Infrared self-consistent solutions of bispinor QED3

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by  Tomasz Radozycki

Infrared self-consistent solutions of bispinor QED
3
∗
Tomasz Radoz˙ycki
Faculty of Mathematics and Natural Sciences, College of Sciences,
Cardinal Stefan Wyszyn´ski University, Wo´ycickiego 1/3, 01-938 Warsaw, Poland
Quantum electrodynamics in three dimensions in the bispinor formulation is considered. It
is shown that the Dyson-Schwinger equations for fermion and boson propagators may be self-
consistently solved in the infrared domain if on uses the Salam’s vertex function. The parameters
definingthebehavior of thepropagators are found numerically for different valuesof coupling con-
stant andgauge parameter. For weak coupling theapproximated analytical solutions are obtained.
The renormalized gauge boson propagator (transverse part) is shown in the infrared domain to be
practically gauge independent.
3 PACSnumbers: 11.10.Kk,11.15.Tk
1
0
2 I. INTRODUCTION forthevertexfunction,whichischosenmostofteninthe
form satisfying Ward-Takahashi identity. Naturally this
n
a identity fixes only longitudinalpartofthe vertexleaving
QED in three space-time dimensions has become a
J the transverse part a subject of further discussion and
testing laboratory for certain nonperturbative aspects
5 improvements [15, 23–27].
of QFT like chiral symmetry breaking [1–10], bound
2 In our previous paper [28] we applied a method elabo-
] satpaptreosx[i1m1a–t1i3o]nosrchceomnfiesn.emDuenetto[8t,h1e4d–i1m8e]nasniodnfaolrityvaorfiothues ratedearlierinQED4 [29]toQED3 withtwo-component
h fermions. It consists on the following five steps:
couplingconstant(as√mass)itisasuperrenormalizable
t
- theorysoitisfreeofinfiniterenormalizationambiguities 1. a certain infrared form of two basic propagators,
p
e typical for four-dimensional theory. S(p)andDµν(k)(suggestedbyperturbativecalcu-
h Itcanbeformulatedintwoinequivalentversions: with lationsorothermethods)asdependentonacouple
[ two- and four-component fermions [19, 20]. The proper- of unknown parameters is assumed,

ties of the theory are different in these two cases. To
1
v the investigation of both versions much attention has 2. the fermion propagator is represented in the spec-
5 been payed over the last twenty years. The work has tral form with one known spectral density ρ(M),
7 been especially concentrated on nonperturbative solu- whichisingeneralpossibleinthe infrareddomain,
1 tions of Dyson-Schwinger (DS) equations with different
6 3. Salam’s form of the vertex function [30], together
approximations incorporated in the theory as quenched
. with S(p) and Dµν(k), is substituted into the first
1 approximation, rainbow approximation, 1/N expansion
0 and various models of vertex function [1–4, 6, 10]. Par- two of the set of DS equations,
3
ticularly often the multi-flavor theory with the limit
1 4. from these two equations the set of self-consistent
N has been used, since it avoids infrared prob-
: → ∞ equations for parameters is derived,
v lems, which usually become troublesome in lower num-
i ber of dimensions. The other important point in this
X 5. the obtained equations are solved numerically or
analysis has been the unpleasant gauge dependence of
r analytically.
a the nonperturbative results which constitute the com-
mon problem of the approximated studies based on DS This method proved to be relatively effective both in
equations [6, 9, 10, 16, 21, 22]. QED , where all parameters were correctly found with-
4
As it is well known, DS equations constitute an in- outthenecessityofinfiniterenormalization,andinspinor
finite set of relations involving Green’s functions which version of QED3, where obtained results stay in general
form the ‘inverted ladder’ type structure: the n-point agreement with other works. In the present paper we
functions depend on n+1-point ones and so on up to would like to extend its application to four-component
infinity. This set cannot be solved without the trunca- QED3, without Chern-Simons term, i.e. when gauge
tionofsuchahierarchy. Butevensuchtruncation,which bosons (‘photons’) remain massless in spite of interac-
turns the infinite set of equations into only a couple of tion.
them, leads to the system which is far from being trivial The common effect of various simplifications of DS
andrequiresfurther simplifications. Usually this trunca- equations is the gauge dependence of the results (even
tion is accomplished by performing certain assumption ofthe physicalobservables). The sourcesofthis undesir-
able behavior are the approximations made to the prop-
agators and to the vertex. It seems therefore valuable
to test various possible approaches with regard to that
∗Electronicaddress: [email protected] particularfeature andmuch work has alreadybeen done
2
in this direction (see the references above). This point where we choose for the metric tensor:
lies in the scope of interest of the present work too.
This paper is organizedas follows. In the next section g00 = g11 = g22 =1. (6)
− −
wedefinethemodelitselfandgivetheresultingsetofDS
The trace of the product of an odd number of gamma
equations. In sectionIII the infraredGreen’s function in
matrices equals zero as in four dimensions (this was not
question are formulated up to several unknown parame-
the case in the spinor representation, where it was pro-
ters. In section IV we substitute these Green’s functions
portional to the antisymmetric tensor εµνρ). Addition-
into DS equations and obtain the set of relations for the
ally we have the identity:
introduced parameters. In the last section we present
analytical and numerical results and some conclusions.
γµγνγ = γν . (7)
µ
−
Using the Lagrangian density (1), one can derive in
II. FORMULATION OF THE MODEL.
the standard way – for instance through Feynman path
integral – the Dyson-Schwinger equations for propaga-
The model is defined through following Lagrangian
tors [31]. For boson propagatorwe obtain the relation
density :
1 kµkα 1kµkα
L(x)= Ψ(x)(iγµ∂µ−m0−e0γµAµ(x))Ψ(x) (1) Dµν(k)= k2 −gµα+ k2 − λ k2 δαν −ie20×
1 λ (cid:18) (cid:19)(cid:20)
−4Fµν(x)Fµν(x)− 2 (∂µAµ(x))2 , Trγ d3p S(p)Γ (p,p k)S(p k)Dβν(k) ,
× α (2π)3 β − −
where λ is the gauge parameter. The quantities m and Z (cid:21)
0
(8)
e denote here the bare fermion mass and the bare cou-
0
pling constant respectively. As mentioned in the Intro-
which may be given the graphical form shown in Fig-
duction, the latter for D = 2+1 is a quantity with the
ure 1. The propagator Dµν(k) on the right hand side is
dimensionality of √mass. That means that in the quan-
not integrated over three-momenta and therefore it can
tumtheoryhighertermsofperturbationexpansionshave
fully be represented through fermion functions (and the
betterultravioletmomentumdependenceinloopintegra-
vertex).
tions and the model is superrenormalizable.
Asalreadytold,inthepresentpaperwedealwithfour-
componentfermionfield,choosingthefollowingrepresen- = +
tation for gamma matrices used also in four-dimensional
QED:
FIG.1: Dyson-Schwingerequationforthegaugebosonprop-
σ 0 iσ 0
γ0 = 3 , γ1 = 1 , agator Dµν(k). Light lines represent free propagators and
0 σ 0 iσ
(cid:18) − 3 (cid:19) (cid:18) − 1 (cid:19) heavyonesdressedpropagators. Thefullcirclestandsforthe
iσ 0 full fermion-boson vertex.
γ2 = 2 . (2)
0 iσ
2
(cid:18) − (cid:19) The DS equation for the fermion propagator does not
There are two other gamma matrices, which anticom- allow for such a separation, and has the form
mute with all above, and which can serve for defining
chiral transformations: 1
S(p)= 1+ie2γµ (9)
p m 0 ×
γ3 = 0 11 , γ5 = 0 i11 . (3) 6 − 0(cid:20)
11 0 i11 0 d3k
(cid:18) (cid:19) (cid:18)− (cid:19) × (2π)3S(p+k)Γν(p+k,p)S(p)Dµν(k) ,
The fermion mass term, that was chosen in (1) in the Z (cid:21)
form m ΨΨ, breaks the chiral symmetry defined by any
0
of the matrices (3). There is, however, the possibility of
′
the other choice for the mass term [3]: m0ΨτΨ, where = +
i 11 0
τ = γ3,γ5 = . (4)
2 0 11 FIG. 2: Dyson-Schwinger equation for the fermion propaga-
(cid:18) − (cid:19) tor. As in Figure 1, heavy lines stand for full functions and
(cid:2) (cid:3)
This term does not break chiral symmetry, since matrix light for free ones.
γ0τ commutes with both γ3 and γ5, but it does violate
parity symmetry (contrary to the choice made in (1)). Its graphical representation is shown in Figure 2.
For matrices (2) we have ordinary relations: While introducing the fermion self-energy Σ(p), it can
be rewritten in a simpler manner
γµ,γν =gµν , trγµ =0, Tr [γµγν]=4gµν ,
{ }
Tr [γµγνγργσ]=4(gµνgρσ gµρgνσ+gµσgνρ) , (5) (p m )S(p)=1+Σ(p)S(p), (10)
0
− 6 −
3
where The free fermion propagator has the usual form:
Σ(p)S(p)= (11) 1
S(0)(p)= . (17)
d3k p m
=ie20γµ (2π)3S(p+k)Γν(p+k,p)S(p)Dµν(k). 6 − 0
Z
Due to the masslessness of the gauge boson and the ab-
Higher equations are not considered in our present
sence of the mass gap for emission of soft photons, the
approach. The vertex function Γµ is not taken from
pole at mass m should, in the full propagator, turn
0
DS equations, but postulated in the form proposed by into a branch point at p2 = m2, where m is a phys-
Salam [30] and used afterwards in the so called ‘gauge
ical mass. This is expected also on the basis of gen-
technique’[24,25,32,33]. Theeventualextensionofour
eralconsiderationsonthe analyticalstructureoffermion
method on higher Green’s functions will be considered
propagator [17, 35] and of confinement, which prohibits
elsewhere.
S(p) from having a simple pole. The numerical results,
performedin 1/N expansionwith bare vertex and in eu-
clidean space, suggesting the existence of complex sin-
III. INFRARED GREEN’S FUNCTIONS
gularities rather than real may as well constitute the
effect of the coarse approximations made while solving
The free propagator of massless vector boson has the DS equations: the approximations that are inevitable in
standard form: any nonperturbative approach. Such singularities may
be a signal of confinement, but are not prerequisite. In
1 kµkν 1 kµkν
D(0)µν(k)= gµν + . (12) the Schwinger Model, massless electrodynamics in two
k2 − k2 − λ(k2)2
(cid:18) (cid:19) space-time dimensions, which exhibits confinement of
fermions, the real singularity in the infrared domain in
As is well known, the interaction with fermions does
the fermion propagator has been found (it has the form
notchangethelongitudinalpartofDµν(k). Thisisguar-
of 1/( p2)5/4) [36, 37].
anteed by the following Ward-Takahashi (WT) identity −
Our infrared assumption for S(p) is then:
1 kν
k Dµν(k)=k D(0)µν(k)= . (13)
µ µ −λ k2 1
S(p)= . (18)
(p m)(1 p2/m2)β
Consequently the nonperturbative propagator may be 6 − −
then written as
Wewillseelaterthatitwillturnouttobeself-consistent.
Z kµkν 1 kµkν
Dµν(k)= 3 gµν + , (14) The values of the exponent β and of mass renormal-
d(k2) − k2 − λ(k2)2 izationconstantδm=m m willbe establishedby the
(cid:18) (cid:19) − 0
requirementofconsistency. The presenceofβ in denom-
withcertainunknownfunctiond(k2). Intheinfrareddo- inatorimprovesthe ultravioletbehaviorofloopintegrals
mainweassumethisfunctiontohaveaTaylorexpansion (we assume that 0<β <1, to be verified a posteriori).
starting from k2, since for bispinor fermions the boson
For our further purposes S(p) has to be written in the
field remains massless [16] and we expect d(0)=0. This
spectral form
is not the case for massive two-component fermions, for
which the topological boson mass is generated.
1 1
Taking the first two terms of this expansion, we have S(p)= dMρ(M) , (19)
p m − p M
Z (cid:18)6 − 6 − (cid:19)
k2
d(k2)=k2 1+ , (15)
κ2 with the one spectral density :
(cid:18) (cid:19)
whereκ2,togetherwiththerenormalizationconstantZ ,
3 sin(πβ) 1
will be determined from consistency conditions. From ρ(M)=
π (M m)(M2/m2 1)β
the unitarity we expect the value of Z3 to satisfy the re- − −
[Θ(M m) Θ( M m)], (20)
striction 0< Z 1 [34]. This expectation was actually
3 × − − − −
≤
confirmedinourpreviousworksonspinorQED [28]and
3
QED [29]. sufficient to define the infrared behavior of the propaga-
4
With the above assumptions the inverse of Dµν may tor. ΘisheretheHeavisidestepfunction. Theparticular
be written as form of ρ(M) was given in our previous works [28, 29].
Using ρ(M), one can write the vertex function, which
D−1(k)µν = (16)
we need to put into DS equations (8) and (9). As men-
Z−1 1+ k2 ( k2gµν +kµkν) λkµkν . tionedinthe Introductionweuse the Salam’svertex[30]
3 κ2 − − (with slight and obvious modification resulting from the
(cid:18) (cid:19)
4
form(19)ofS(p),whichinourcasecontainstwoterms): M in (24) with the use of (20), similarly as it was done
in [28], we obtain:
S(p+k) Γµ(p+k,p)S(p)
e2Γ(β+1/2)
1 1 Πµν(k)= 0 (27)
= dMρ(M) γµ π3/2Γ(β+1)m ×
p+ k m p m
"
Z 6 6 − 6 − 1
x(1 x)
1 γµ 1 . (21) × −k2gµν +kµkν dx(1 k2/m2x−(1 x))β+1/2 .
−6p+6k−M 6p−M# (cid:0) (cid:1)Z0 − −
ItautomaticallyguaranteesthecompliancewiththeWT The integral over Feynman parameter x leads to the
identity: hypergeometric (Gauss) function
kµS(p+k)Γµ(p+k,p)S(p)=S(p) S(p+k). (22) 1 x(1 x)
− dx − =
(1 yx(1 x))β+1/2
Z0 − −
1
IV. SELF-CONSISTENT INFRARED F (2,β+1/2;5/2;y/4), (28)
2 1
EQUATIONS 6
and the DS equation (23) may be given the form
A. Gauge boson propagator
D−1(k)µν = λkµkν +( k2gµν +kµkν) (29)
− − ×
Inserting (19) and (21) into the right hand side of (8) e2Γ(β+1/2)
1+ 0 F (2,β+1/2;5/2;k2/4m2) .
we obtain × 6π3/2Γ(β+1)m 2 1
(cid:20) (cid:21)
Afterthesubstitutionoftheexpression(16)fortheleft
D−1(k)µν =−k2gµν +kµkν −λkµkν +ie20Trγµ× hand side and cancellation of the tensor structures, we
d3p 1 1 areleftwiththescalarequation,forwhichwerequirethe
dMρ(M) γν
× (2π)3 p m+iε p k m+iε adjustment the first two terms of the Taylor expansion
Z Z (cid:16)6 − 6 −6 − in k2. In that way we get two equations for Z and κ2:
1 1 3
γν , (23)
−6p−M +iε 6p−6k−M +iε(cid:17) Z−1 = 1+ e20Γ(β+1/2) , (30)
where we have rewritten this equation for inverse propa- 3 6π3/2mΓ(β+1)
gator,whichiseasiertohandle,becauseinthiscaseDµν Z−1 1 = e20Γ(β+3/2) , (31)
decouples from other functions. 3 κ2 30π3/2m3Γ(β+1)
The vacuumpolarizationtensormaybe definedasthe
spectral integral whereweexpandedtheGaussfunctionforsmallmomen-
tum according to the formula:
Πµν(k)= dMρ(M)(Πµν(k) Πµν(k)) . (24)
m − M ab
Z 2F1(a,b;c;z) 1+ z+ (z2),
≈ c O
where Πµν(k) denotes the usual perturbative tensor:
m
and Γ is the Euler function. These are two of the set of
Πµmν(k)= (25) four equations for parameters of the model, to be solved
d3p 1 1 in section V.
ie2Trγµ γν .
0 (2π)3 p m+iε p k m+iε
Z 6 − 6 −6 −
It may be evaluated in the standard way with the use B. Fermion propagator
of identities (5), and for instance by performing Wicks
rotation and introducing Feynman parameters. Passing Togetothertwoequationsforparameters,weput(19),
back to Minkowski space, we have (21)and(14)togetherwith(15)intotheDSequation(9).
We obtain, after some simplifications
Πµν(k)= (26)
m
1 (p m )S(p)= (32)
e2 x(1 x) 6 − 0
0( k2gµν +kµkν) dx − . 1+[Σ(p)S(p)] +[Σ(p)S(p)] +[Σ(p)S(p)] ,
π − (m2 k2x(1 x))1/2 A B C
Z0 − −
where we divided fermion self-energy into pieces com-
For k2 < 4m2 the x integral is well defined. The ing from different tensor structures in Dµν(k): gµν and
transversalityofΠµν(k)isafavorableconsequenceofus- kµkν fromthetransversepartandagainfromthegauge-
m
ing the vertex function in the form (21), satisfying the dependent longitudinal part. After small rearrangement
WT identity (22). Performing the spectral integral over they are:
5
[Σ(p)S(p)] = (33)
A
d3k 1 1
iZ κ2e2 dMρ(M) γµ γ (m M)
3 0 (2π)3 p+ k m+iε µ(k2+iε)(k2 κ2+iε)(p m+iε) − →
Z Z (cid:20) 6 6 − − 6 − (cid:21)
[Σ(p)S(p)] = (34)
B
d3k 1
iZ κ2e2 dMρ(M) k (m M)
3 0 (2π)3 6 p+ k m+iε)(k2+iε)2(k2 κ2+iε) − →
Z Z (cid:20) 6 6 − − (cid:21)
[Σ(p)S(p)] = (35)
C
ie2 d3k 1
0 dMρ(M) k (m M) .
λ (2π)3 6 (p+ k m+iε)(k2+iε)2 − →
Z Z (cid:20) 6 6 − (cid:21)
A comment should be made here. To avoid technical the integrals, we will come back to real values of κ. The
difficulty while performing Wick’s rotation in the above procedure was discussed in [29]. With this trick each
momentumintegrals,weanalyticallycontinuedthevalue of the above momentum integrals can be performed in
of κ to imaginaryvalues on the upper half plane of com- anordinarywayknownfromperturbationtheory. Itcan
plex κ. Now the deformation of the integration contour easilybeseenthatallintegralsarefinitewithouttheneed
as is requiredby passing into euclidean space,is not dis- of any regularization. Omitting details of this standard
turbedbytheinappropriatelocationofpolessinceallsin- calculation, we find (in Minkowski space)
gularitieshavethe ‘Feynman’position. Afterperforming
d3k 1 1
I = ie2 γµ γ = (36)
A 0 (2π)3 p+ k m+iε µ(k2+iε)(k2 κ2+iε)
Z 6 6 − −
1
e2 1 1
0 dx(3m p(1 x)) ,
8πκ2 −6 − (m2x p2x(1 x))1/2 − (m2x p2x(1 x)+κ2(1 x))1/2
Z0 (cid:20) − − − − − (cid:21)
1
d3k 1 e2 p(p+m)
I = ie2 k = 0 dx 1 x6 6 (37)
B 0 (2π)3 6 (p+ k m+iε)(k2+iε)2(k2 κ2+iε) 8πκ2 − κ2 ×
Z 6 6 − − Z (cid:20)(cid:18) (cid:19)
0
1 1 p(p+m) x(1 x)
+ 6 6 − ,
× (m2x p2x(1 x))1/2 − (m2x p2x(1 x)+κ2(1 x))1/2 2 (m2x p2x(1 x))3/2
(cid:18) − − − − − (cid:19) − − (cid:21)
1
d3k 1 e2 1
I = ie2 k = 0 dx
C 0 (2π)3 6 (p+ k m+iε)(k2+iε)2 −8π (m2x p2x(1 x))1/2
Z 6 6 − Z0 (cid:20) − −
p(p+m) x(1 x)
+6 6 − . (38)
2 (m2x p2x(1 x))3/2
− − (cid:21)
Now the contributions to Σ(p)S(p) may be written as All the abovespectral integrals canbeen found similarly
as in [28], therefore we omit the technicalities.
[Σ(p)S(p)] = (39)
A
In the infrared domain, when p2 m2, the left hand
1
Z3κ2 dMρ(M) IA (m M) , sideoftheDSequation(32),aftersu→bstituting(18),con-
p m+iε − →
Z (cid:20) 6 − (cid:21) tainstwotypesofsingularterms(theonlytermsthatare
[Σ(p)S(p)] = (40)
B important):
Z κ2 dMρ(M)[I (m M)] ,
3 B
− →
Z
1 m2βδm(p+m) m2β
[Σ(p)S(p)] = dMρ(M)[I (m M)] . (41) 6 + . (42)
C λ C − → − (m2 p2)β+1 (m2 p2)β
Z − −
6
Therefore, to avoid lengthy expressions we will not give V. SOLUTIONS AND CONCLUSIONS
theresultsfor(39),(40),and(41)intheirfullcomplexity,
because it is sufficient for our goal to only pick out from The four equations we have obtained, i.e (30), (31),
them the identical singular terms. We find: (49) and (50) are sufficient to determine all parameters.
e2Z 1 1 Werewritethemwiththeuseofrenormalizedquantities:
[Σ(p)S(p)]A ≈ 08π3 m2 I1(m2,κ2)− m3 × fermion mass m = m0 + δm, gauge coupling constant
(cid:26)(cid:20) (cid:21) e = Z1/2e and gauge parameter λ = Z λ. Besides, it
p(p+m) p(p+m) 1 3 0 R 3
×(1 6 p62/m2)β+1 + 6 6 2 I2(m2,κ2)− m3 isusefultointroduceadimensionlessparameterζ = 4πe2m.
− (cid:20) (cid:18) (cid:19) Afterexecutingtheparametricintegralsin and and
1 3
3 1 I I
, (43) performing the reverse analytical continuation in κ, we
−4βm (1 p2/m2)β get
(cid:21) − (cid:27)
e2Z 1 p(p+m)
[Σ(p)S(p)]B ≈ 08π3 m3 (1 6 p62/m2)β+1 (44) δm = ζ 1 1 4m2−κ2 ln(4m2/κ2+1) ,(52)
(cid:20) − m 2 λ − − 4m2
p(p+m) 1 β 1 1 (cid:20) R (cid:21)
6 6 − , ζ 3 1 4m2 κ2
− 2m3 β − 4βm (1 p2/m2)β 1= 2 2 −
(cid:18) (cid:19) − (cid:21) 2 4β λ − − 4m2+κ2
(cid:20) (cid:18) R (cid:19)
e2 p(p+m) 1 κ2
[Σ(p)S(p)]C ≈−8π0λ 6 6 m3 (1 p2/m2)β+1 −2m2 ln(4m2/κ2+1) , (53)
(cid:20) − (cid:21)
p(p+m) β 1 1 1 2ζ Γ(β+1/2)
+ 6 6 − + ,(45) Z = 1 , (54)
2m3 β 4βm (1 p2/m2)β 3 − 3 √πΓ(β+1)
(cid:18) (cid:19) − (cid:21)
where refers to diverging terms, when p2 m2. The κ2 15√πΓ(β+1)
≈ → = . (55)
functions 1 and 2 have the following form: m2 2ζ Γ(β+3/2)
I I
1
x+2 Thissetofequationsforδm/m,β, Z3 andκ2/m2 may
(m2,κ2)= dx , (46)
I1 (m2x2+κ2(1 x))1/2
Z0 −
1 Κ
x2(x+2)
(m2,κ2)= dx . (47)
I2 (m2x2+κ2(1 x))3/2 30
Z0 −
If the solution is to be self-consistent close to the
20
fermion mass shell, the divergent terms on both sides
of DS equation (10) must be identical. Equating them,
and making use of the fact that up to finite terms we
10
have
p(p+m) 2m2
6 6 , (48)
(1 p2/m2)β ≈ (1 p2/m2)β Ζ
− − 0.1 0.2 0.3 0.4
p(p+m) m(p+m) m2
6 6 6 ,
(1 p2/m2)β+1 ≈ (1 p2/m2)β+1 − (1 p2/m2)β Β
− − −
wederivethefollowingtworelationsforunknownparam-
eters δm and β:
δm=e20Z3 m (m2,κ2)+ 1 , (49) 0.2
1
8π − I λZ
(cid:20) 3(cid:21)
e2Z 3 1
1= 0 3 κ2 3(m2,κ2)+ 2 . 0.1
8π − I 4βm λZ −
(cid:20) (cid:18) 3 (cid:19)(cid:21)
(50)
The function (m2,κ2) is defined as follows Ζ
I3 0.1 0.2 0.3 0.4
1
= (m2,κ2) m2 (m2,κ2) =
I3 κ2 I1 − I2
FIG. 3: The dependence of κ in units of m (upper plot) and
1 (cid:0) (cid:3)
(1 x)(x+2) of power β (lower plot) on the parameter ζ. The dashed
dx − . (51) line corresponds to approximated solutions defined by equa-
(m2x2+κ2(1 x))3/2
Z0 − tions (56)-(59). The gauge parameter is chosen as λR =0.3.
7
be solved numerically for certain chosen values of renor- Now, from (55) it becomes obvious that for small ζ the
malizedgaugeparameterλ andtheresultsplotasfunc- left hand side must be large. This is in agreement with
R
tions of parameter ζ. Let us consider the case of weak our expectations concerning the subsequentterms in the
coupling, when ζ is small. Since the applicability of our Taylorexpansion(15). Butforκ2 m2 thefirsttermin
≫
method requires 0 < β < 1, then from equation (55) squarebracketsonthe righthand side of(53) dominates
we deduce that κ2/m2 should be large. It may be jus- overallother,whichmaybeeasilyverified. Theexpected
tified by the elementary estimation given below. First positivity of β requires then considering only gauges for
we rewritefollowingexpressioninterms ofbeta function which 0 < λ < 1/2. This is not surprising, since β is
R
B(x,y) and its integral representation: a strongly gauge dependent quantity. For our numerical
calculations we have then chosenthe values of the gauge
1 parameters from that range.
√πΓ(β+1) 1 1
= B(β+1,1/2)= t−1/2(1 t)βdt,
2Γ(β+3/2) 2 2 − In figure 3 we show the dependence of parameters κ
Z
0 and β on ζ for exemplary value of λ =0.3 (solid lines).
R
Infigure 4 we similarlyplotthe dependence oftwo other
and next use the inequalities valid for 0<β <1:
parameters: fermion mass renormalization constant δm
and gauge field renormalization constant Z . It is nice
1 1 3
1 t−1/2(1 t)βdt < 1 t−1/2dt=1, to observe that we have 0<δm<m and 0<Z3 <1, as
2 − 2 we expected. The dashed lines in these figures represent
Z0 Z0 approximate solutions of the set (52)-(55), as described
1 1 below.
1 1 2
t−1/2(1 t)βdt > t−1/2(1 t)dt= .
2 − 2 − 3 In the weak coupling regime,assuming that we do not
Z Z
0 0 choose the gauge parameter λ approaching zero (but
R
stillto be less than1/2),the equationsmaybe giventhe
∆m
0.6
Κ
30
0.4
20
0.2
10
Ζ
0.1 0.2 0.3 0.4
Z3 0.1 0.2 0.3 0.4 Ζ
1
Β
0.9
0.8
0.8
0.5
0.7
Ζ 0.2
0.1 0.2 0.3 0.4
Ζ
0.1 0.2 0.3 0.4
FIG.4: Thedependenceofmassrenormalization δminunits
of m (upper plot) and charge renormalization constant Z3
(lowerplot)ontheparameterζ. Thedashedlinecorresponds FIG. 5: The comparison of the behavior of κ (upper plot)
toapproximatedsolutionsdefinedbyequations(56)-(59). On and parameter β (lower plot) for different values of gauge
the upper plot the dashed line is almost identical with the parameter: dotted line – λR = 0.1, dashed line – λR = 0.2,
solidoneandthereforeitisnotvisible. Thegaugeparameter solid line – λR =0.3, mixed line – λR =0.5 (Yennie gauge).
is chosen as λR =0.3. Theplotsofκforvariousgaugesfollowalmostthesamecurve.
8
The terms m2/κ2 in the first two equations arise from
∆m
the expansion of and for large κ2, but in fact they
1 3
I I
may be omitted, since they are of order ζ, as results
0.8 from the last equation. As already told, the solutions of
thesesimplifiedequationsaredrawninfigures3and4as
dashedlines. Onthe plots forκ and δm these curvesare
0.5 not visible since they are almost identical with the full
solutions, but also on the other two they do not deviate
from the ‘exact’ results too much.
0.2
From (57) we see that the exponent β in this approxi-
mation may be written as
Ζ
0.1 0.2 0.3 0.4
Z3 3ζ 1
β = 2 , (60)
8 λ −
1 (cid:18) R (cid:19)
0.9 which means that λR = 1/2 is a kind of Yennie
gauge.
0.8 It is interesting to observe, if and how the values of
the parameters depend on the gauge. This is shown in
figures 5 and 6 which are performed for the following
0.7
values from the range [0,1/2]: λ = 0.1, 0.2, 0.3, 0.5.
R
Ζ The Landaugaugecannotbe used because it wouldlead
0.1 0.2 0.3 0.4
to negative value of β .
Theparticularstressdeservestheobservationthatthe
FIG. 6: The comparison of the behavior of fermion mass
gaugedependence ofthe parameterκ is extremely weak.
renormalization δm (upper plot) and gauge field renormal-
ization constant Z3 (lower plot) for different vales of gauge This means that the renormalizedgauge boson propaga-
parameter: dotted line – λR = 0.1, dashed line – λR = 0.2, tor (strictly speaking its transverse part) is practically
solid line – λR =0.3, mixed line – λR =0.5 (Yenniegauge). gauge independent, as it should be. Please note that
inother approximationschemesapplied to DS equations
in QED one obtains the gauge dependent value of the
3
following approximated form: polarization scalar [16].
δm ζ 1 6m2 ThedependenceofZ3onλRisrelativelyweakforsmall
= , (56) ζ too. These results are worth noticing, since the full
m 2 λ − κ2
(cid:20) R (cid:21) gauge independence should appear in the exact theory,
ζ 3 1 12m2 andone oughtnotto expect too muchfromthe approxi-
1= 2 , (57)
2 4β λ − − κ2 mated model, where infrared forms of Green’s functions
(cid:20) (cid:18) R (cid:19) (cid:21)
2ζ arepostulatedinasimpleform. Gaugedependenceofthe
Z3=1 , (58) ‘physical’fermionmassobtainedinourworkisrelatively
− 3
strongbutitisacommonfeatureofnonperturbativecal-
κ2 15
= . (59) culations in this model [6, 7, 21].
m2 ζ
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