unbreakable
1 of the string connecting a static quark-anti-quark pair and have a k-dependent string tension, which may or

withchargesinthe fundamental N and N represen- may not be proportional to the Casimir operator of the
] { } { }
t tations are described by a low-energy effective bosonic corresponding representation [20, 21].
a
string theory. While the string tension σ determines
l
- the quark-anti-quark potential V(r) =σr at asymptotic Since it is easiest to simulate numerically, in this let-
p
ter we study the dynamics of strings in (2+1)-d SU(2)
e distances, the massless modes corresponding to trans-
h verse fluctuations of the string give rise to the universal Yang-Millstheory whichhas the center Z(2). Other the-
[ ories in (3+1) dimensions or with other gauge groups
Lu¨scher term proportional to 1/r [1, 2], as well as to a

are expected to show similar behavior. Here we investi-
1 diverging string thickness proportional to logr [3]. The
v effectivestringtheoryalsomakesdetailedpredictionsfor gate the strings connecting two static charges Q in the
0 the excited states of the string [4]. Lattice gauge the- SU(2) representation 2Q + 1 , which we refer to as
{ }
1 2Q+1 -strings,notto be confused with k-strings. The
ory provides us with a powerful tool for investigating
5 { }
2Q+1 -stringswithintegerQhavek =0andwilleven-
the string dynamics using Monte Carlo simulations. In
2 { }
tuallybreakatlargedistances,whilethe 2Q+1 -strings
1. this way, the linearly rising quark-anti-quark potential with half-integer Q have k = 1 and are u{nbreak}able. At
has been calculated at large distances [5]. By develop-
0
asymptoticdistancesall 2Q+1 -stringsconnectinghalf-
9 ing ahighlyefficientmulti-levelsimulationtechnique [6], { }
integer charges have the same tension σ as the funda-
0 Lu¨scher and Weisz have studied the universal 1/r-term
: in the quark-anti-quarkpotential at large distances [7]. mental 2 -string. Since SU(2) Yang-Mills theory has
v { }
no dynamical fundamental charges, the static charges Q
i
X In theories with dynamical fundamental charges, the atthetwoendsofa 2Q+1 -stringcanonlybescreened
r confining string connecting two static color charges can by dynamical gluons{. When} a pair of gluons is created
a breakduetothecreationofdynamicalcharge-anti-charge from the vacuum, the external sources are screened and
pairswhichscreentheexternalstaticsources. Numerical thus reduced to Q 1. As a consequence, the 2Q+1 -
evidence for charge screening was obtained in a lattice − { }
string decays to a 2Q 1 -string and abruptly reduces
gauge-Higgs model in which the dynamical fundamen- its tension accordi{ngly−[22}, 23]. While some numerical
tal charges are scalars [8]. Direct evidence for string evidenceforstringdecaywaspresentedin[24], usingthe
breaking was first observed in the SU(2) gauge-Higgs
multi-level simulation technique of [6], we are able, for
model [9, 10] and later also in the Z(2) gauge-Higgs the first time, to investigate string decay in detail.
model [11]. While the fundamental string is unbreak-
able in SU(N) Yang-Mills theory, the string connecting We consider (2+1)-d SU(2) Yang-Mills theory on a
static adjoint charges can break due to pair-creation of cubic lattice using the standard Wilson plaquette action
dynamical gluons. This effect has been investigated in forlinkvariablesinthefundamentalrepresentation. The
[12, 13, 14, 15, 16, 17, 18]. Numericalevidence for string external color charges Q at the two ends of a string are
breakinginlatticeQCDwithdynamicalquarkshasbeen represented by Polyakov loops Φ (x) in the 2Q+ 1
Q
{ }
observed in [19]. The center symmetry of SU(N) Yang- representation wrapping around the Euclidean time di-
Mills theory is Z(N). Consequently, each SU(N) rep- rection. The corresponding potential V (r) between the
Q
resentation and hence each external static charge can static sources is extracted from the Polyakovloop corre-
2
lator
4
Φ (0)Φ (r) exp( βV (r)). (1)
h Q Q i∼ − Q 3.5
In orderto ensure a goodprojectiononthe groundstate
3
of the string, we have simulated at inverse temperatures
as large as β = 64 in lattice units. The spatial lattice
2.5
size was L = 32 and the bare gauge coupling was cho-
sen as 4/g2 = 6.0 which puts the deconfinement phase
2
transition at β 4. While this is a moderate coupling,
c
≈
we are confident that our results remain unchanged, at 1.5 fit
least qualitatively, in the continuum limit. The values Q = 1/2
Q = 3/2
of the simulated Polyakov loop correlators range from 1
10−8 to 10−135. Measuring such small signals would be 2 4 6 8 10 12 14 16
5
completely impossible without the Lu¨scher-Weisz multi-
level simulation technique. We have slightly refined this 4.5
method by applying the segmentation of the lattice not
only to slabs in time, but also to blocks in space. By 4
carefully tuning the parameters of the multi-level algo-
rithm, we havebeen able to extractthe potentials V (r) 3.5
Q
forthe 2 -, 3 -, 4 -,and 5 -strings. Asshowninthe
{ } { } { } { } 3
top panel of figure 1, at distance r 8, the 4 -string
≈ { }
decays, thus reducing its tension to the one of the fun-
2.5 fit
damental 2 -string. Similarly, the bottom panel shows
Q = 1
{ }
that,atdistancer 6,the 5 -stringdecaysandreduces Q = 2
≈ { } 2
its tension to the one of the adjoint 3 -string. Only at 2 4 6 8 10 12 14 16
{ }
r 10 the string breaks completely, at about the same
≈ FIG. 1: Top: Potential V(r) of two static color charges with
distanceastheadjoint 3 -string. Notunexpectedly,the Q = 1 (squares) and Q = 3 (stars), shifted by a constant
{ } 2 2
tensionofastringisthesame,nomatterwhetheritcon- foramoreconvenientcomparisonoftheslopes. Bottom: The
nects screened or unscreened external charges Q. same for Q=1 (squares) and Q=2 (stars). The lines are a
A fit of the fundamental potential to fit of the multi-channel model to the Monte Carlo data. The
π horizontal band at 2M0,2 = 4.84(2) corresponds to twice the
V (r)=σr +2M + (1/r3), (2) mass of a source of charge Q=2 obtained from the measure-
1/2 − 24r O ment of a single Polyakov loop.
works very well and yields the asymptotic string tension
σ = 0.06397(3). In particular, the Monte Carlo data
Here c is the coefficient of a sub-leading 1/r correc-
are in excellent agreement with the predicted coefficient Q
π of the Lu¨scher term. The “mass” contribution of tion which does not necessarily assume the asymptotic
−an24external charge Q = 12 to the total energy of the Lcou¨nstcrhiberutvioanlueof−an2π4o.rigTinhael c“hmaarsgse”QM+Q,nn tdheastcrhibasesbethene
system is given by M = 0.109(1). This “mass” itself
screened to the value Q by n gluons. Just as the
is not physical because it contains ultra-violet divergent
“mass” M = M , the “masses” M themselves
pieces. Since string decay occurs at moderate distances, 1/2,0 Q,n
are not physical, because they again contain ultra-violet
its typical energy scale is not wellseparatedfrom Λ .
QCD
divergent contributions. However, the mass differences
Consequently, unlike the string behavior at asymptotic
distances, string decay can not be addressed in a fully ∆Q,n = MQ−1,n+1−MQ,n are physical since the diver-
gent pieces then cancel. The 3 - and 4 -string are de-
systematiclow-energyeffective stringtheory. Inparticu- { } { }
scribed by the two-channel Hamiltonians H and H ,
lar,unlikethe stringtensionσ ofthe unbreakablefunda- 1 3/2
while the 5 -string is described by the three-channel
mental string, the tension σQ of an ultimately unstable { }
Hamiltonian H with
2Q+1 -string (with Q 1) is not defined unambigu- 2
{ } ≥
ously. Herewedefineσ byafitoftheMonteCarlodata
Q E (r) A
tcoonasisdiemrptlheeph2eQno+m1en-osltorginicgalams aodmelu.ltIin-chthainsnmelosdyeslt,ewme. H1(r)=(cid:18) 1,A0 E0,1(r) (cid:19),
A channel co{ntaining}a 2Q+1 -string connecting two H (r)= E3/2,0(r) B ,
chargesQ, whichresulted{ fromsc}reening a largercharge 3/2 (cid:18) B E1/2,1(r) (cid:19)
Q+n by n gluons, has the energy E (r) C 0
2,0
cQ H2(r)= C E1,1(r) A . (4)
E (r)=σ r +2M . (3)
Q,n Q − r Q,n 0 A E0,2(r)
 
3
Q σQ σQ/σ 4Q(Q+1)/3 0.4
fit
1/2 0.06397(3) 1 1 0.35 Q = 1/2
Q = 3/2
1 0.144(1) 2.25(2) 8/3
0.3
3/2 0.241(5) 3.77(8) 5
2 0.385(5) 6.02(8) 8 0.25
0.2
TABLE I: Fitted values of the string tensions σ . The ratio 0.15
Q
σ /σ with σ =σ is compared with the value 4Q(Q+1)/3
Q 1/2 0.1
representing Casimir scaling.
0.05
Q MQ,0 MQ−1,1 MQ−2,2 ∆Q,0 ∆Q−1,1 0
2 4 6 8 10 12 14
1/2 0.109(1) — — — — 0.7
fit
1 0.37(3) 1.038(1) — 0.67(3) — 0.6 Q = 1
3/2 0.72(5) 1.32(5) — 0.60(5) — Q = 2
0.5
2 1.04(3) 1.71(3) 2.42(1) 0.67(3) 0.71(3)
0.4
TABLE II: Fitted values of the “mass” M of an original 0.3
Q,n
chargeQ+nthathasbeenscreenedtothevalueQbyngluons,
0.2
together with the differences ∆Q,n =MQ−1,n+1−MQ,n.
0.1
0
Here A, B, and C are decay amplitudes which we as-
-0.1
sume to be r-independent. The potential V (r) is the
Q 2 4 6 8 10 12 14
energy of the ground state of H . Figure 2 compares
Q
FIG. 2: Top: Forces F(r) that the {2}- and {4}-string ex-
the forces F(r) = dV(r)/dr in the 2 -, 3 -, 4 -,
− { } { } { } ert on the external charges Q = 1 (squares) and Q = 3
and 5 -stringcaseswiththeresultsofthemulti-channel 2 2
{ } (stars), respectively. Bottom: Thesameforthe{3}-and{5}-
model. The tensions σ listed in table 1 have been de-
Q stringconnectingexternal chargesQ=1(squares) andQ=2
termined by a fit to the Monte Carlo data. The sim- (stars), respectively. The lines represent the fit of the multi-
ple model works rather well. It is interesting to note channel model to the Monte Carlo data.
that the ratios σ /σ do not obey Casimir scaling, i.e.
Q
they are not equal to 4Q(Q+1)/3. The “masses” M
Q,n
are listed in table 2. Remarkably, within the error bars, stretched further and further, individual strands eventu-
the mass differences ∆Q,0 =MQ−1,1 MQ,0 all take the allyrupture,therebyabruptlyreducingthetensionofthe
−
same value M = 0.65(5), independent of Q. We in- cable. While strand rupture is well-known in the mate-
G
terpret M as a constituent gluon mass which in units rialscienceofcentimeterthicksteelcableswithatension
G
of the string tension takes the value M /√σ = 2.6(2). of about 105 Newton, we have seen that a similar pro-
G
It should be pointed out that, in contrast to the string cessoccursfortheconfiningstringsinnon-Abeliangauge
tension, M is not unambiguously defined. It just re- theories which have about the same tension but are 13
G
sults from the fit parameters of the phenomenological orders of magnitude thinner. Whether a strand picture
model. The value ∆ = M M = 0.71(3) in- may correctly describe the actual anatomy of decaying
1,1 0,2 1,1
−
dicates that the addition of a second constituent gluon 2Q+1 -strings is an interesting question that will re-
{ }
costs an energy slightly larger than M . Interestingly, quirefurtherinvestigationswhichgobeyondthescopeof
G
the mass of two constituent gluons 2M = 1.3(1) is the present letter.
G
close to the 0+ glueball mass M0+ = 1.198(25) ob- It wouldbe interesting to investigate string decay and
tained in [25] at the same value of the bare coupling. string breaking for other SU(N) gauge theories. In
MG also sets the distance scale for string decay and SU(3) Yang-Mills theory the 3 -string connecting a
{ }
string breaking. A leading order estimate for the dis- quark in the 3 with an anti-quark in the 3 repre-
{ } { }
tance at which the 4 -string decays into the 2 -string sentationis unbreakable,whilethe 8 -stringconnecting
{ } { } { }
is r 2MG/(σ3/2 σ1/2) = 7.3(6), while the distance two adjointsourcescan breakby pair creationof gluons.
≈ −
at which the 3 - and the 5 -string ultimately break is Whena 6 -stringisstretched,theexternalsourceinthe
{ } { } { }
estimated to be around r 2MG/σ1 =9.0(7). 6 -representationwilleventuallybescreenedtoa 3 by
≈ { } { }
String decay can be viewed as a quantum analogue of a gluon. The correspondingstring decay should be anal-
the classical process of strand rupture in a cable con- ogoustothe decayofthe 4 -stringinSU(2)Yang-Mills
{ }
sisting of a bundle of strands. When such a cable is theory discussed above. In analogy to the 5 -string in
{ }
4
SU(2),the 10 -stringinSU(3)Yang-Millstheoryisex- Using the Lu¨scher-Weisz multi-level algorithm, study-
{ }
pectedtodecaytoanadjoint 8 -string,beforeitbreaks ingstringdecayinSU(3),SU(4),Sp(2),G(2),andother
{ }
completely at larger distances. In QCD with dynami- Yang-Mills theories is interesting and definitely feasible.
cal quarks, strings can also decay by quark-anti-quark One may also ask whether string decay can be studied
pair creation. Due to its Z(4) center symmetry, SU(4) analytically in supersymmetric theories.
Yang-Mills theory has two distinct unbreakable strings,
connecting external charges either in the 4 and 4 M.P. acknowledges useful discussions with F. Gliozzi
{ } { }
or in the 6 -representation. For external sources with and J. Greensite. This work is supported in part by
{ }
non-trivial N-ality, one then expects cascades of string funds provided by the Schweizerischer Nationalfonds
decays down to the 4 -string for k = 1,3 and down to (SNF). The “Centerfor ResearchandEducationin Fun-
{ }
the 6 -string for k =2. damental Physics” at Bern University is supported by
{ }
Studying gauge groups other than SU(N) would also the “Innovations-und KooperationsprojektC-13”of the
be interesting. For example, all Sp(N) gauge theories Schweizerische Universit¨atskonferenz (SUK/CRUS).
have the same center Z(2). The first Lie group in this
sequence is Sp(1) = SU(2) = Spin(3), while the sec-
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{ }
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