Instantons and the infrared behavior of the fermion propagator in the Schwinger Model

0.2 MB·English

by  Tomasz Radozycki

Instantons and the infrared behavior of the fermion propagator in the Schwinger
Model
Tomasz Radoz˙ycki
Department of Mathematical Methods in Physics,
Warsaw University, Hoz˙a 74, 00-682 Warsaw, Poland∗
Fermion propagator of the Schwinger Model is revisited from the point of view of its infrared
behavior. The values of anomalous dimensions are found in arbitrary covariant gauge and in all
contributing instanton sectors. In the case of a gauge invariant, but path dependent propagator,
the exponential dependence, instead of power law one, is established for the special case when the
path is a straight line. The leading behavior is almost identical in any sector, differing only by the
slowly varying, algebraic prefactors. The other kind of the gauge invariant function, which is the
amplitudeofthedressedDiracfermions,maybereduced,bytheappropriatechoiceofthedressing,
8
tothe gauge variant one, if Landau gauge is imposed.
0
0
PACSnumbers: 11.10.Kk,11.15.-q
2

n
a I. INTRODUCTION Wilson’spapers[7],wheretheauthordiscoveredthisphe-
J nomenon in the Thirring model [8] and φ4 theory. Since

9 that time much work has been done to understand the
2 Two dimensional and fully solvable field theoretical behavior of the Green’s or correlation functions in vari-
modelshavebecomeefficienttestinglaboratoriesforvar-
ous systems, with special focus, at least in field theory,
] ious aspects of nonperturbative quantum field theory as
h on the ultraviolet. The calculations have been mainly
wellasinotherbranchesofphysics. Oneofthemostpro-
t based on perturbation theory with further application
- ductive examples of that kind is the so called Schwinger
p of the renormalisation group methods. QED and QCD
Model (SM) [1] i.e. a system consisting of a massless
e mayservehereasimportantexamples[9]. Inthepresent
h fermioninteractingwithabeliangaugefieldintwospace-
paper we would like to consider the SM fermion propa-
[ time dimensions, together with its subsequent modifica-
gator(a fermioncorrelationfunction) and its anomalous
tions (for instance N fermionic flavors, fermion given a
2 dimension in the infrared.
nonzero mass or chiral version of the model). Among
v Oneoftheknownandimportantfeaturesofthismodel,
9 its nontrivial and noteworthy features one can mention making of it a valuable testing layoutfor certain aspects
9 confinement,theexistenceoftopologicalsectors,sponta-
of QCD, is the existence of nontrivial topological sec-
3 neouschiralsymmetrybreakingandfermioncondensate,
tors [10, 11, 12, 13, 14, 15]. These topological sectors
4 gauge boson mass generation and axial anomaly. Be-
. orinstantonsrevealtheirinfluenceonexpectationvalues
1 side these peculiarities, essential mainly to high energy
of products involving fermion fields. This is connected
0 physics, the SM, either massless or massive, of one fla-
with the fact, that objects of the kind Ψ(x)Ψ(y) are not
8 vor or more (as well as other models), has attracted the
chirally invariant and have nonzero matrix elements be-
0 otherresearchersattention,forinstanceinthecontextof
: tween distinct topological vacua [16]. Hence, if the true
v condensedmatter physics [2, 3]. Here, much interesthas
vacuumstate(thesocalledθ-vacuum)isdefinedthrough
i also been payed to three dimensional models as QED .
X 3 the sum, a Bloch-like superposition of topological vacua
But QED is of certain importance for the investigation
2 n> needed to maintain the cluster property [17]:
ar of electrons in metals as well [3]. |
There are plenty of papers dealing with different as- ∞
θ >= einθ n>, (1)
pects of the SM, but relatively small concern [4] has so | |
far been devoted to the analysis of infrared anomalous n=X−∞
dimensions of the fermionic correlation functions, which the off-diagonal terms arising when one calculates the
belongs to the both above areas of interest: confinement expectation values <θ Oˆ θ > not necessarily vanish.
in quantum field theory and long-range correlations in The instanton backg|ro|und not only gives additional
condensed matter. The situation here is the more inter- contributions to the propagator (or other fermion func-
esting,thatallGreen’sfunctions,atleastinthemassless tions)[10,18]butislikelytoessentiallychangethelong-
case, may be found explicitly thus allowing for the ana- range correlations between charged particles, due to the
lytic and exact results [1, 5, 6]. appearance of fermionic zero modes.. It is then worth-
The question of field operators products in QFT hav- while toexaminethe fermionpropagatorforlargespace-
ing other than canonical dimensions, dates back to the time or euclidean separations to find out, how values of
anomalousdimensionsdependonthetopologicalsectors.
Even if they are not of direct physical meaning because
of their gauge dependence, the relative values between
∗Electronicaddress: [email protected] different sectors are interesting and their differences do
2
not vary with the gauge parameter. As far as we know, In section IV we consider another type — although
no such calculation (which seems necessary for the com- not entirely unrelated — of the gauge independent func-
plete infraredanalysisofthe propagator)hasso farbeen tion,definedasthevacuumamplitudeofthedressed(and
performed. It is worth recalling here that for the four- gauge invariant) fermion fields of the Dirac type [26]:
point(purelyfermion)functiontheessentialmodification
oftheboundstatesingularityintwo-momentumspace— Ψ˜(x)=e−i d2zFµ(z,x)Aµ(z)Ψ(x), (3)
whichisasimplepoleask =0—hasbeenestablishedin
R
dhiitgihoenrailnnsotannptoolnarse—ctobrrsa:nkch=p±oi1n,t±—2(ctohnetrreibauptpieoanrse)d[1a9d]-. wofherµerFeµduscaetsisfithees∂pzµroFpµa(gza,txo)r,=aggδa(i2n)(wzi−thx)a1ll. tOopurolcohgoicicael
F
contributions taken into account, to that considered in
Although anomalous dimensions for fermion fields are
section II, provided the Landau gauge was chosen. This
gauge dependent quantities, they, as mentioned, even-
is due to the fact that the function used, is purely
tually might be distinct in different topological sectors. µ
F
longitudinalandcouplesonlyto∂A,whichdisappearsin
Therefore, a natural question arises whether one is able
the Landau gauge [21]. As we will see in section IV, the
to make all noncanonical exponents vanish, by the ap-
choice of the function is related to the definition of
propriate choice of gauge. Does a kind of Fried-Yennie µ
F
the string in (2).
gauge exist for all sectors or for each sector separately?
The SM is a perfect tool to answer these questions since
the fermion propagator is known in all sectors, thanks
II. GAUGE VARIANT PROPAGATOR
to the calculations performed in the euclidean space. To
that aspect we devote section II.
Another interesting task is to find a gauge indepen- As written in the Introduction the original Schwinger
dent function and analyse its long-range behavior. Such Model consists of a massless fermion field Ψ interacting
functions are of certain importance in condensed matter with a gauge boson Aµ. The Lagrangiandensity for this
physicsbutshouldalsoconstitute essentialobjects inin- model has the following form:
vestigations concerning fundamental problems of QCD,
(x) = Ψ(x)(iγµ∂ gγµA (x))Ψ(x)
like confinement for instance, similarly as Wilson loop L µ− µ
1
does. It is well known, that analytical properties of the Fµν(x)F (x), (4)
µν
ordinary (gauge variant) function are gauge dependent. − 4
For example in QED, the mass-shell singularity of the with a gauge-fixing term, if needed.
electron propagator in p-space which, in one gauge, is a Thespace-timehasonetimeandonespacedimension.
simple pole, in another becomes a branch point. It is then natural to choose the fermion field in the sim-
Gauge invariant function, however,is not an unequiv- plest form of the two-component spinor, this size being
ocal notion. In section III we use the function deriving determined, for the even number of dimensions D, by
fromSchwinger[20](seethefootnoteinthequotedwork) the value 2D/2. In this case, for the 2 2 Dirac gamma
and [1] in the form: matrices one can choose the representa×tion given below:
0 1 0 1
<0T(Ψ(x)exp ig dzµAµ(z) Ψ(y))0>, (2) γ0 = 1 0 , γ1 = 1 −0 , (5)
| "− Zγ(x,y) # | (cid:18) (cid:19) (cid:18) (cid:19)
where γ(x,y) is a certain space-time path from x to y. 1 0
γ5 =γ0γ1 = .
The function (2) has been widely used in the literature 0 1
(cid:18) − (cid:19)
forvarioussystems[4,21,22,23,24,25],sometimeswith
For the metric tensor we use the convention:
contradictoryresults. Onemayhopethatthiskindofan
amplitudedescribesthepropagationofphysicalparticles. g00 = g11 =1,
However, one has to keep in mind that (2) is a path de- −
pendent object and, therefore, not unique. Considering and the antisymmetric symbol εµν is defined by:
the simplest case of a straight line, (probably more suit-
able for the ultraviolet asymptotics) which is tractable ε01 = ε10 =1, ε00 =ε11 =0, (6)
−
analytically, we find the change of the infrared behavior
frompowerlawdecaytoexponentialone. Thiseffectwas
showntobeexistent(butcutoffdependent)andactually
path insensitive in QED [21] and also demonstrated in 1 Dirac’soriginalsuggestionforthestaticchargeinthe3+1dimen-
3
sional electrodynamics, was to take the dressing in the Lorentz
the Schwinger Model [4], although without considering
noncovariant —butlocalintime—form:
other than k = 0 instanton sectors. We show, yet, that
it appears independently of the gauge-field topology. A Ψ˜(t,x)=exp −ig d3y 1 ∇A(y) Ψ(t,x).
kindofanaverageoverdifferentpaths(butstillfork =0) 4π |x−y|
(cid:20) Z (cid:21)
wasperformedin[4]showingaslightlyfasterdecay,how-
Later this formula was modified to account for moving elec-
ever without changing the leading exponential behavior. trons[27]
3
but whenever it is required by the mathematical rigor, at y:
the considering of the corresponding euclidean space is
assumed.
∂z µ(z;x,y)=g(δ(2)(z x) δ(2)(z y)). (9)
The nonperturbativeformulaforthe fermionpropaga- µJ − − −
tor was given already in the first Schwinger papers [1]
and later rederived by other methods. It may be given The symbol (x) denotes the free fermion propagator,
0
S
the following form: i.e. S0(x) = −21πx26xiε, and index (0) on the l.h.s of (7)
refers to the trivial−(k =0) instanton sector.
S (x y)= (7)
(0)
−
= 0(x y)e−2i d2ud2wJµ(u;x,y)△µν(u−w)Jν(w;x,y) ,
S −
R
where (u w)isamassive(ofmasssquaredequalto
µν A. Landau gauge
g2/π) v△ector −boson propagator, in general gauge depen-
dent, i.e. the propagator of the Schwinger boson. The
current µ may be written as: If one chooses µν(x) to be transverse function, i.e.
J satisfying ∂µ (△x) = 0, as Schwinger in his original
x△µν
µ(z;x,y) = gγµγν∂z[D(z x) D(z y)]= (8) papersdid,onearrivestothefollowing,wellknown,form
J ν − − −
= g(gµν +γ5ǫµν)∂z[D(z x) D(z y)]. of the propagator:
ν − − −
It can be easily checked, with the use of: S (x y)= (x y)exp ig2β(x y) , (10)
(0) 0
− S − − −
2 D(z)=δ(2)(z), (cid:2) (cid:3)
z
where we introduced a function β defined by the two-
thatthisauxiliarycurrentisnotconservedanddescribes momentum integral and expressible through cylindrical
the creationofachargedparticleatxandits anihilation functions:
d2p 1
β(x y) = 1 eip(x y) =
−
− (2π)2 − (p2 g2/π+iǫ)(p2+iǫ)
Z (cid:16) (cid:17) −
i iπ +γ +ln g2(x y)2/4π+ iπH(1)( g2(x y)2/π) x y timelike
= 2g2 − 2 E − 2 0 − − (11)
 i hγ +ln g2(px y)2/4π+K ( g2(xpy)2/π) i x y spacelike.
 2g2 E − − 0 − − −
h p p i

Symbolγ denotesheretheEulerconstantandfunctions behavior. The obtained value, negative in the Landau
E
H(1) and K are Hankel function of the first kind, and gauge,may,however,befreelymodifiedbyagaugetrans-
0 0
Basset function respectively. formation.
From (10) and (11) one can easily deduce the IR By the proper modification and extension of the for-
asymptotics, considering euclidean or spacelike infinity. mula (7) (spoken of in detail in [18]) for the case with
The canonicaldimension is associated with the free field instantons, where zero modes of the Dirac operator con-
and, therefore, appears already in the coefficient (x). tribute to the propagator (for the sector k, the number
0
S
The anomalous one comes from the infrared behavior of of these modes is equal to k [30]), one arrives to the
the function β. For (x y)2 this behavior is dic- formula for the k = 1 secto|rs|:
− →−∞ ±
tated by the logarithmic function, since K is exponen-
0
tially decreasing for large arguments. The leading term ig
takes the form: S(1)(x−y)= 4π3/2 cosθ−iγ5sinθ eγE+ig2β(x−y) .
eγE/2 (cid:0) (cid:1) (13)
S (x y) (x y)( g2(x y)2/π)1/4 . (12) The angle θ comes fromthe definition ofthe true, physi-
(0) − ∼ √2 S0 − − −
cal vacuum of the theory given in (1). The formula (13)
In this case the anomalous dimension is equal to 1/2 with the use of the infrared approximation of (11) leads
−
and this reflects the known infrared behavior of S (p) to:
(0)
in momentum space: const [28, 29]. The long-range
( p2)5/4
correlation is then enha−nced with respect to the nonin- igeγE/2e−iγ5θ 1
S (x y) . (14)
teractingcase,contraryto,whatiscalled,Luttinger-type (1) − ∼ (2π)3/2 ·( g2(x y)2/π)1/4
− −
4
Although in the spinor space the matrical structures ( g2(x y)2/π)(1 ξ)/4 ,
−
× − −
of (12) and of (14) are different, their scaling properties (18)
areidentical,sotheanomalousdimensioninthissectoris
stillequalto 1/2. Theasymptotic(noncanonical)scale eγE/2 1−ξ ige−iγ5θ
invariance is−then maintained. One should emphasize S(1)(x−y)∼ √2 2π3/2
(cid:18) (cid:19)
here, that apart from (12) and (14), no other algebraic 1
.
termsarise. Thelong-rangeexpansionof(11)shows,that
×( g2(x y)2/π)(1+ξ)/4
subleading terms have an exponential characterderiving − −
from the behavior of the Basset function. As may be seen above, thanks to the universality of the
factor (16), there exists a gauge ξ = 1, for which in
both sectors the anomalous dimensions disappear and
B. Longitudinal contributions the propagator recovers its canonical dimension. What
is more, in this gauge S (x y) = (x y). This is
(0) 0
− S −
In this section we verify, how gauge transformations easily understandable on the basis of perturbation cal-
influencetheobtainedvaluesofanomalousdimensionsin culation [29]. For ξ = 1, the gauge boson propagator
all topological sectors. For the trivial sector k = 0, the becomes proportional to g . Therefore, if one draws
µν
simplest idea is to moveback to (7) andreplace (x), any Feynman diagram with two external fermion ‘legs’,
µν
△
which previously was chosen to be transverse, with the eachinternallinecontainingthisfunctionandjoiningtwo
quantity: vertices leads to the summation of the kind
△µν(x)=△Tµν(x)→△Tµν(x)+△Lµν(x), (15) γµγα1γα2 ·...·γα2k+1γµ , (19)
and see the dependence of the results upon different where γα1,γα2,...γα2k+1 come from either vertices or
choices of L (x). Using L (x) = ∂ ∂ F(x), one can
△µν △µν µ ν fermion propagators (remember that the free fermion
easily find from (7) and (9), that there would appear
propagatoris proportional to γ) squeezed between these
in (10) an additional, multiplicative factor, which reads:
twovertices. The number ofthose γ’sis alwaysodd (the
number of fermion propagators always exceeds by one
exp[ ig2(F(x y) F(0))]. (16)
− − − the number of vertices). Consequently, the correspond-
ingexpressionvanishes,sinceanyoddnumberofgamma
Its form is dictated by the longitudinal part of the cur-
rent µ. If we analyze the formula (37) of [18] together matrices may, in two dimensions, always be reduced to
J only one, by subsequent applications of the formula:
with (9) and (30), it becomes obvious that (16) will be
common for each instanton sector, since, while passing
γαγβγρ =gαβγρ+ǫαβγ5γρ =gαβγρ+ǫαβǫρλγλ . (20)
from k = 0 to k = 1, all terms additionally appearing
±
in the exponent are transverse and, therefore, not sensi-
After the reduction we simply apply γµγαγ =0, which
ble to the substitution (15). µ
can also be deduced from (20).
TheaboveresultagreeswiththatobtainedbyLandau
Onthe other hand S (x y)remains nontrivialeven
and Khalatnikov[31] and later rederivedby Zumino [32] (1) −
in this gauge. Due to its nonperturbative and topolog-
for ordinary QED. The topological sectors are absent in
ical nature, this simple reasoning based of perturbative
QED and, obviously, were not consiered at the time,
3+1 Feynman diagrams may not be applied.
but the transformation property described by (16) turns
out to be general.
Naturally, there exists a wide class of functions F(x),
III. GAUGE INVARIANT, PATH DEPENDENT
onecanselect,butwerestrictourselvestothosearisingin
PROPAGATOR
R gauges,withanunessentialmodification. We assume
ξ
then:
Oneofthecandidatesforagaugeinvariantonefermion
F(x y)=ξ d2wD(x w) (w y), (17) Green’sfunctionisthatoriginatingfromtheSchwinger’s
− − △ − paper [20] (some kind of ‘string expression’ appeared in
Z
the Dirac’s paper too [26]):
where (w y) is a 2D Klein-Gordon propagator for a
△ −
particle of certain mass, which, without a real change Sinv(x y)= (21)
of the infrared asymptotics, may be chosen to be equal −
to the mass of the Schwinger boson: g/√π. With this =<0T(Ψ(x)exp ig dz Aµ(z) Ψ(y))0> .
µ
choice, it can easily be verified that the additional fac- | "− Zγ(x,y) # |
tor (16) reduces to exp[iξg2β(x y)] leading to the
−
asymptotics: The gaugeindependence ofS (x y) my be easily ver-
inv
−
ified, yet this quantity depends now on the chosen path
eγE/2 1−ξ γ(x,y). Twoquestionscomeimmediatelyintoview. One
S (x y) (x y)
(0) − ∼ √2 S0 − is practical: for which path is it possible to analytically
(cid:18) (cid:19)
5
evaluate the vacuum expectation value (21)? The other sion in the source form:
is more fundamental: is there any physical justification
for the choice of one γ before another?
exp ig dz Aµ(z) = (22)
In practice the calculation of (21) is possible only for "− Zγ(x,yµ) #
the simplestchoices,as straightline for instance. Such a
pathiscertainlyagoodoptionfortheultravioletasymp- =exp i d2z (z;x,y)Aµ(z) ,
µ
totics, since then x is close to y and the curve may be − C
(cid:20) Z (cid:21)
approximatedwith a short sectionof a straightline. For
where we introduced a certain new current:
the infrared asymptotics we need infinitely long path,
but, on the other hand, the long-range behavior should
not depend too strongly on the contour peculiarities for µ(z;x,y)=g dwµδ(2)(z w), (23)
C −
finite values of z. This is to some extent supported by Zγ(x,y)
satisfying:
the results of [4].
The choice of the path γ, as well as of any other form
∂z µ(z;x,y)=g(δ(2)(z x) δ(2)(z y)). (24)
of the invariant propagator,is related to the selection of µC − − −
the gauge for the noninvariant function. For instance
Itis worthnoting thatthis divergenceis identicalto (9),
the straight line path corresponds to the Fock gauge which means that the difference of the two currents ν
(zµ − xµ)Aµ(z) = 0 [33] since, in this case, the string and µ is transverse: J
in (21) disappears and both invariant and noninvariant C
functions equal each other. This means that the gauge µ(z;x,y) µ(z;x,y)= (25)
independence of (21) becomes somewhat illusory. This J −C
is as if one told that the propagator calculated in cer- =g γ5ǫµα∂z∂β (2 gµβ ∂µ∂β) du D(z u).
α z − z − z z β −
tain fixed gauge is gauge independent. Obviously it is! Zγ(x,y)
(cid:2) (cid:3)
Similarly as the position of the moving body in a fixed
As we will see below, this transversality guarantees the
moment t=t is time independent. Therefore the main
0
gauge invariance of the final expression. Now we cal-
point is to ascertain which path is the most convenient
culate the fermion propagator using functional integral
one(whichmightalsomeanthatcertaingaugeorclassof
formalism, where it is defined as the derivative over
gaugesis privileged). Onthe other hand, this choice can
fermionic sources η and η of the generating functional:
beambiguous. Itislikelythateachtypeofexperimentor
measurement has its own preferable path and, therefore,
oneisunabletodefine itinaversatileway. Onlyasetof Z(η,η,J)= ∞ eikθZ(k)[η,η,J]= (26)
universalfeatures foraunconfinedparticleis guaranteed
k=
to be preparation independent. One should not expect X−∞
then, that all measurements performed over an electron = ∞ eikθ DΨDΨDA(k)exp i d2x( (k)(x)
(chargedparticle) obtained by the ionization of an atom L
k= Z (cid:20) Z
withstronglaserimpulsegiveidenticaloutcomeasthose X−∞
over a trapped particle in the solid body. +η(x)Ψ(x)+Ψ(x)η(x)+Jµ(x)A(k)(x)) ,
µ
Nevertheless the propagator defined in (21) has found (cid:21)
its applications (see for instance [22, 25, 34]) and, cer-
tainly, is worth being investigated with the topological whereA(µk) isthegaugefieldrestrictedtothek-instanton
contributions taken into account. Without going too sector,and (k) isthe SchwingerModelLagrangianden-
L
deeply into above questions, we restrict ourselves, in the sity with the substitution A A(k). One immedi-
µ µ
→
presentwork,tothesegmentofthestraightlineenclosed ately sees, that the only modification introduced by the
between x and y, for which (21) may formally be calcu- string(22)fork =0is the replacementofthe currentJ
µ
lated in all instanton sectors. with J . This results in the following change of the
µ µ
−C
It will be convenient to write down the string expres- formula (7):
i
Sinv(x y)= (x y)exp d2ud2w( µ(u;x,y) µ(u;x,y)) (u w)( ν(w;x,y) µ(w;x,y)) . (27)
(0) − S0 − −2 J −C △µν − J −C
(cid:20) Z (cid:21)
The evaluating of the exponent leads to the familiar expression [4]:
Sinv(x y)= (x y) (28)
(0) − S0 −
g2
exp 2ig2β(x y) i dz dwµ (z w)+ig2γ5 d2z dw (D(z y) D(z x))ǫµα∂z (z w) ,
× " − − 2 Zγ(x,yµ)Zγ(x,y) △ − Z Zγ(x,y)µ − − − α△ − #
6
In the instanton sectors other than zero, there appear and again satisfying:
contributionsfromzeromodesoftheDiracoperator. We
againwrite the string expressionin the source form (22) ∂z µ(z;x,y)=g(δ(2)(z x) δ(2)(z y)), (30)
µK − − −
and consider (26) for k = 1. We will not give here
details of this calculation, sin±ce it is similar to what was with µ µ. Both currents µ and µ have the same
K −C J K
donein[10]and[18](seeformula(37)ofthelatter). The divergence, so we again obtain ∂z( µ µ) = 0. An
µ K −C
obvious difference now is the replacement of the current inspection of the exponent of the formula (37) in [18]
µ defined then as: shows that all ‘currents’ there become now transverse
K and the longitudinal — i.e. gauge dependent — part of
µ(z;x,y)=gγνγµ∂zD(z x) gγµγν∂zD(z y)], doesnotcontribute. Theevaluationoftheresultant
K ν − − ν − △µν
(29) expression gives:
Sinv(x y)= igeγEe−iγ5θ (31)
(1) − 4π3/2
g2
exp i dz dwµ (z w)+ig2 d2z dw (D(z y)+D(z x))ǫµα∂z (z w) .
× "− 2 Zγ(x,yµ)Zγ(x,y) △ − Z Zγ(x,y)µ − − α△ − #
TtoocfionndsidtehreailnlftrearrmedsaaspypmeaprtiontgicisnotfhSeienxvp(xo,nye)n,tswoefh(a2v8e) S(i1n)v(x−y) ∼ ige(γ4Eπe)3−/i2γ5θe−g√4π√−(x−y)2 .
and(31),whicharesimilar,althoughnotidentical. Hap-
pily, for γ(x,y) being a segment of the straight line, the As we see, both terms decrease exponentially, with the
expressions: sameeuclideancorrelationlengthequalto 4 ,whereκis
πκ
a mass of the Schwinger boson, but with different power
d2z dwµ(D(z−y)±D(z−x))ǫµα∂αz△(z−w) (32) prefactors. In QED3 such an exponential behavior was
Z Zγ(x,y) suggestedtobeuniversalandunavoidableintheinfrared,
for the gauge invariant propagators defined by the Wil-
turnouttobezero. Thismaybeprovedbyadirectcom-
sonstrings[21]. In the SchwingerModel without instan-
putation,butcanalsoeasilybeseeninthefollowingway.
tons, the same result has been obtained in [4]. We have
If we substitute wµ =(yµ xµ)t+xµ (where parameter
t [0,1]),whichmeanstha−tdwµ =(yµ xµ)dt,andshift now shown, that this phenomenon appears for k = ±1
∈ − too. Theasymptoticscaleinvarianceisthenbrokenboth
the integration variable zµ zµ +xµ, the only vector
→ by the exponential factor and by the various powers of
appearing in (32), after executing the z integration, is
(x y)2 indifferenttopologicalsectors. Oneshouldre-
yµ xµ. This vector has to saturate Lorentz indices of − −
ǫµα−, so we obtain a factor ǫµα(y x )(y x )=0. member, however,that fermions in the SM are confined,
µ− µ α− α and do not exist as asymptotic particles.
As regards the first integral in (28) and (31), it may
be expressed, for the path in question, by the Meijer’s
function G:
IV. AMPLITUDE FOR THE DRESSED
dz dwµ (z w)= (33) FERMIONS
µ
△ −
Zγ(x,y)Zγ(x,y)
iπ g2(x y)2 3 The alternative form of a gauge invariant propagator
= 2+G2,1 − 2 ,
g2 1,3 − 4π 0,1,1 is connected with the amplitude for the dressed fermion
(cid:18) (cid:18) (cid:12)(cid:12) 2 (cid:19)(cid:19) fields as suggested by Dirac [26]:
and if (x y)2 , it behaves like(cid:12)(cid:12)−i√π (x y)2.
This term−is res→po−ns∞ible for the significa2ngt cpha−nge−of the Ψ˜(x) = e−i d2zFµ(z,x)Aµ(z)Ψ(x),
character of the infrared asymptotics: from power decay
R (35)
toexponentialdecay. Thisreferstoallinstantonsectors.
Now we are in a position to write down the full form of Ψ˜(x) = ei d2zFµ(z,x)Aµ(z)Ψ(x).
the IR terms of the propagatorSinv: R
To ensure the gauge independence of the dressed field, a
S(i0n)v(x−y) ∼ ( 2ge2−(γxES0y()x2/−πy)1)/2e−g√4π√−(x−y)2 , function Fµ has to satisfy the relation:
− −
(34) ∂µ (z,x)=gδ(2)(z x). (36)
zFµ −
7
For the particular choice: where A(0)µ is certain potential restricted to the instan-
tonsectork =0andbistheexternalscalarfunction,the
(z,x)=g dw δ(2)(z w), (37) choice of which is dictated by the topology of the gauge
µ µ
F −
Zγ(x,ζ) field. In that way, the whole nonzero winding number of
Aµ may be attributed to εµν∂ b, A(0)µ being the trivial
where ζ is arbitrary (in this case the field Ψ˜(x) is gauge ν
topology field. Aµ(x) standing in the exponents of (40)
independentuptotheglobaltransformation,becausethe
reduces actually to A(0)µ(x).
divergence of has an extra term canceling, however,
Fµ Since (41) constitutes a simple shift we cannow easily
for bilinears ΨΨ), one obtains the string version of the
passin(26)fromthefunctionalintegrationoverAtothat
invariant propagator (21). Yet, we are free to consider over A(0). It is known that the coupling term gΨA(0)Ψ
other choices, as for instance: 6
may be gauged away if we introduce new fermion fields
(z,x)=g∂zD(z x). (38) defined by the relations:
Fµ µ −
The above form of the dressing is of certain importance, Ψ(x) = e−ig6∂x d2zD(x−z)γµA(µ0)(z)Ψ′(x),
sinceitiscloselyrelatedwiththeproblemofpathdepen-
R (42)
dence in (21). If one wished to construct a string, which
would simultaneously be path independent and ensuring Ψ(x) = Ψ′(x)eigγµ6∂x d2zD(x−z)A(µ0)(z) .
the gauge invariance, it should be set up of the longi-
R
tudinal part of the field Aµ only [27]. Such a definition ThistransformationisanelementofU(1) U (1)group,
A
eliminates the γ dependence because Aµ constitutes the which leaves Lagrangian(4) invariant but⊗the same can-
L
total derivative of certain scalar function, and the string not be told about fermionic measure in (26) [35, 36, 37].
integral is expressible through the values at its ends x This is a reflection of the anomaly present in the model,
and y. In this way we arrive at certain particular choice leading to the gauge boson mass generation. The other
of (z,x), for: effect of the transformation (42) is the modification of
µ
F
the source terms, which now become:
exp ig dz Aµ(z) = (39)
"− Zγ(x,yµ) L # Ψ(x)eigγ5 d2z∂µzD(z−x)ǫµνA(ν0)(z)η(x),
R (43)
=exp ig dz d2w∂µ∂zD(z w)Aν(z) =
"− Zγ(x,yµ)Z z ν − # η(x)eigγ5 d2z∂µzD(z−x)ǫµνA(ν0)(z)Ψ(x).
R
=exp ig d2w∂w(D(y w) D(x w))Aν(w) , Thealterationoftheaboveexpressionsistheonlydiffer-
ν − − −
(cid:20) Z (cid:21) ence as compared to [18]. This allows us to skip most of
and this is just, what (38) with (35) mean.2 the calculations. In the k =0 sector the effect of (43) is
to replace µ in (7) with its transverse part:
It should be noted here, that our calculation is fully J
nonperturbative and fermions are confined. For such a
µ(z;x,y) µ(z;x,y) (44)
nonperturbativeevaluationoftheamplitudewithdressed J → JT
fields (35)viaFeynmanpathintegralwe donotneedthe = gγ5ǫµν∂z[D(z x) D(z y)].
ν − − −
existence of asymptotic free fields, and the time nonlo-
cality spoken of in [27] is not an obstacle here. The whole procedure becomes then equivalent to tak-
Thesimplestwaytocalculatethepropagatorwithsuch ing µν in the Landau gauge. It turns out, that the
△
dressed fields is to replace fermionic source terms in the same happens alsofor nontrivialtopologicalsectors. For
generating functional (26) with: k = 1,thestartingpointwillbetheformula(37)in[18].
Ifwe±recallthatbothcurrents µ and µ havethe same
Ψ(x)eig d2z∂µzD(z−x)Aµ(z)η(x), divergence, i.e. the same longJitudinalKparts, we imme-
diately see that the transformations (43) turn µ into
R (40) µ: K
η(x)e−ig d2z∂µzD(z−x)Aµ(z)Ψ(x). KT
µ(z;x,y) µ(z;x,y) (45)
This allows us to appRly directly the formulas from [18] K → KT
= gγ5ǫµν∂z[D(z x)+D(z y)].
with obvious modifications only. Assume the following − ν − −
form of the gauge field:
The other term multiplying in the exponent of the
µν
△
Aµ(x)=A(0)µ(x)+εµν∂ b(x), (41) quotedformula(37)of[18]isalreadytransverse,soagain
ν
the whole story reduces to the choosing of the Landau
gauge. Weconcludethen,thatthewholepropagatorand,
naturally, its infrared asymptotics is the same as found
2 Thechoice(38)isalsominimalinguaranteeinggaugeinvariance in section IIA and, after all, the anomalous dimensions
ofthefields(35),inthissense,thatitispurelylongitudinal. found in (12) and (14) turn out to have some meaning.
8
For the choice of µ other than (38) one would have seems to be a natural result.
F
to do with the following substitutions: For the gaugeindependent function defined via Dirac-
like dressed fields, the result is dressing dependent. For
µ(z;x,y) µ(z;x,y) µ(z,x)+ µ(z,y),
the dressing chosen in section IV, which is equivalent to
J → J −F F
(46) constructing the string of the longitudinal part of the
µ(z;x,y) µ(z;x,y) µ(z,x)+ µ(z,y). gauge field only, we find the propagator to be identical
K → K −F F tothe gaugevariantonecalculatedintheLandaugauge.
Itshouldbeemphasized,thatthesenewcurrentsarestill This again refers to all instanton sectors.
transverse, although in general other than those given Undoubtedly, the principal question to be answeredis
by (44) and (45), so again only T contributes. Thus, theproperdefinitionoftheinvariantpropagator. Ingen-
△µν
the neglecting of L is universal for all fields (35), but eralthisdefinitionwillprobablybenotunique,butthere
△µν
the transverse contribution is dependent. shouldbe certainprescriptionsforthe specific cases. For
F
the‘string’propagatorthefollowingquestions,worthbe-
ing answered, arise. Firstly, it would be interesting to
V. SUMMARY AND OUTLOOK calculate propagatorfor paths γ other than the segment
of the straightline and find whether infraredbehavior is
Inthissectionwewouldliketosummarizetheobtained sensibletothatchoice. Secondly,onmightthinkabouta
results. ThesolvabilityoftheSchwingerModelallowsus systematic study of the possible connection between the
to find the infrared behavior and anomalous dimensions fixing of a gauge in the variant function and the choice
forthefermionGreen’sfunctionsinallinstantonsectors. of a specific path in the invariant one. Some work in
In the case of an ordinary, gauge dependent propagator, this direction was done in [38]. Thirdly, one might ask,
the IR anomalous dimension equals to 1/2 when we whether there are any physical reasons for the choice of
choose the Landau gauge or ξ/2 1/2 for−the R gauge. the particular path γ as more convenient than another.
ξ
−
Thevalueofthisexponentisindependentonthetopolog- As higher functions can also be found explicitly
icalsector and may be simply put to zero,if one chooses in the Schwinger Model in all topological sectors, it
ξ =1. would be interesting to construct the two-fermion (i.e.
The gauge independent propagator may be defined in four-point) Green’s function in a gauge-invariant way,
two possible ways: either by the string ansatz adopted and investigate both its infrared anomalous dimensions
from Schwinger or as an amplitude of the dressed and analytical properties. This analysis my be the
fermions suggested by Dirac. In the first case, the cor- more interesting, that this function contains a bound
relation function becomes path dependent. For the sim- state singularity, the character of which varies between
plest case of a straight line it may be found explicitly. topological sectors.
One observes then, the essential change of the infrared
asymptotics: the suppression of the fermion propagator
is no longer algebraicbut becomes exponential, with the
correlationlengthbeingproportionaltotheinverseofthe
Acknowledgments
mass of the fermionic bound state — the Schwinger bo-
son. This phenomenon appears in all instanton sectors,
differing only in algebraic prefactors. The k = 0 contri- I would like to thank to Professor J´ozef Namys lowski
butiondecaysslightlyfasterthanthatfork = 1,which for interesting discussions.
±
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