Factorization of integrals, defining the beta-function, into integrals of total derivatives in N=1 SQED, regularized by higher derivatives

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by  K. V. Stepanyantz

Factorization of integrals, defining the β-function,
= 1
into integrals of total derivatives in N SQED,
regularized by higher derivatives
K.V.Stepanyantz
January 18, 2011
1
1 Moscow State University, physical faculty,
0
department of theoretical physics.
2
119992, Moscow, Russia
n
a Abstract
J

5 Somecalculationsinsupersymmetrictheories,madewiththehigherderivativeregulariza-
1 tion, show that the β-function is given by integrals of total derivatives. This is qualitatively
explained for the N =1 supersymmetric electrodynamics in all orders.
]
h
t
- 1 Introduction
p
e
h
[
Quantum corrections in supersymmetrictheories were studied for a long time. In particular,

1
alotofpapersweredevoted tocalculations ofaβ-functionandananomalousdimension,e.g. [1].
v
Mostcalculationsweremadewiththedimensionalreduction[2],whichisaspecialmodificationof
6
5 the dimensional regularization. The matter is that the dimensional regularization [3] breaks the
9
supersymmetryand is not convenient for calculations in supersymmetrictheories. However, it is
2
. wellknownthatthedimensionalreductionisnotself-consistent[4]. Ways, allowingtoavoidsuch
1
problems, are discussed in the literature [5]. Other regularizations are also sometimes applied
0
1 for calculations in supersymmetric theories. For example, in Ref. [6] the two-loop β-function of
1
theN=1 supersymmetricYang–Mills theory was calculated with the differential renormalization
:
v [7].
i
X It is also possible to make calculations with the higher covariant derivative regularization,
r proposed in Ref. [8]. It was generalized to the supersymmetric case in Ref. [9] (another variant
a
was proposed in Ref. [10]). This regularization is self-consistent and does not break the super-
symmetry. A calculation of the one-loop β-function for the (non-supersymmetric) Yang–Mills
theory, regularized by higher covariant derivatives, was made in Ref. [11]. After corrections,
made in subsequent papers [12], the result coincided with the well-known result, obtained with
the dimensional regularization [13]. It was proved that in the one-loop approximation cal-
culations with the higher covariant derivative regularization always agree with the results of
calculations with the dimensional regularization [14]. Some calculations in the one-loop and
two-loop approximations were made for various theories [15, 16] with a variant of the higher co-
variant derivative regularization, proposed in [17]. The structure of the corresponding integrals
was discussed in Ref. [16].
A calculation of the three-loop β-function for the N = 1 supersymmetric electrodynamics
(SQED) regularized by higher derivatives [18] reveals an interesting feature of quantum correc-
tions: loop integrals, defining the β-function, are integrals of total derivatives. (In Ref. [18]
1
three-loop integrals, defining the β-function, were reduced to integrals of total derivatives using
integration by parts.) A two-loop calculation, made with the dimensional reduction in N = 1
SQED and revealing a similar feature, was presented in Ref. [19], where the factorization of
integrands into total derivatives is explained in all loops using a special technique, based on
the covariant Feynman rules in the background field method [20]. Such factorization allows to
calculate one of the loop integrals analytically and obtain the NSVZ β-function [21], which (in
N = 1 SQED) relates the β-function in n-th loop with the anomalous dimension in the previous
loop. Due to this, application of the higher covariant derivative regularization is sometimes
very convenient in the supersymmetric case. The fact that integrals, appearing with the higher
covariant derivative regularization, in the limit of zero external momentum become integrals of
total derivatives, seems to be a general feature of supersymmetric theories. It was verified in
the two-loop approximation in [22] for a general renormalizable N = 1 supersymmetric theory.
A similar results can be obtained with different versions of the higher derivative regularization.
Using a usual supergraph technique for N = 1 SQED, regularized by higher derivatives, a
partial explanation of the factorization of integrands into total derivatives was made in [23] by
substituting a solution of the Ward identity into the Schwinger–Dyson equation. Using this
method it is possible to extract a contribution, giving the exact NSVZ β-function in all orders.
However, there is also another contribution. In N = 1 SQED this contribution is nontrivial
starting from the three-loop approximation. In the three-loop approximation it is given by an
integral of a total derivative and is equal to 0. It was conjectured in [23] that this occurs in all
loops. A partial four-loop verification of this statement was made in Ref. [24].
In this paper for N = 1 SQED using usual supergraphs we try to explain, why integrals,
defining a β-function, are integrals of total derivatives in all loops. Although the explanation
is not completely rigorous, we demonstrate that a key ingredient of the proof is a special alge-
braic identity, obtained in this paper. Moreover, the simple arguments, presented here, can be
formulated in a rigorous form, and the results can be compared with explicit calculations. We
will describe this in the forthcoming paper.
The paper is organized as follows:
In Sec. 2 we recall basic information about N = 1 SQED, regularized by higher covariant
derivatives. In Sec. 3 we explain how the three-loop β-function is related with the two-loop
anomalous dimension due to the factorization of integrands into total derivatives. A qualitative
explanation of this feature is given in Sec. 4 for N = 1 SQED. The results are briefly discussed
in the Conclusion. In Appendix A we present explicit expressions for the three-loop integrals,
defining the β-function and the anomalous dimension in N = 1 SQED. An identity, which is a
key ingredient of our arguments, is proved in the Appendix B.
2 N=1 supersymmetric electrodynamics, regularized by higher
covariant derivatives
Let us consider N = 1 SQED, regularized by higher derivatives. In the massless case it is
described by the action1
1 1
S = Re d4xd2θW CabR(∂2/Λ2)W + d4xd4θ φ∗e2Vφ+φ∗e−2Vφ , (1)
4e2 a b 4
Z Z (cid:16) (cid:17)
where R(∂2/Λ2) is a regulator, e.g. R = 1+∂2n/Λ2n. This action is invariaent undeer the gauge
transformations
1In ournotation ηµν =diag(1,−1,−1,−1); θa ≡θbCba; θa and θ¯a denotetheright and left components of θ,
respectively.
2
i
φ→ eiΛφ; φ → e−iΛφ; V → V + (Λ∗−Λ), (2)
2
where Λ is an arbitrary chiral superfield. A gauge is fixed by adding
e e
1
S = − d4xd4θ VD2D¯2R(∂2/Λ2)V +VD¯2D2R(∂2/Λ2)V (3)
gf 64e2
Z (cid:16) (cid:17)
to the action. The term with higher derivatives does not remove divergences in the one-loop
approximation [25]. In order to cancel the remaining one-loop divergences, the Pauli-Villars
determinants should be inserted into the generating functional
cI
Z[J,j] = Dµ detPV(V,M ) exp i(S +S +S +Sources) , (4)
I Λ gf
Z YI (cid:16) (cid:17) n o
where c = 1 and c M2 = 0. It is useful to present the Pauli–Villars determinants in the
I I I
I I
followinPg form: P
−1
detPV(V,M ) = DΦ DΦ eiSI , (5)
I I I
(cid:16)Z (cid:17)
where SI is the action for the Pauli–Villars fields,2 e
1 1
S = d4xd4θ Φ∗e2VΦ +Φ∗e−2VΦ + d4xd2θM Φ Φ +h.c. . (6)
I 4 I I I I 2 I I I
Z (cid:16) (cid:17) (cid:16) Z (cid:17)
We assume that MI = aIΛ, where aI aere consetants. (Thus, there isethe only dimensionful
parameter Λ in the regularized theory.)
The generating functional for connected Green functions and the effective action are defined
by the standard way. Terms in the effective action, correspondingto the renormalized two-point
Green function of the gauge superfield, have the form
1 d4p
Γ(2) = − d4θV(−p,θ)∂2Π V(p,θ)d−1(α,µ/p), (7)
V 16π (2π)4 1/2
Z
where α is a renormalized coupling constant, and
1
∂2Π = − DaD¯2D (8)
1/2 a
8
is a supersymmetric transversal projector. We calculate
d d−1(α ,Λ/p)−α−1 =−dα−01 = β(α0), (9)
dlnΛ 0 0 p=0 dlnΛ α2
0
(cid:16) (cid:17)(cid:12)
where α and Λ are considered as independent va(cid:12)riables. The anomalous dimension is defined
(cid:12)
similarly. Ifthetwo-point Greenfunctionofthematter superfieldinthemasslesslimitiswritten
as
1 d4p
Γ(2) = d4θ φ∗(−p,θ)φ(p,θ)+φ∗(−p,θ)φ(p,θ) (ZG)(α,µ/p), (10)
φ 4 (2π)4
Z (cid:16) (cid:17)
then the anomalous dimension is defined by e e
2Thisaction differsfrom theone,usedin [18],becauseherethequotientofthecoefficientsin thekineticterm
and in the mass term does not contain the factor Z. Using terminology of Ref. [26], one can say that here we
calculate the canonical coupling αc, while in Ref. [18] the holomorphic coupling αh was calculated. Certainly,
after the renormalization the effective action does not depend on the definitions. However, the definitions, used
here, are much more convenient.
3
d
γ α (α,Λ/µ) = − lnZ(α,Λ/µ) . (11)
0
dlnΛ
(cid:16) (cid:17) (cid:16) (cid:17)
The two-point Green functions are calculated using a standard supergraph technique, described
e.g. in textbooks [27, 28].
3 Three-loop calculation for SQED
A calculation of the three-loop β-function gives the following result:
d d β(α )
d−1(α ,Λ/p)−α−1 = − α−1(α,µ/Λ) = 0
dlnΛ 0 0 p=0 dlnΛ 0 α2
0
(cid:16) (cid:17)(cid:12)
1 d (cid:12) 1 1 d
= 1− lnG(α ,Λ/q(cid:12)) = − lnZG(α,µ/q)−lnZ(α,Λ/µ) =
0
π dlnΛ q=0 π πdlnΛ q=0
1(cid:16) (cid:12) (cid:17) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
= 1−γ α0(α,Λ/µ) . (cid:12) (cid:12) (12)
π
(cid:16) (cid:16) (cid:17)(cid:17)
Theexplicit three-loop expressionforthelefthandsideandthetwo-loop expressionfortheright
hand side are presented in Appendix A. It should be noted that they are given by well defined
integrals. In particular, the higher derivative regularization and the differentiation with respect
to lnΛ ensure that there are no IR divergences. This agrees with the results of Ref. [29] that
the IR region does not affect to the β-function.
In the three-loop approximation the integrals, obtained with the higher covariant derivative
regularization, can not be calculated analytically. However, it is possible to find relation (12)
between them. It is easy to see that this relation appears due to the factorization of integrands
into total derivatives.
4 Factorization of integrands into total derivatives in N = 1
SQED: qualitative explanation
NowletustrytoexplainthefactorizationofintegrandsintototalderivativesinN = 1SQED,
described by action (1). We will calculate the two-point Green function of the gauge superfield.
Due to the Ward identity this function is transversal. In order to obtain the β-function we make
calculations in the limit p → 0 after differentiation with respect to lnΛ:
d
d4θV∂2Π V × (Momentum integral) . (13)
1/2
dlnΛ p=0
Z (cid:12)
Therefore, in order to find a β-function it is possible to make the sub(cid:12)stitution
(cid:12)
V → θ¯aθ¯ θbθ ≡ θ4, (14)
a b
so that
d4θV∂2Π V → −8. (15)
1/2
Z
This is possible, because we make calculations in the limit p → 0, where p is an external
momentum. (In this limit the gauge superfield V does not depend on the coordinates xµ.)
4
We will try to reduce a sum of Feynman diagrams for the considered theory to integrals of
total derivatives. In the coordinate representation such an integral can be written as
Tr [xµ,Something] = 0. (16)
(cid:16) (cid:17)
In order to extract such commutators first we consider diagrams, containing a vertex to which
only one external line (and no internal lines) is attached. We can sum such a diagram with the
diagram, in which the external line is shifted to the nearest vertex [30]. This gives a sum of the
subdiagrams, presented below. Making the substitution V → θ4 we obtain
+ =
D¯ D2 D2 D¯2D2∂ D¯ D2∂ D¯2D2
= −θaθ θ¯b b +θaθ +iθ¯b(γµ) aθ µ −iθa(γµ) b b µ + . (17)
a 4∂2 a4∂2 b a 8∂4 a 4∂4 16∂4
Only the first and the third terms give nontrivial contributions to the two-point function of
the gauge superfield, because they contain θ¯. Really, finally it is necessary to obtain
d4θθaθ θ¯bθ¯,
a b
Z
while calculation of the supergraph can not increase degrees of θ or θ¯. Therefore, we should
have θ¯aθ¯ from the beginning.
a
Now let us proceed to calculation of Feynman diagrams. In the one-loop approximation
all calculations have been already done, so that we can ignore this case. (In the one-loop
approximation contributions of the Pauli–Villars field are very essential, but here we do not
consider them. For investigation of the Pauli–Villars contribution it is necessary to consider the
massive case. In principle, this is made similarly, butall expressions become more complicated.)
Firstweconsiderdiagrams,inwhichexternalV-linesareattachedtodifferentloopsofmatter
superfields. Letusdenoteachain of thematter superfieldpropagators, connectingvertexes with
quantum gauge field, by ∗.3 Then each loop will be proportional to
D¯2D2∂ D¯ D2 D¯ D2
Tr iθ¯c(γν) dθ ν ∗−θcθ θ¯d d ∗ ∼ Tr −θcθ θ¯d∗ d ∗
c d 8∂4 c 4∂2 c 4∂2
(cid:16) D¯2D D¯ D2 D¯(cid:17)2D2∂ (cid:16)
−θ¯dθc∗ c ∗ d ∗+iθ¯c(γν) dθ ∗ ν ∗+θ2,θ¯1,θ1,θ0 terms . (18)
4∂2 4∂2 c d 8∂4
(cid:17)
(Herewedonotdistinguish∗and∗2. Thisispossibleonlyqualitatively. However, thecalculation
can be repeated using a rigorous approach. In this case the result should be slightly modified.)
After some simple algebra this expression can be written as
Tr −2θcθ θ¯d[θ¯ ,∗]+iθ¯c(γν) dθ [y∗,∗]+... = 0+..., (19)
c d c d ν
where y∗ = x − iθ¯a((cid:16)γ ) bθ , and dots denote terms, proportio(cid:17)nal to θ2, θ¯1, θ1, θ0. In order
ν ν ν a b
to calculate the two-point function we should multiply two such expressions. Then the terms,
3It is possible to give a rigorous definition of ∗. This will be done in [31]. Below for simplicity we omit some
details, needed in therigorous approach.
5
denoted by dots, vanish dueto the integration over d4θ. Therefore, the sumof all such diagrams
is given by an integral of a total derivative and is equal to 0.
Now let us consider diagrams, in which external V-lines are attached to a single loop of the
matter superfields.
We can shiftθ-sto an arbitrarypointof theloop, commutingthem withmatter propagators.
Then we obtain
i d D¯2D2∂µ D¯2D2∂
∼ Tr θ4 ∗ µ ∗ ;
θaθ¯b θcθ¯d 32dlnΛ ∂4 ∂4
(cid:16) (cid:17)
d D¯2D ∂ D¯dD2
∼ 4(γµ) c Tr θ4 c µ ∗ ∗
θaθ¯b θcθ θ¯d d dlnΛ 16∂4 ∂2
c (cid:16) (cid:16)
D¯2D2∂ D¯2D D¯dD2
µ c
+ ∗ ∗ ∗ ;
16∂4 16∂2 ∂2
(cid:17)(cid:17)
d D¯ D2 D¯2 D¯dD2 D¯ D¯dD2
∼−4i Tr θ4 − d ∗ ∗ ∗− d ∗ ∗
θaθ θ¯b θcθ θ¯d dlnΛ 4∂2 8∂2 4∂2 2∂2 4∂2
a c (cid:16) (cid:16)
D¯ Dc D¯2D D¯dD2 D¯ D2 D¯2Dc D¯2D D¯dD2
d c d c
+ ∗ ∗ ∗+ ∗ ∗ ∗ ∗ .
2∂2 8∂2 4∂2 4∂2 8∂2 8∂2 4∂2
(cid:17)(cid:17)
In order to write the sum of these diagrams as an integral of a total derivative, we will start
with the calculation of the following sum:
1
+
θaθ¯b θcθ¯d 2 θaθ¯b θcθ θ¯d
c
Using the identity
∂ ∂ ip
[xµ, µ]= [−i ,− µ]= −2π2δ4(p )= −2πiδ4(p) (20)
∂4 ∂p p4 E
µ
after some algebra we obtain that this sum is given by
d iπ2 D¯2D2∂µ π2
4i Tr θ4 ∗D¯2D2δ4(∂ )+ y∗, ∗ = −Tr θ4∗D¯2D2δ4(∂ ) . (21)
dlnΛ 8 α µ 16∂4 2 α
(cid:16) (cid:16) h i(cid:17)(cid:17) (cid:16) (cid:17)
δ-function allows to perform integration over the momentum of the considered matter loop.
This corresponds to cutting the diagram and gives diagrams for the two-point Green function
of the matter superfield [19]. For example,
- + + ...
For diagrams, which can not be made disconnected by two cuts of the matter loop, this gives
the corresponding contribution to the anomalous dimension.
If a method, based on substitutinga solution of the Ward identity into the Schwinger–Dyson
equation, is used, a sum of the considered diagrams is contained in the effective diagrams that
can be expressed in terms of the two-point function of the matter superfield. These diagrams
give
6
α2
β(α) ← − γ(α). (22)
π
(A part of this contribution, correspondingto higher terms in the expansion of lnZ, comes from
diagrams, considered below.)
Now let us calculate the remaining diagrams
1
+
θaθ θ¯b θcθ θ¯d 2 θaθ¯b θcθ θ¯d
a c c
The considered sum is given by
d 1 D¯2D D¯dD2 1 D¯2D2∂ D¯2D D¯dD2
Tr θ4 (γµ) c y∗, c ∗ ∗ + (γµ) c µ ∗ c ∗ ∗
dlnΛ 32 d µ ∂2 ∂2 16 d 16∂4 ∂2 ∂2
1 (cid:16) D¯(cid:16)2D2∂ Dh¯dD2 D¯2D iD¯2 D¯dD2 D¯ D2 D¯ Dc D¯2D
+ (γµ) c µ ∗ ∗ c ∗−4i ∗ ∗ d ∗+ d ∗ c ∗
16 d 16∂4 ∂2 ∂2 2∂2 8∂2 8∂2 2∂2 8∂2
D¯dD2 D¯ D2 D¯2Dc D¯2D D¯dD(cid:16)2
d c
× ∗+ ∗ ∗ ∗ ∗ . (23)
8∂2 4∂2 4∂2 8∂2 8∂2
(cid:17)(cid:17)
The first term in this expression is an integral of a total derivative. The other terms have
not yet been written as an integral of a total derivative.4 Let us qualitatively explain, how this
can be made. For this purpose we cut the diagram by any chain of internal lines:
Then according to Eq. (23) (we omit the first term, which is the integral of a total derivative)
the following variants are possible:
1. There is
D¯aD2 D2Da
∗ ∗= [∗,θ¯a] or ∗ ∗ = [∗,θa] (24)
8∂2 8∂2
on the left (right) side.
D¯aD2 D¯ D2
a
2. There is ∗ ∗ ∗ on the left (right) side. Then on the other side we have
8∂2 8∂2
D¯2 D¯2Db D¯2D
8i{θb,[θ ,∗]} = 8i −∗ ∗+∗ ∗ b ∗ . (25)
b 4∂2 4∂2 8∂2
(cid:16) (cid:17)
D¯aD2 D¯2D
b
3. There is ∗ ∗ ∗ on the left (right) side. Then on the other side we obtain
8∂2 8∂2
D¯2D2∂ D¯2Dc D¯ D2
2(γµ) bθ4[y∗,∗] = 2(γµ) bθ4 ∗ µ ∗−i(γ ) d∗ ∗ d ∗ . (26)
a µ a 8∂4 µ c 4∂2 8∂2
(cid:16) (cid:17)
(The last term in (23) is contributed by this expression and by the expression in variant 4. That
is why this expression gives only one half of this term.)
D¯2D D¯aD2
b
4. There is ∗ ∗ ∗ on the left (right) side. Then on the other side we have
8∂2 8∂2
4Inthiscasethemethod,basedonsubstitutingasolutionoftheWardidentityintoSchwinger–Dysonequation,
does not work, because here transversal parts of theGreen functions are essential.
7
D¯2D2∂ D¯ Dd D¯ D2 D¯2Dd
2(γµ) b[y∗,∗]θ4 = 2(γµ) b ∗ µ ∗−i(γ ) c∗ c ∗−i(γ ) c∗ c ∗ ∗ θ4. (27)
a µ a 8∂4 µ d 2∂2 µ d 4∂2 8∂2
(cid:16) (cid:17)
5. All modified propagators are on the same side of the diagram.
Graphically the commutators (or anticommutators), which are obtained on one side of the
diagram, correspond to one of subdiagrams
or
∗
with θ-s or y attached to the vertexes. This subdiagrams can be calculated by a standard
µ
supergraph technique. Using this method θ-parts of the graphs can be presented as effective
lines with all external lines attached to one of the vertexes. (For this purpose it is necessary to
calculate θ-integrals.)
- (cid:27)
Then a number of loops in the θ-part of the considered diagram is evidently decreased. The
procedure can be repeated, if one perform the following substitutions:
D¯2D2
→ A, (P = 0 or P = 1);
16∂2 A A
D¯2Da D¯aD2
→ −[θa,A}; → −[θ¯a,A};
8∂2 8∂2
D¯2 D¯2D2∂
− → [θa,[θ ,A}}; θ4 µ → θ4[y∗,A];
4∂2 a 8∂4 µ
D¯2D2∂ D¯ Dd
µ −i(γµ) c c θ4 → [y∗,A]θ4. (28)
8∂4 d 2∂2 µ
(cid:16) (cid:17)
(Also it is necessary to include the factors (−1)P, but we will not discuss this here. Using
another method, based on this arguments, this signs can be easily taken into account.) The
procedure can be repeated using the identities
[θ ,AB} = [θ ,A}B +(−1)PAA[θ ,B};
a a a
[y∗,AB]θ4 = [y∗,A]θ4B +A[y∗,B]θ4−2i(−1)PA(γ ) a[θ¯ ,A}[θb,B}θ4+O(θ3);
µ µ µ µ b a
θ4[y∗,AB]= θ4[y∗,A]B +Aθ4[y∗,B]−2i(−1)PA(γ ) aθ4[θb,A}[θ¯ ,B}+O(θ3);
µ µ µ µ b a
[θa[θ ,AB}} = [θa[θ ,A}}B +A[θa[θ ,B}}+2(−1)PA[θa,A}[θ ,B}. (29)
a a a a
It is easy to see that at the last step we can obtain no more than three subdiagrams.
Let us denote expressions for these subdiagrams by A, B, and C. They can be written as
differential operators, constructed from the supersymmetric covariant derivatives D, D¯ and the
usual derivatives ∂ . (It is important that they do not explicitly depend on θ.) Then according
µ
to Eq. (23) finally we obtain
8
d
−2 Tr θ4 (γµ)ab[y∗,A][θ¯ ,B}[θ ,C}+(γ )ab(−1)PA[θ ,B}[θ¯,C}[y∗,A]
dlnΛ µ b a µ a b µ
(cid:16) (cid:16)
−4i[θa,[θ ,A}}[θ¯b,B}[θ¯ ,C} +cyclic perm. of A, B, C. (30)
a b
(cid:17)(cid:17)
In the Appendix we prove the identity
Tr θ4 (γµ)ab[y∗,A][θ¯ ,B}[θ ,C}+(γ )ab(−1)PA[θ ,B}[θ¯,C}[y∗,A]
µ b a µ a b µ
(cid:16) (cid:16)
−4i[θa,[θ ,A}}[θ¯b,B}[θ¯ ,C} +cyclic perm. of A, B, C
a b
1 (cid:17)(cid:17)
= Tr θ4(γµ)ab y∗,A[θ¯ ,B}[θ ,C}+(−1)PA[θ ,B}[θ¯ ,C}A
3 µ b a a b
(cid:16) h i
+cyclic perm. of A, B, C. (31)
Thus, the sum of all remaining diagrams is given by an integral of a total derivative. However,
this integral is not equal to 0, because singularities, proportional to δ-functions, can appear.
Taking them into account, it is possible to obtain Eq. (22) [31].
5 Conclusion
In this paper we investigate a conjecture that in supersymmetric theories, regularized by
higher covariant derivatives, all integrals, defining the β-fucntion, are integrals of total deriva-
tives. This fact allows to calculate at least one of the integrals analytically, so that the fac-
torization of integrands into total derivatives is in the close connection with the NSVZ exact
β-function.
For N = 1 SQED, regularized by higher derivatives, we presented qualitative arguments,
which demonstrate an origin of the factorization of the integrands into total derivatives. Al-
though they are applicable in all orders, the proof, presented in this paper, was not made by
a completely rigorous method. Also, we have not considered contributions of the Pauli–Villars
fields. Nevertheless, this proof allows to understand, why the factorization of integrands into to-
tal derivatives takes place. (Another proof was given in Ref. [19].) The complete rigorous proof
(forN = 1SQED),which, inparticular, includescalculation of Pauli–Villars fieldscontributions
will be described in a future paper.
Acknowledgements.
This work was supported by RFBR grant No 08-01-00281a. I am very grateful to A.L.Kataev,
A.A.Slavnov, and A.V.Smilga for valuable discussions.
A Integrals, defining the three-loop β-function
Herewepresentexpressionsforthethree-loop integrals. Actually, they wereobtainedin Ref.
[18], but here we write them in a more beautiful form. The β-function is given by the following
integrals:
9
β(α0) = d d−1(α ,Λ/p)−α−1 = 16π(A +A +A ), (32)
α2 dlnΛ 0 0 p=0 1 2 3
0
(cid:16) (cid:17)(cid:12)
(cid:12)
where5 (cid:12)
1 d4q 1 d d M2
A = − c ln(q2+M2)+ I ; (33)
1 2 I (2π)4q2dq2dlnΛ I q2+M2
XI Z (cid:16) I (cid:17)
d4q 1 d d d4k 1 1 q4
A = −2e2 − c
2 (2π)4q2dq2dlnΛ (2π)4k2R2 (k+q)2 J(q2+M2)2
Z Z k J J
X
1 e2 Λ d4t 2e2
× R 1+ ln −
((k+q)2+M2) k 4π2 µ (2π)4 t2(k+t)2
J !" (cid:16) (cid:17) Z
d4t 2e2
+ c ; (34)
I (2π)4 (t2+M2)((k+t)2+M2)
I Z I I #
X
d4q 1 d d d4k d4l 4e4 1 1
A = −
3 (2π)4 q2dq2dlnΛ (2π)4(2π)4 k2R l2R (q+k)2 2(q+l)2
k l( "
Z Z
2k2 1 q4 1
+ − − c ×
(q+k+l)2(q+l)2 (q+k+l)2 I(q2+M2)2((q+k)2 +M2)
# I I I
X
1 2k2
× − +
2((q +l)2+M2) ((q+k+l)2+M2)((q+l)2+M2)
" I I I
1 2M2
− + I
((q+k+l)2+M2) ((q+k)2+M2)((q+k+l)2+M2)
I I I
2M2 2M2
+ I + I . (35)
(q2+M2)((q+l)2+M2) ((q+l)2+M2)((q+k+l)2+M2)
I I I I #)
Here for simplicity we denote R ≡ R(k2/Λ2). A is a one-loop result. A is a sum of two-
k 1 2
loop diagrams, three-loop diagrams with two loops of the matter superfields, and diagrams with
insertions of counterterms, arising from renormalization of the coupling constant. A is a sum
3
of three-loop diagrams with a single loop of the matter superfields.
Thetwo-pointGreenfunctionofthemattersuperfieldinthetwo-loop approximation isgiven
by the following integrals:
d4k 2e2 1 d4t 2e2 1 d4t
lnG= − 0 1− 0 + c
(2π)4 k2R (k+q)2 R (2π)4 t2(k+t)2 IR (2π)4
k " k k
Z Z I Z
X
2e2 d4k d4l e4 2
× 0 + 0 −
(t2+M2)((k+t)2+M2) (2π)4(2π)4 k2R l2R (q+k)2(q+l)2
I I # Z k l
4 8k2−4q2
− + . (36)
(q+k)2(q+k+l)2 (q+k)2(q+k+l)2(q+l)2
!
5This result was also presented in Ref. [32], but some sign in A3 were written incorrectly.
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