Chiral Condensate and Short-Time Evolution of QCD(1+1) on the Light-Cone

4 Pages·0.12 MB·English

by  Matthias Burkardt

Chiral Condensate and Short-Time Evolution of QCD on the Light-Cone
1+1
Matthias Burkardt
Department of Physics
New Mexico State University
Las Cruces, NM 88003-0001
U.S.A.
Frieder Lenz and Michael Thies
Institut fu¨r Theoretische Physik III
Universit¨at Erlangen-Nu¨rnberg
Staudtstraße 7
D-91058 Erlangen
Germany
(Dated: February 1, 2008)
2
0 Chiral condensates in the trivial light-cone vacuum emerge if defined as short-time limits of
0 fermionpropagators. Ingaugetheories,thenecessaryinclusionofagaugestringincombinationwith
2 thecharacteristic light-coneinfrared singularities contain therelevant non-perturbativeingredients
n responsible for formation of the condensate, as demonstrated for the’t Hooft model.
a
J
I. INTRODUCTION the triviality of the light-cone vacuum on the one hand
9
2 and the non-perturbative nature of the short-time limit
of light-cone correlationfunctions on the other.
The triviality of the light-cone vacuum is the originof
1
most of the simple properties of field theories if quan-
v
tized onthe light-cone(for a recentreview, see Ref. [1]).
5 II. CONDENSATE AND SHORT-TIME
3 Forpurelykinematicalreasonsthereisnodistinctionbe- EVOLUTION
2 tween the ground state of free and interacting theories.
1 In view of the many physical consequences usually at-
0 tributedtonon-trivialvacua,thishasraisedconsiderable We define the condensate with respect to that of the
2 non-interacting theory,
concern about the equivalence of light-cone and equal-
0
time quantization schemes. While for bosonic theories
/ ψ¯ψ = lim ψ¯ψ ψ¯ψ 0 , (2.1)
h the common belief is that the key lies in the constrained h i ε 0 h iε−h iε
-t zero-mode dynamics [2], in fermionic theories a way out → (cid:2) (cid:3)
p of this apparent contradiction has been sought in pre- with
he sscurcihptaiosntshefo“rpraergiutylarinizvaatriioannto”ftrhegeudlaivreizragteinotncroenldateinnsgatIeRs ψ¯ψ ε = 0ψ¯(ε)Peig 0εdxµAµψ(0)0 (2.2)
: h i h | | i
v and UV cutoffs [3]. It is doubtful that universal, kine- R
Xi matical prescriptions exist which describe the formation regularized in a gauge invariant way. For evaluating the
point-split condensate (2.2), we first consider a generic
of condensates in theories such as gauge theories where
r matrix element of the type
a chiral symmetry breaking is not tantamount to fermion
mdeafisnsecorerdateiropna.raInmtehteerspraessvenactuwuomrke,xtpheectpartoiopnosvaallu[4e]stoof M(ε)= 0A(ε)Peig 0sds′ddxsµ′ Aµ(x(s′))B(0)0 (2.3)
h | | i
products of Heisenberg operators,infinitesimally split in R
wherex(s)isastraightpathwithx(0)=0andx(s)=ε.
light-conetimedirection(inadditiontoaspacedirection)
ByshiftingtheargumentofthegaugefieldA andusing
willbe showntoyieldthecorrectcondensateinQCD µ
1+1
translationalinvariance of the vacuum, we can represent
— the ’t Hooft model [5]. Unlike in standard quantiza-
M(ε) as
tion, the short-time limit of Heisenberg operators differs
non-perturbatively from the corresponding Schr¨odinger
M(ε)= 0A(0)W(s)B(0)0 (2.4)
operators. Thisconnectionbetweencondensatesandthe h | | i
non-trivialityofthesingularbehaviorofcorrelationfunc-
with the point splitting now specified by the operator
tions at short light-cone times will be the focus of this
W(s) (momentum operator P ),
µ
work.
ateTdhienc[o6n,d7]e.nTsahteesoefsttuhdei’etsHmoaokfetumseodoeflgheanserbaelernelaevtiaolnus- W(s)=e−iεµPµPeig 0sds′ddxsµ′Aµ(x(s′)) . (2.5)
between properties of the excited states and the conden- R
Differentiation with respect to s,
sate,derivedinstandardquantization(Oakes-Rennerre-
lation and sum rules). Here we will present a direct cal- dW(s) dxµ
culation of the condensate which makes use explicitly of = i (Pµ gAµ(0))W(s) , (2.6)
ds − ds −
2
and integration of this differential equation with the ini- with the self-energy given in principal value prescription
tial condition W(0) = 1 allows one to simplify this ex- by
pression, yielding
Ng2
W(s)=e−iεµ(Pµ−gAµ(0)). (2.7) m2r ≡m2− 2π . (2.15)
With this form of the operator W the chiral condensate The right hand side of (2.14) combines with the remain-
of Eq. (2.2) is seen to be given by the space-time evolu- derofthematrixelementin(2.13)toyieldagainC(p,t).
tionofa systemoflight quarkscoupledto the currentof ThetermproportionaltoA+(0)inHeff canbeexpressed
an infinitely heavy quark. Thus, on the light-cone with via the Poisson equation in terms of the fermion color
its kinematical vacuum, the dynamics of a heavy-light charge density,
quark system determines the chiral condensate. So far,
everything is rather general and applies equally well to [A (0)] = g dp′dp′′ϕ†j(p′)ϕi(p′′) . (2.16)
QCD in 4 dimensions. We now specialize to the ’t Hooft + ij −2 −2π 2π (p p )2
Z ′− ′′
model [5] and write Eq. (2.2) as
Withthisresult,Eq. (2.13)canbesimplifiedbyreplacing
ψ¯ψ ε = 0ψ¯i(0)(e−iε+Heff)ijψj(0)0 (2.8) in the large N limit the operator
h i h | | i
w0)i,ththeeffectiveHamiltonianinlight-conegauge(A− = ϕ†i(p)ϕi(p′)→2πδ(p−p′)Nθ(−p), (2.17)
by its expectation value. Here the triviality of the light-
(H ) =(P +λP )δ g[A (0)] . (2.9) conevacuumisexplicitlyused. Choosingunits suchthat
eff ij + ij + ij
− −
We have denoted the slope of the path, ε /ε+, by λ Ng2
− =1 , (2.18)
and will take ε+ and λ as independent parameters from 2π
now on; due to the subtraction of the free value, the
factoring out a step function θ( p) from C(p,t) and
condensatewillturnoutto be independentofλ. P+ and −
changing p into p, the time evolution of C can finally
P aretheHamiltonianandmomentumoperatorforthe −
’t−Hooft model, respectively. be cast into the form of a typical light-cone Schr¨odinger
equation,
For evaluation of the chiral condensate, we represent
the spinor ψ in terms of the unconstrained right-handed m2 1 1
component ϕ, iC˙(p,t) = − +λp C(p,t)+ C(0,t)
2p 2p
(cid:18) (cid:19)
1 1 1 ∞ 1 ∂C(p′,t)
ψ(x)= m ϕ , (2.10) dp′ . (2.19)
21/4 (cid:18) i√2∂− (cid:19) − 2Z−0 p′−p ∂p′
and find (after Fourier transforming the fermion fields) This evolution equation for C(p,t) at short times to-
gether with the initial condition (cf. Eq. (2.12))
dpm
ψ¯ψ = C(p,ε+) (2.11)
h iε 2π p C(p,0)=N (2.20)
Z
with determines the condensate. We note that
dq m2
C(p,t)= 2πh0|ϕ†i(p)(e−iHefft)ijϕj(q)|0i . (2.12) iC˙(p,t=0)=N 2p +λp .
Z (cid:18) (cid:19)
In order to compute this correlation function we derive For non-interacting fermions,
its equation of motion. In
m2
dq C0(p,t)=Nexp it +λp (2.21)
iC˙(p,t)= 2πh0|ϕ†i(p) Heffe−iHefft ijϕj(q)|0i , (cid:26)− (cid:18)2p (cid:19)(cid:27)
Z
(cid:0) (cid:1) (2.13) solvestheevolutionequationwiththecorrectinitialcon-
we treat the two terms of H (cf. Eq. (2.9)) separately. dition. Due to the presence of singularities, C(p,t) devi-
eff
Inthefirstterm,theproductoftheoperators(P +λP ) ates significantly for arbitrarily small times from its ini-
+
and ϕ†i(p) can be replaced by their commutator. In t−he tialvalue N. Characteristicforthe short-timelight-cone
large N limit, this commutator generates a combination dynamics is the infrared singularity which implies
ofthequarkself-energyandmomentumwhere,following
limC(0,t)=0= limC(p,0) . (2.22)
’t Hooft’s original paper [5], t 0 6 p 0
→ →
m2 The short-time behavior for large momenta is the same
h0|[ϕ†i(p),P++λP−]= 2pr +λp h0|ϕ†i(p) , (2.14) in the interacting and in the free theory. It is therefore
(cid:18) (cid:19)
3
irrelevant for the evaluation of the condensate and we valuefor the condensate(see below),i.e., the condensate
drop in the following the ultraviolet regulator, is directly connected to the changein the infraredsingu-
larity of the quark propagator.
λ=0 . (2.23) As a consequence of the light-cone singularity in the
infrared, determination of the short-time behavior of
Inthis case the evolutionequationtogether with the ini-
C(p,t) requires a non-perturbative calculation of the
tial condition implies that C(p,t) (as well as C (p,t))
0 function K(q). In terms of K (q), the condensate is
(0)
depends only on the ratio of the variables p and t,
written (cf. Eqs. (2.1),(2.11)) as
C(0)(p,−iτ)=NK(0)(p/τ) , (2.24) ψ¯ψ = N m ∞ dq(K(q) K (q)) . (2.30)
0
h i − 2π q −
where we have switched to imaginary time. With Eq. Z0
(2.24) the time evolution is converted into the integro- Due to the scaling property (2.24) the dependence on
differential equation the regulator ε+ has disappeared entirely from the
expression of the condensate.
dK(q) m2 1 1 ∞ 1 dK(q′)
q = − K(q) dq′ (2.25)
dq 2q − 2 − q q dq
Z0 ′− ′
and the asymptotic behavior of K(q) is determined by III. CALCULATION OF THE CONDENSATE
the initial condition for C(p,t),
Wenowbrieflysketchtheevaluationofthecondensate
lim K(q)=1 . (2.26)
q by Mellin transformation of the equation for K. The
→∞
techniques developed in [9] will be used. We define
We can also determine the infrared behavior of K(q).
Using results for Hilbert transforms of powers (cf. [8]) it
γ(κ)= ∞dqqκ 1K(q) . (3.1)
is seen that for small q −
Z0
K(q) qβ0 (2.27) The Mellin transform is only well defined if
∼
where βn denote the solutions of the ’t Hooft boundary −β0 <κ<0 . (3.2)
condition
For non-negative κ the integral (3.1) diverges at large
πqcotπq =1 m2 (2.28) values of q, while the infraredbehavior(2.27)entails the
− lower limit. The Mellin transform converts Eq. (2.25)
ordered according to into the recursion relation (cf. [10])
1
q = βn , n=0,1,2... with βn [n,n+1]. (2.29) γ(κ)= m2 1+π(κ 1)cotπκ γ(κ 1) . (3.3)
± ∈ −2κ − − −
Thus the (confining) interaction in the ’t Hooft model (cid:2) (cid:3)
As can be easily verified,
changes the essential singularity of K (q) of the non-
0
interacting theory κ u+1sin2πu
exp 2π du 2
π κ − 0 sin2πu
γ(κ)=
K0(q)=e−m2/2q N 2 nβ0cosRπ(κ+(cid:16)β0+1/2) (cid:17)o
(cid:16) (cid:17)
toabranchpoint. Thisremarkablephenomenonisdueto ∞ 1+ (m2π−β1n)−ta1nπκ πβ0−(m2−1)tanπκ
tthioenpCre(spe,ntc)e. oFfotrhecogmapuagreissotnrinwgeinretmhaerckortrhealtattihone fsuanmce- · nY=11+ (m2π−(κ1)+tnan)πκ πκ+(m2−1)tanπκ
  (3.4)
calculation in the Gross-Neveu model, involving a sim-
ilar large N approximation, yields the non-interacting
solves the recursion relation (3.3). Condition (2.26) de-
form of the correlation function with the fermion mass
termines the normalization . The solution has the ex-
modified by the interactions. N
pected poles at the endpoints of the regularity interval
For the general case, we have not been able to ex-
(3.2) and is free of singularities within this interval.
press the solution of Eq. (2.25) via the Mellin trans-
According to Eq. (2.30) the condensate is directly
form (see below) in simple terms. In the chiral limit
given by the Mellin transform
(β √3m/π 0),
0
≈ → m
ψ¯ψ = N lim(γ(κ) γ (κ)) (3.5)
0
C(p,−iτ)≈N(p/τ)β0θ(1−(p/τ)). h i − 2π κ→0 −
where
It can be verified that Eq. (2.25) is satisfied up to terms
of O(β ). This expression displays the subtleties of the m2 κ
0 γ (κ)= Γ( κ) (3.6)
p,t 0 limit. It reproduces in the chiral limit the exact 0 2 −
→ (cid:18) (cid:19)
4
is the Mellin transform of K (q) for non-interacting Heisenberg operators [4]. In this way, the condensate is
0
fermions. Expansion of the Mellin transforms (3.4, 3.6) obtained as the short light-cone time limit of an appro-
around κ = 0, with the normalization factor chosen priate correlation function. In light-cone quantization,
N
such that the singular pieces ( 1/κ) cancel, yields non-vanishing order parameters reflect the non-triviality
∼
of the short-time limit of the relevant Heisenberg opera-
ψ¯ψ = N m 1+γ+ln m2 (3.7) tors.
h i − 2π π We have carried out a study of the chiral condensate of
(cid:26) (cid:18) (cid:19)
two dimensional QCD within light-cone quantization. A
1 ∞ 1 1
+(1 m2) + . direct and explicit calculation of the condensate within
− "β0 n=1(cid:18)βn − n(cid:19)#) the ’t Hooft model has been presented. The kinematical
X
structure of the light-cone vacuum has been an essential
This agrees with the result obtained in [7]. In particular ingredientinthiscalculation. Ourcalculationshowsthat
in the chiral limit where the 1/β0 term dominates, the the short-time limit of the quark propagator is afflicted
result by non-perturbative physics. On the light-cone this cor-
relationfunction is singularin the infrared;the singular-
N
ψ¯ψ = , (3.8) ity depends on the dynamics. The essential singularity
h i −√12
of the non-interacting theory is converted by the inter-
actions in QCD to a branch point — a phenomenon
first derived in [6] is reproduced. The calculations [6, 7] 1+1
whichdefiesaperturbativedescriptionandisresponsible
are based on low energy theorems and connect the con-
for generation of the condensate in the chiral limit. The
densate with properties of the mesons of the ’t Hooft
resultingfermioncorrelationfunctiondifferssignificantly
model (in particular the “Goldstone boson”in the chiral
atshortlight-conetimes fromthe correlationfunctionof
limit). Inourapproachwiththequarkpropagatorasthe
the Gross-Neveu model. In this two dimensional model
essentialingredient,the condensateis determinedby the
a chiral condensate emerges in the process of mass gen-
short-time behavior of a heavy-light quark system. The
eration [4]. Here the essential singularity in the infrared
equivalenceofthesequitedifferentapproachessuggestsa
persists,itsparametersaremodifiedbyinteractions. The
relationin light-conequantizationsimilar to the relation
successfuldescriptionofthecondensatesinthesedynam-
in ordinary coordinates based on chiral Ward identities
ically very different models strongly supports the idea of
whichconnectsquarkpropagatorsandtheGoldstonebo-
reconciling the non-trivial vacuum properties with the
son Bethe-Salpeter wave function (cf. [11]).
kinematical nature of the light-cone vacuum by the dy-
namical, non-perturbative short light-cone time limit of
IV. CONCLUSIONS Heisenberg operators.
In quantum field theory, condensates are calculated in Acknowledgments M.B. was supported by a grant
generalasexpectationvaluesofSchr¨odingeroperatorsin from DOE (FG03-95ER40965) and through Jefferson
the corresponding vacua. Non-vanishing order parame- Lab by contract DE-AC05-84ER40150 under which the
tersreflectthenon-trivialityofthevacuum. Inlight-cone Southeastern Universities Research Association (SURA)
quantization, the kinematical structure of the light-cone operates the Thomas Jefferson National Accelerator Fa-
vacuum gives rise to trivial vacuum expectation values cility.
ofSchr¨odingeroperatorswhich thereforecannotserveas
order parameters. On the light-cone, condensates have
to be defined as vacuum expectation values of limits of
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