Instanton modifications of the bound state singularity in the Schwinger Model

0.19 MB·English

by  Tomasz Radozycki

Instanton modifications of the bound state singularity in the Schwinger Model
Tomasz Radoz˙ycki
Department of Mathematical Methods in Physics,
Warsaw University, Hoz˙a 74, 00-682 Warsaw, Poland∗
We consider the quark-antiquark Green’s function in the Schwinger Model with instanton con-
tributions taken into account. Thanks to the fact that this function may analytically be found,
we draw out singular terms, which arise due to the formation of the bound state in the theory —
the massive Schwinger boson. The principal term has a pole character. The residue in this pole
containscontributionsfrom variousinstantonsectors: 0,±1,±2. Itisshown,thatthenonzeroones
change the factorizability property. The formula for the residue is compared to the Bethe-Salpeter
wave function found as a field amplitude. Next, it is demonstrated, that apart from polar part,
there appears in the Green’s function also the weak branch point singularity of the logarithmic
and dilogarithmic nature. These results are not in variance with the universally adopted S-matrix
7 factorization.
0
0 PACSnumbers: 11.10.Kk,11.10.St,11.55.-m
2

n
I. INTRODUCTION lated [3, 4]. This equation is, however, extremely com-
a
J plicated. Firstly, it is a multidimensional integral equa-
tionandthesearealwaysmorechallengingthandifferen-
9 The Schwinger Model (SM) [1], describing a massless
2 fermion field in interaction with a gauge boson in two tial ones. Secondly, being an equation for the unknown
space-time dimensions, has become a very fruitful sys- bound state wave function, it requires, as an input, the
3 knowledge of the nonperturbative propagatorsfor ingre-
tem in field theory. Above all it possesses many of the
v dient particles and the interaction kernel between them.
features, one is convinced to find in the theory of strong
2
Naturally no such quantities are known in realistic field
2 interactions. Therefore, thanks to its relative simplic-
theories. Fromtheverybeginningoneisthendoomedto
2 ity, SM is expected to be a perfect testing layout for
extremely strong approximations. As to the calculations
1 the various nonperturbative aspects of QCD. They in-
0 volve confinement, topological sectors, instantons, con- in field theoretical models, there are only few examples
7 (for instance the so-called Wick-Cutkosky Model [5]),
densates. Other interesting and nontrivial features one
0 whereitispossibletofindsolutionsbuteveninthatcase
should mention, are the existence of anomaly and gauge
/ oneisforcedtosimplifytheequationeitherbyneglecting
h bosonnonzeromassgenerationwithouttheneedofintro-
t ducing an auxiliary Higgs field. Naturally some of these relative time dependence or simplifying the interaction
-
kernel together with one-particle propagators.
p properties are relatedwith one another being in fact dif-
e ferent manifestations of the same physical phenomena. The otherpossibilityininvestigatingboundstates,in-
h
Another aspect, that should be appreciated in SM, stead of solving Bethe-Salpeter equation, is to consider
:
v is the exact solvability of the equations, which allows four-point Green’s function in the appropriate channel,
Xi one to perform many calculations without unpleasant andfind the residue ofthe pole correspondingto a given
approximations and additional assumptions or simplifi- bound particle [6, 7]. This residue is constructed just
r
a cations. One example is the quark propagator(the term fromtheBethe-Salpeterwavefunctions. However,know-
‘quark’ means here the fundamental fermion, due to its ing the full four-point function is usually as unlikely as
confinement property),which has alreadybeen found by solving exactly the Bethe-Salpeter equation itself. One
Schwinger,butalsootherfunctionsstandwideopen. For should then all the more appreciate the SM, for which
instance, in our previous work [2], we were able to ob- the exact form of the Green’s function in question, was
tain the explicit formulae for the whole set of four-point found [2, 8]. This is the main reason, why we decided
Green’s functions. to look for the bound state singularity in this model, to
analyze the nature of this singularity, and to derive the
A very troublesome and long-standing question in
formulaforthe Bethe-Salpeterfunction. Thecalculation
Quantum Field Theory is the consistent description of
performed primarily in section III shows the complete
bound states. The problem becomes particularly severe
agreement with general considerations [6, 7, 9].
when such a state has a relativistic character and re-
tardation effects in the interactions among constituent Acertainnewaspectappearshowever,ifonetakesinto
particles cannot be neglected. Over half a century account that the SM is a topologically nontrivial the-
ago, the equation for the bound state wave function ory [10, 11, 12, 13, 14, 15]. As it is well known, fermion
— the so called Bethe-Salpeter equation — was formu- Green’s functions gain contributions from nonzero in-
stanton sectors. This is connected with the existence
in the theory of the (infinite) set of topological vacua,
the superposition of which constitutes the true ground
∗Electronicaddress: [email protected] state. The expectation values for field operators in this
2
newstate,becomenowsumsofmatrixelementstakenbe- residue of the factorizable part of the preceding section.
tween various topological vacua. If a given operator has
non-vanishing off-diagonal elements, then they have to
enter into the calculation,apartfromthe main, diagonal II. BASIC DEFINITIONS AND NOTATION
contributions. Distinct topological vacua correspond to
different, and topologically inequivalent, configurations The Lagrangian of the Schwinger Model, describing a
of the gauge field, and the corresponding transition be- fermion(quark)fieldΨininteractionwithagaugeboson
tween them is described by the instanton field. There Aµ, has the following form:
is a proportionality between topological index of a vac-
uum and the chiral charge and, therefore, off-diagonal (x) = Ψ(x)(iγµ∂µ gγµAµ(x))Ψ(x)
L −
elements may appear only for the chirality variant oper- 1 λ
Fµν(x)F (x) (∂ Aµ(x))2 , (1)
ators. To that category belong the products of fermion µν µ
− 4 − 2
fields and, consequently, we may expect the appearance
where λ fixes the gauge and g is the coupling constant.
of certain new terms in the Green’s functions involving
The space-time has one time and one space dimension.
quarks. The explicit calculations for the two- and four-
In this world, the most simply is to choose the quark
point functions has been done in [16], where these sup-
field in the two-component form. In this case, the 2 2
plementary terms were explicitly given.
×
Diracgammamatricesmayhavetherepresentationgiven
Now the question arises if, and how, these contribu-
below:
tions modify the behaviour of the two-fermion Green’s
function close to the bound state singularity. To this 0 1 0 1
γ0 = , γ1 = − , (2)
questionwedevotesectionIVA(oneinstantoncontribu- (cid:18)1 0 (cid:19) (cid:18)1 0(cid:19)
tions) and section IVB (two instantons contributions).
Our results indicate two modifications. Firstly, we see
that the residue in the polar term acquires new ingredi- γ5 =γ0γ1 = 1 0 .
ents: theBethe-Salpeterwavefunctionismodified—and (cid:18)0 1(cid:19)
−
thisisquiteobvious,sinceitisdefinedthroughtheprod-
For the metric tensor we choose the convention:
uctofthetwoquarkfields,whichissensitivetoinstanton
background—andtheothertermappears,whichmakes g00 = g11 =1,
the residue nonfactorizable. Yet, it may still be written −
as a sum of several factorizable terms. Secondly, apart and the antisymmetric symbol εµν is defined by:
fromthepole,wefindterms,whichshowanothertypeof
singularity—branchpointsingularityofthelogarithmic ε01 = ε10 =1, ε00 =ε11 =0. (3)
−
and dilogarithmic nature. These contributions are small
All the Green’s functions may be obtained by differenti-
in comparison with the principal term, and in the usual
atingoverexternalcurrentsofthe generatingfunctional,
bound state pole derivations [6, 7], are neglected by the
which is, as usually, defined by the Feynman path inte-
proof construction itself. In the SM, we happily dispose
gral:
the full Green’s function without being obliged to make
any addition assumptions or simplifications, which usu-
allyremainbeyondourcontrol,andthoseterms survive. Z[η,η,J]= Ψ Ψ Aei d2x[L+ηΨ+Ψη+JµAµ] , (4)
Z D D D R
On the other hand, the calculated instanton con-
tributions disappear for transition amplitudes between Aswehavealreadywritteninthe Introduction,this the-
asymptotic states and to that extent our results are ory is topologically nontrivial. The true vacuum, which
in agreement with the analyticity properties of the S- is usually called the θ-vacuum, becomes a superposition
matrix [6, 17, 18, 19, 20, 21, 22]. For the confined parti- of the infinite set of degenerate vacua, bearing different
cles one cannot require the cluster decomposition on the topological indices n [23, 24]:
level of the Green’s function. The main argument refer-
∞
ring to the possible separation of certain subgroups of θ >= einθ n> . (5)
particles into the distant space-time regions, may be ap- | |
n=X−∞
plied only for hadrons, but neither for individual quarks
nor even qq structures, if they do not bear the complete Inthisnewvacuum,thegeneratingfunctional(4),which
set of quantum numbers of a meson. It is then not sur- constitutes the Feynman form of < θ θ > , breaks
η,η,J
|
prising, that the quark Green’s function reveals much into pieces:
richer analytical structure, than the hadronic S-matrix
∞
does.
Z[η,η,J]= eikθZ(k)[η,η,J]. (6)
Finally, in section V we calculate directly the Bethe-
k=X−∞
Salpeter wave function as a qq field amplitude between
one-particlestateofmomentumP andthevacuumstate, Inthisformula—contraryto(5)—thesummationruns
and find that it corresponds to that obtained from the over instanton number, i.e. k corresponds to the change
3
of the topologicalindices in the two vacua in the expres- this sumisnowlimited onlyto the lowvaluesofk. Con-
sion: < θ θ > . Henceforth we will use the com- trary to the expression exp i d2x ηΨ+Ψη , which
η,η,J
|
monsymbol 0>,for denoting the true vacuum. Ineach contains arbitrary high powe(cid:8)rsRof Ψ(cid:2)and Ψ an(cid:3)(cid:9)d, there-
of Z(k)’s the|functional integration with respect to the fore, contributes to all sectors in (6), the four-fermion
gaugefield Aµ, is restrictedto a specific class ofconfigu- function gets terms with at most k = 2. This is due to
rations bearing the Pontryaginindex k. the fact, thatfour fermionfields canch±angethe chirality
NowtheqqGreen’sfunctionmayberepresentedasthe of the state by (at most) 4, and the topological vacuum
appropriate functional derivative of Z: index is a half of the chiral charge [23]. In the following
sections we investigate in detail all possible terms.
G (x ,x ;x ,x )= (7)
ab,cd 1 2 3 4
=<0T(Ψ (x )Ψ (x )Ψ (x )Ψ (x ))0>=
a 1 b 2 c 3 d 4
| |
δ4Z III. QUARK-ANTIQUARK BOUND STATE
=N ,
δη (x )δη (x )δη (x )δη (x )(cid:12) WITHOUT TOPOLOGICAL CONTRIBUTIONS
a 1 b 2 c 3 d 4 (cid:12)η,η,J=0
(cid:12)
(cid:12)
where N is a constant ensuring that < 00 > = 1. In this section we will concentrate on the 0-instanton
η,η,J=0
|
In the following, we are only interested in singularities term of the function G (x ,x ;x ,x ). It is a great
ab,cd 1 2 3 4
that may be associated with bound states, and hence, merit of the Schwinger Model, that this function may
throughoutthis work,byGwe understandonly the con- be found exactly, in an explicit and analytic way. The
nected part of this function. details of our calculation have already been given in [2]
Due to the fact, that Z expands in the sum of Z(k)’s, andwewillnotelaborateonthemhere,butinsteadrecall
the same refers also to the Green’s function. However, only the final result:
G(0) (x ,x ;x ,x )=
ab;cd 1 2 3 4
1
= S (x x )S (x x )+[S(x x )γ5] [S(x x )γ5] exp ig2[β(x x ) β(x x )
ac 1 3 bd 2 4 1 3 ac 2 4 bd 1 2 1 4
2{ − − − − } { − − −
1
β(x x )+β(x x )] + S (x x )S (x x ) [S(x x )γ5] [S(x x )γ5]
2 3 3 4 ac 1 3 bd 2 4 1 3 ac 2 4 bd
− − − } 2{ − − − − − }
x x
×exp{−ig2[β(x1−x2)−β(x1−x4)−β(x2−x3)+β(x3−x4))]}−(cid:26) c3 ↔ d4 (cid:27) , (8)
↔
where the index (0) refers to the instantonnumber. The mustlater getcontributionsfromthe nontrivialtopolog-
functionβ(z)isdefinedthroughthetwo-dimensionalmo- ical configurations [16].
mentum integral:
Now we will concentrate on the structure of the qq
d2p 1 eipz function (8) and try to reveal, whether it has a pole in
β(z)= − , (9) the t-channel. To this goal we introduce new space-time
Z (2π)2(p2 g2(cid:0)/π+iǫ)((cid:1)p2+iǫ)
− variables: X, Y, x and y, defined in the following way:
and may be expressed in terms of MacDonald or Hankel
function, depending on whether z is space- or time-like,
1
but the above representationis more suitable for us. We
X = (x +x ), x=x x ,
1 3 1 3
recallthatin2Dthecouplingconstantg isdimensionfull 2 −
and the quantity µ2 = g2/π plays a role of the mass 1
Y = (x +x ), y =x x . (11)
2 4 2 4
squared of the so called Schwinger boson – a qq bound 2 −
state we are interested in.
The object S(x) in (8) is the full quark propagator,
Thanks to the translational invariance, which manifests
which has the known form:
itself through the dependence of G on x x ’s only, we
i j
−
S(x)= (x)exp ig2β(x) , (10) have three independent variables: Z = Y X, x and
S0 − −
(cid:2) (cid:3) y. From the four terms present in (8), the two explic-
with (x) = 1 6x being the free propagator. The itly written are of interest for us, since the two other,
S0 −2πx2−iε
term‘full’usedabove,meanshere‘fullinthe0-instanton obtained by the antisymmetrization, contribute to the
sector’, i.e. it is given in the form of the original singularities not in t, but eventually in the u-channel.
Schwinger works [1]. However S – bilinear in Ψ’s — First of the two interesting terms, denoted by I, takes
4
the form: eventual singularity is cancelled by the sine func-
tions present in the numerator in (16), either by
1
G(0)I(Z;x,y)= S(x) S(y)+[S(x)γ5] [S(y)γ5] that explicitly written or by the one hidden in the
2{ ⊗ ⊗ }
formula for I . If both parameters are nonzero, on
exp ig2[β( Z +(x y)/2) β( Z +(x+y)/2)) 1 2
× { − − − − can easily obtain q2 = α+β p2 =0. Thus, there
β( Z (x+y)/2))+β( Z (x y)/2))] , (12) β
− − − − − − } may only arise a contri(cid:16)butio(cid:17)n to the singularity at
where we made use of the evenness of β: β(x) =β( x). q2 = 0. It cannot, however, be worse than that in
−
For brevity we omitted the spinor indices, and used the I .
1
tensorproductnotation,whichshouldnotcauseanycon-
2. α(p2 µ2)=0andβ(p q)2 =0. Ifα=0,wehave
fusion. Thepolecorrespondingtotheboundstateshould − −
q =p and, ingeneral,one mightgeta contribution
befoundinthecomplexplaneofP2,whereP isthetwo-
to the pole at q2 = µ2 (actually it would become
momentum canonically conjugated to Z. The matrical
a branchpoint), but it does nothappen because of
partdependsonlyontherelativecoordinatesxandyand
the sine functions becoming zero for q p = 0. If
entersintotheresidueofthispole,butdoesnotinfluence −
β = 0, we obtain p = 0, which stays in contradic-
its position. Let us then perform the Fourier transform
tion with the assumption p2 = µ2. For α,β = 0,
over Z of the exponent containing β functions. Making we find α2p2 = β2(p q)2, which is incompa6tible
use of (9) we my write the quantity in question as: −
with p2 =µ2 and (p q)2 =0.
−
FI(P,x,y)= d2ZeiPZe−4ig2 (2dπ2)p2e−ipZI1(p,x,y) , 3. αp2 = 0 and β((p q)2 µ2) = 0. This case is
− −
Z R almost identical with the previous one and need
(13)
not be separately considered.
where I (p,x,y) = sin(px/2)sin(py/2). Expanding the ex-
1 (p2−µ2+iǫ)(p2+iǫ) 4. α(p2 µ2)=0andβ[(p q)2 µ2]=0. Ifoneofthe
ponent in series, we can execute the Z integration ob- − − −
parameters is zero, then we find a contradiction of
taining:
αp+β(p q)=0,witheitherp2 =µ2 or(p q)2 =
FI(P,x,y)= ∞ (−4ig2)nI (P,x,y)+(2π)2δ(2)(P), wµ2e.haWvehaα−t2pr2em=aβin2s(pis tqo)2asasnudmaeftαer,βca6=nce0l.l−inIgftshoe,
n
nX=1 n! arisingfactorsµ2onb−othsides,wegetα=β. This
(14) leads to q = 2p and a singularity emerges now at
The In’s satisfy the following recurrent relation: q2 =4µ2. This is the two-particle singularity.
In (q,x,y)= (15) Summarizing,weobservethatinI2 thepoleatµ2 disap-
pears and a new singularity arises at (2µ)2. Now we are
d2p sin(px/2)sin(py/2)
= I (q p,x,y). in a position to carry out a proof by induction: the as-
Z (2π)2(p2−µ2+iǫ)(p2+iǫ) n−1 − sumedsingularitiesundertheintegralin(15)are: p2 =0,
It is obvious that I1(p,x,y) has a pole at p2 = µ2. The p2 =µ2,(p−q)2 =0and(p−q)2 =((n−1)µ)2. Wehave
other singularity is at p2 = 0. But what about terms of again four possibilities, the analysis of which is identical
higher n? Do they alter the behaviour at p2 = µ2? Let as above, except the last point, where we now get the
equation α2 = (n 1)2β2. This leads to q = np and
us firstconstructI2(p,x,y). Accordingto (15) ithasthe q2 =n2µ2. Hence,−the conclusion is, that the singularity
form:
corresponding to n intermediate particles (q2 =n2µ2) is
I2 (q,x,y)= (16) present only in In. This means, that the pole at µ2 may
be simply picked up from the first term of the expan-
d2p sin(px/2)sin(py/2)
= I (q p,x,y). sion (14).
Z (2π)2(p2 µ2+iǫ)(p2+iǫ) 1 −
− Now we need to apply the same to the second expres-
Thepossiblesingularitiesinq2,exhibitedinI ,cancome sion in the Green’s function (8). Again we take out the
2
exponent and perform the Fourier transform, the result-
from the singularities of the integrand functions. There
are four of them: obliviously p2 = 0, p2 = µ2, and the ing expression being called FII(P,x,y). Happily both
othertwooriginatingfromI . Theseare: (p q)2 =0and exponents differ only by the sign, so we are able to im-
1
(p q)2 = µ2. It is not possible to satisfy −all these four mediately write down the final result. The total contri-
− bution (i.e. I+II) to the bound state pole in G(0) is:
or even three conditions simultaneously, so the singular
points may appearonly asa resultofthe mergingoftwo G(0)I(P;x,y)= 4iπ S(x)γ5 S(y)γ5
singularities. In accordance with Landau procedure [6, b.s. − ⊗
sin((cid:2)Px/2)si(cid:3)n(P(cid:2)y/2) (cid:3)
25,26],weintroducetwoparameters(αandβ)satisfying , (17)
α,β 0 and α2 +β2 = 0. With the assumption αp+ × P2 µ2+iǫ
≥ 6 −
β(p q)=0, the following possibilities occur: We see the perfect factorization of the residue, and the
−
quantity:
1. αp2 =0andβ(q p)2 =0. Ifoneoftheparameters
equalszero,then−we getp=0 or p q =0 andthe Φ (x)= 2√πS(x)γ5sin(Px/2), (18)
P
− −
5
plays a role of the Bethe-Salpeter amplitude [3, 4, 7] or dom, corresponding to the transition amplitude in ques-
the wave function. The minus sign is chosen for later tion, is then zero and the tunnelling phenomenon disap-
convenience. Note, that this quantity is found explicitly pears. This does not mean, however, that the notion of
and exactly, without any simplification referring neither θ-vacuum is useless, since topologicalvacua (contrary to
to the relative time x0 nor to the structure on the in- the θ-vacuum) violate cluster property [30]. We are the
teraction kernel. It is an essential point, since even in fated to use (5) and must put up with modification in
the simplified model theories, one usually is limited to the fermion Green’s functions
the case of the so called ‘equal time wave function’ (i.e.
The contributions to the four-point function were al-
defined only on the hypersurface x0 = 0 in the center
ready found in our previous work [16]. Below we will
of mass frame), which neglects retardation effects and
concentrate on the eventual bound state pole in the for-
consequently leads to the nonrelativistic bound state, or
mulae.
ladder approximation for the interaction kernel. On the
otherhand,thefunction(18)isafullyrelativisticobject.
A. 1-instanton sector
IV. TOPOLOGICAL CONTRIBUTIONS TO
THE QUARK-ANTIQUARK BOUND STATE
By virtue on the Atiyah-Singer index theorem [24, 27,
Asithasbeenmentioned,thevacuumintheSchwinger 28, 29, 31], the massless Dirac operator in the external
Model is nontrivial: it is a superposition (5) of vacua fieldbearingnonzeroinstantonnumberkhaszeromodes.
correspondingto varioustopologicalnumbers. As a con- In the present section we consider the case of k = 1,
±
sequence, all chirality changing operators— and to that for which there is only one such mode. Its detailed form
classthe productoffermionfields belongs—whichhave may be found in the literature [10, 13, 32]) and we will
nonvanishingmatrix elements betweendifferent topolog- not recall it here. This mode gives now its contribution
ical vacua, acquire contributions from nonzero instanton while calculating the Feynman path integral (4). Since
sectors. We recallhere, thatif the theorycontainsmass- it corresponds to the zero eigenvalue, it disappears from
less fermions (at least one) — and this is just the case the quadratic part of the Lagrangian and remains only
of the Schwinger Model — the amplitude of the spon- in source terms. Due to the Grassman character of the
taneous tunnelling between different topological vacua quark fields, of the four differentiations in (7), two must
vanishes[23,27]. Mathematicallyitisexpressedthrough be saturated by sources multiplying zero modes. The
zero value of the euclidean Dirac operator determinant: other two are performed in a common way. The result
det[i∂ eA],becauseofthe appearanceofzeroeigenval- forthefour-pointfunction,withbothk=+1andk= 1
uesw6he−nA6µbearstheinstantonindex[24,27,28,29,31]. instanton numbers taken into account, obtained in [1−6],
The functional integral over fermionic degrees of free- is:
ig
G(1)(x1,x2;x3,x4)= −8π3/2eγE(cosθ−iγ5sinθ)⊗S0(x2−x4) (11⊗11−γ5⊗γ5)
(cid:2)
eig2[β(x1−x3)−β(x2−x4)−β(x1−x4)+β(x1−x2)+β(x2−x3)−β(x3−x4)]+(11 11+γ5 γ5) (19)
× ⊗ ⊗
eig2[β(x1−x3)−β(x2−x4)+β(x1−x4)−β(x1−x2)−β(x2−x3)+β(x3−x4)] + antisymmetrization.
×
(cid:3)
The antisymmetrization runs here both over ‘initial’, again contributes to the pole in the t-channel, the other
i.e. x and x and ‘final’ i.e. x and x (accompanied twoonlyintheu-channelandwillbe omitted. Introduc-
1 2 3 4
by the appropriate spinor indices change) variables in ing center of mass and relative variables (11), similarly
G(x ,x ;x ,x ). Thus(19)getsthreemoreterms,which as it was done in section III, we obtain for the first term
1 2 3 4
are not explicitly written here for brevity. One of them in (19):
(with the simultaneous substitution 1 2 and 3 4)
↔ ↔
ig
G(1)Ia(Z;x,y)= eγE (cosθ iγ5sinθ)eig2β(x) (y) (11 11 γ5 γ5) (20)
−8π3/2 − ⊗S ⊗ − ⊗
(cid:2) (cid:3)
exp ig2[ β( Z+(x+y)/2)+β( Z+(x y)/2))+β( Z (x+y)/2)) β( Z (x y)/2))] ,
× { − − − − − − − − − − }
6
The last exponent in the above, per analogiam to (13), the exponent has only the sign invertedand need not be
may now be written in the form: elaboratedon. Their joint effect is then simply to cancel
11 11 in (19) and to double γ5 γ5. Their contribution
e4g2 (2dπ2)p2e−ipZJ1(p,x,y) , (21) to⊗the bound state is therefore:⊗
R
where J1(p,x,y) = (cpo2s(−pµx2/+2)iǫsi)n(p(p2y+/i2ǫ)). Now we might in- G(b1.s).I(P;x,y)= iµceoγEs((cid:2)Peixg/2β2()xs)ien−(iPγ5yθ/γ25)(cid:3)⊗(cid:2)S(y)γ5(cid:3)
troduce J ’s and consider the recurrent relation similar
n , (22)
to (15), but it is unnecessary. Due to the presence of × P2 µ2+iǫ
−
thesinefunctioninthenumerator,wecandirectlyapply
theanalysisofsectionIIIandcometotheresultthatthe where,insteadofcosθ iγ5sinθ,wehavewrittene−iγ5θ.
−
contributiontotheboundstatecomesonlyfromthefirst The antisymmetrized term, which is left so far, has an
termoftheseriesfortheexponent(21). Astothesecond analogous form, but with x and y swopped, and tensor
term (in our nomenclature it would be called Ib) in (19) products inverted. Our final result in this sector is:
G(1)(P;x,y)= iµeγE eig2β(x)e−iγ5θγ5 S(y)γ5
b.s. P2 µ2+iǫ ⊗
− (cid:8)(cid:2) (cid:3) (cid:2) (cid:3)
cos(Px/2)sin(Py/2)+ S(x)γ5 eig2β(y)e−iγ5θγ5 sin(Px/2)cos(Py/2) . (23)
× ⊗
(cid:2) (cid:3) (cid:2) (cid:3) (cid:9)
Thankstothequarkfieldbeingmassless,whichresultsin As we shall see in the following section, this is actually
thechiralinvarianceoftheinitialLagrangian,thedepen- the case, but it contributes also other terms that modify
dence on the parameter θ may be gauged away, so one the factorization and even change the character of the
might simply put θ = 0. Comparing the two obtained singularity at P2 =µ2.
equations (17) and (23) we see that introducing a new
Bethe-Salpeter amplitude in the form:
Φ (x)= 2√πS(x)γ5sin(Px/2) (24)
P
−
µ
+ eγEeig2β(x)e−iγ5θγ5cos(Px/2),
2√π
the residue factorization in the pole will be restored i.e.
one will have:
B. 2-instanton sector
Φ (x) Φ (y)
P P
G (P;x,y) ⊗ (25)
b.s. ∼ P2 µ2+iǫ
−
In the 2-instanton sector there are two zero modes of
if the 2-instanton sector contributes to the bound state
thetheDiracoperator[10,13,27,28,32]. Nowallofthe
a term (with the appropriate constant coefficient):
differentiationsin(7)aresaturatedwithsourcesstanding
with zero modes. The 2-instanton contribution to the
[eig2β(x)e−iγ5θγ5] [eig2β(y)e−iγ5θγ5] Green’s function, which was evaluated in [16], has the
⊗
cos(Px/2)cos(Py/2). form:
×
g4
G(2)(x1,x2;x3,x4)=−256π4e4γE e2iθ 11−γ5 ⊗ 11−γ5 (x02+x12) (26)
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)
×(−x04+x14)+e−2iθ 11+γ5 ⊗ 11+γ5 (−x02+x12)(x04+x14) exp ig2 β(x1−x4)
+β(x x )+β(x (cid:0)x )+(cid:1)β(x(cid:0) x )+(cid:1) β(x x )+β(x (cid:3)x ) (cid:8)+(cid:2)antisymmetrization.
2 3 1 2 3 4 1 3 2 4
− − − − −
(cid:3)(cid:9)
Using again the variables Z, x and y, performing the required antisymmetrization, and replacing β functions with
integral formula (9), we obtain, with the additional use of the identity e−iθγ5(11 γ5)=e∓iθ(11 γ5):
± ±
7
µ4 1
G(2)(Z;x,y)= e4γE (eig2β(x)e−iγ5θ) (eig2β(y)e−iγ5θ) (Z2 (x y)2)γ5 γ5
−64π2 ⊗ × − 4 − ⊗
(cid:2) (cid:3) (cid:2)
1 1
+ (Z2 (x y)2) (11 γ5) (11+γ5)+(11+γ5) (11 γ5) (27)
2 − 4 − − ⊗ ⊗ −
(cid:0) (cid:1)
1 d2p 1 e−ipZcos(px/2)cos(py/2)
−2εαβZα(x−y)β(11⊗γ5+γ5⊗11) exp 4πiµ2Z (2π)2 −(p2+iǫ)(p2 µ2+iǫ) ,
(cid:3) (cid:2) − (cid:3)
with εαβ being the antisymmetric symbol defined in (3). This is exactly what we needed for (25). This is not,
Now we will show that the first term (i.e. Z2 times however,thewholestory. Thesimilarcontributioncomes
γ5 γ5) gives just the contribution needed to restore from the second matrical term in (27), which also con-
the⊗factorization (25). tains Z2. Since we have already saturated (25) this sup-
One could suspect that there might be some cancel- plementary part violates the factorization. It has the
lations, among various expressions in the formula (27). form:
Below we shall argue that the cancellation that would
iµ2
lead to complete disappearance of any term, is not pos- − e2γE(eig2β(x)e−iθγ5 eig2β(y)e−iθγ5)
8π ⊗
sible due to their independent matrical structure. Only
cancellationsamongaspecificmatrixelements—butnot [(11 γ5) (11+γ5)+(11+γ5) (11 γ5)]
× − ⊗ ⊗ −
matrices as a whole — are admissible. Consider namely cos(Px/2)cos(Py/2)
. (31)
the expression: × P2 µ2+iǫ
−
aγ5 γ5 + b[(1 γ5) (1+γ5)+(1+γ5) (1 γ5)] Due to the above mentioned lack of cancellations among
⊗ − ⊗ ⊗ −
+ c(11 γ5+γ5 11) (28) independent matrix structures, this term must survive.
⊗ ⊗ Moreover it cannot cancel with the (x y)2 term with
andmultiplyitfirstlyby(11+γ5) (11+γ5)andsecondly −
⊗ the same matrical coefficient since, contrary to (31), the
by(11 γ5) (11 γ5). Weobtain(a+2c)(11+γ5) (11+
γ5)+b−0an⊗d(a−2c)(11 γ5) (11 γ5)+b 0respec⊗tively. latter disappears for x = y. Besides, it does not con-
· − − ⊗ − · tribute to the bound state pole at all. This can be
On the other hand if we take the double trace (i.e over
easily argued as follows. After rearrangements identi-
firstmatrixandoversecondmatrixinthetensorproduct calto (29)one getsthe additional1/Z2 comingfromthe
indices), we get a 0+8 b+0 c. This shows that the
· · · logarithm in the exponent. This can be neutralized, if
three tensors listed in (28) are really independent.
we act with the two-dimensionald’Alambert operator in
Letusobservenow,thatthequantityinthelastexpo-
variable P. When this is done, we obtain a first order
nent of (27), after rearranging the terms under the inte- pole at P2 = µ2. Now the question is: given a func-
gral(the integrandfunction willbe calledbelowM) and tion f satisfying: ∂2f(P) 1 . Does f also dis-
partially executing it, may be rewritten in the following P ∼ P2−µ2
play a pole? The answer is negative. It may easily be
way:
found that f(P) Li (P2/µ2) and dilogarithm has a
2
d2p limit π2/6 when i∼ts argument goes to unity. Therefore
4πiµ2 M(Z,p,x,y)= 2γ ln( µ2Z2/4)
Z (2π)2 − E− − it may be neglected in comparison with the polar term.
One should emphasize, however, that although diloga-
d2p 1
4πi e−ipZ (29) rithmic function has a well defined limit, it is not an an-
− Z (2π)2 p2 µ2+iǫ
− alyticfunction inthis point. Itisa branchpointandthe
d2p 1 cos(px/2)cos(py/2) function has a cut on the half line [µ2, [. The presence
+4πiµ2 e−ipZ − , ∞
Z (2π)2 (p2+iǫ)(p2 µ2+iǫ) ofcos(Px/2)cos(Py/2)does not changethe conclusions,
−
since cosine is an analytic function, and may only make
If we expand, as before, the exponent of the last two
things better. On the other hand for x = y = 0 cosines
terms and take the Fourier transform over Z, it be-
can be replaced with unities and a branch point arises.
comes obvious that both of them contribute to the pole
Such terms are usually lost in the common approach,
at P2 = µ2. The further analysis is again identical to
where one fishes out only the polar contribution of the
that of Sec. III, except for the difference, that now there
bound state.
is additional cancellation between the integrals in (29).
ForthetermproportionaltoεαβZ (x y) ,similarar-
Finally all the factors combine together in (27) and we α − β
gumentationis applicable, with a difference that we now
obtain the expected term:
have to do with logarithmic function, which again may
−iµ2e2γE(eig2β(x)e−iθγ5γ5) (eig2β(y)e−iθγ5γ5) beneglectedifcomparedtothepolarterm. Hence,aswe
4π ⊗ see, due to the topological background,not only nonfac-
cos(Px/2)cos(Py/2) torization appears, but the character of the singularity
. (30)
× P2 µ2+iǫ in P2 =µ2 is slightly changed.
−
8
Theseresultsdonot,however,contradicttheS-matrix we now have to do with an object billinear in quark
pole factorization [17, 18, 19, 20, 21]. The S-matrix de- fields). We now pass to the CM and relative coordi-
scribesthetransitionamplitudesamongphysicalasymp- nates: x = X + x/2 and x = X x/2, and intro-
1 2
−
totic states. In the Schwinger Model these are the Fock duce a wave function by separating the uniform motion
states of Schwingerbosons. Consequently we are limited of the bound particle as a whole, from the internal mo-
onlytothecurrent-currentGreen’sfunctionsandthisre- tion: Φ (x)=eiPXχ(x ,x ). Finally we obtain:
P 1 2
quire putting x,y 0, and taking the trace over spinor
indices with γµ γ→ν matrices. Since the trace of an odd ΦP(x)= 2√πS(x)γ5sin(Px/2)
number of γ’s is⊗zero, it is easy to observe that the only −+ µ eγEeig2β(x)e−iθγ5γ5cos(Px/2).(37)
contributiontothepolecomesnowfrom(17)andfactor- 2√π
ization reappears [10]. This is what should be expected
since the vector current does not couple to instantons. This amplitude wave function entirely agrees with that
Summarizing,thatpartofthefullqq Green’sfunction, found in (24) from the factorizable part of the bound
in which one is limited to the polar terms, has the form: state residue.
1
iG (P;x,y)= [Φ (x) Φ (y) (32)
b.s. P2 µ2+iǫ P ⊗ P VI. SUMMARY
−
+Ψ(−)(x) Ψ(+)(y)+Ψ(+)(x) Ψ(−)(y)],
P ⊗ P P ⊗ P
In conclusion we would like to recapitulate our main
with results. We have found analytically the polar contribu-
µ tiontotheqq Green’sfunction. Inthetopologicallytriv-
Ψ(±)(x)= eγEeig2β(x)e−iθγ5(11 γ5), (33)
P 2√2π ± ialcase,theresidueturnsouttobe factorizable,whichis
commonlytakenforgranted. Outofthisweobtainedthe
and Φ (x) given by (24). formulafortheBethe-Salpeterwavefunction. Thesitua-
P
tionturnedouttobedifferent,ifoneconsidersoneortwo
instanton background. We have obtained expression for
V. WAVE FUNCTION AS A FIELD the residue in this case and found, that the BS function
AMPLITUDE has been modified by the θ-vacuum structure. The full
polar term may now be written as a sum of several fac-
The question arises, how the obtained results harmo- torizableterms. Oneseesthat,apartfromthe formation
nizewiththeBethe-Salpeterwavefunctiondefinedasan oftheboundstate(Schwingerboson),thereappearsalso
amplitude of fields. Fortunately in the Schwinger Model another way for a pair qq, not bearing meson quantum
wehavealltools,whichallowustofinditexplicitlyeven numbers, to travel over the space. They may jump from
inthenontrivialtopologicalsectors. Letusthenconsider one instanton to the other. This phenomenon is possible
the amplitude in question: in there are at least two instantons.
Moreover if one takes into account also these compo-
χP(x1,x2)=<0T(Ψ(x1)Ψ(x2)P > . (34) nents, that are usually neglected for P2 µ2 as being
| | →
smallin comparisonwiththatrevealinga pole,one finds
Thecalculationofthisquantityturnsouttobeverysim-
another type of singularity — a branch point connected
ple,ifonealreadyknowstheexpressionforthefullquark
with the appearance of logarithmic and dilogarithmic
propagator in the instanton background[16]:
functions. They arise because of long-range correlations
in the condensed vacuum.
ie
Sfull(x)=S(x)+ 4π3/2 cosθ−iγ5sinθ eγE+ie2β(x) . Alltheseinstantoncontributionsvanish,ifweconsider
(cid:0) (cid:1) (35) theS-matrixelements,i.e. onlytransitionamplitudesfor
the‘physical’asymptoticstates. These,intheSchwinger
The state P > is the one Schwinger boson state of two-
momentum| P, and (34) is expressible,thanks to the for- Model, are Fock states of one or more bosons. The four-
point(quark)function is thenreplacedwith the current-
mulaeoftheLSZformalism,throughthevertexfunction.
current function, for which it is shown, that simple pole
It is known that, thanks to the two gauge symmetries
structureaswellasfactorizationofthe residuereappear.
(U(1) and U (1)), the vertex function may in turn en-
A
TheanalyticalstructureoftheGreen’sfunctionmaythen
tirely be constructedfromthe quark propagator. This is
bemoreintricatethanthatoftheS-matrix,especiallyfor
a miracle of the Schwinger Model. One then gets:
topologically nontrivial theory.
χ (x ,x )= i√π e−iPx1γ5S(x x )
P 1 2 1 2
−
+e−(cid:2)iPx2S(x x )γ5 . (36)
1− 2 Acknowledgments
(cid:3)
If we now substitute for the quark propagator the ex-
pression S , we obtain the formula for the amplitude I would like to thank to ProfessorsJ´ozef Namys lowski
full
χ inallinstantonsectorsinquestion: 0, 1(recallthat and Krzysztof Meissner for elucidating discussions.
P
±
9
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