Do we need Feynman diagrams for higher orders perturbation theory?
ABSTRACT We compute the two and three loop corrections to the beta function for
YangMills theories in the background gauge field method and using the
background gauge field as the only source. The calculations are based on the
separation of the one loop effective potential into zero and positive modes and
are entirely analytical. No two or three loop Feynman diagrams are considered
in the process.
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Article: Quantum Field Theory
Quantum Field Theory: A Modern Perspective, Graduate Texts in Contemporary Physics. ISBN 9780387213866. Springer Science+Business Media, Inc., 2005. 01/2005;  [Show abstract] [Hide abstract]
ABSTRACT: We have calculated the value of the CallanSymanzik beta function to order g5 for nonAbelian gauge theories with fermions. We discuss internal consistency of the calculation, and consider the approach to the aymptotic energy range in such theories.Physical Review Letters 01/1974; 33:244246. · 7.73 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The analytic calculation of the threeloop QCD βfunction and anomalous dimensions within the minimal subtraction scheme in an arbitrary covariant gauge is presented. The result for the βfunction coincides with the previous calculation [O.V. Tarasov, A.A. Vladimirov and A. Yu. Zharkov, Phys. Lett. B 93 (1980) 429].Physics Letters B 01/1993; · 4.57 Impact Factor
Page 1
arXiv:1202.1368v1 [hepph] 7 Feb 2012
Do we need Feynman diagrams for higher orders perturbation theory?
Renata Jora
a ∗
aNational Institute of Physics and Nuclear Engineering, PO Box MG6, BucharestMagurele, Romania.
(Dated: February 8, 2012)
We compute the two loop correction to the beta function for Yang Mills theories in the background
gauge field method and using the background gauge field as the only source. The calculations are
based on the separation of the one loop effective potential into zero and positive modes contributions
and are entirely analytical. No two loop Feynman diagrams are considered in the process.
PACS numbers: 11.10 Ef, 11.10 Gh, 11.10 Hi, 11.15 Bt.
I.INTRODUCTION
The instanton approach for SU(N) gauge theories with or without fermions has been initiated by ’t Hooft [1] and
further developed in [2] and [3]. In this method the separation of quantum degrees of freedom into zero modes (spin
dependent) and positive modes (spin independent) is crucial. Moreoverthe so called zero modes have an ” antiscreening
” effect which is ultimately responsible for asymptotic freedom. The presence of fermions has an opposite effect. In
[4] we suggest that in essence the magnetic properties of the QCD vacuum play a decisive role in the chiral symmetry
breaking. Furthermore we show in [5] that in the process of gluino decoupling from supersymmetric QCD separation
into zero and positive modes is very important.
A very useful method for computing beta functions for the gauge coupling constant is the background gauge field
method [6] which is based on the decomposition of the gauge field into a background gauge field and a fluctuating
field, the quantum gauge field. Even from the dawn of this method the background gauge field was regarded as an
alternate source. However the regular sources J(x) and η(x), η′(x) (corresponding to the quantum gauge fields and
ghost respectively) are introduced and one uses the conventional functional formalism to derive beta function or other
loop corrections. The reason is simple; the background gauge field does not couple linearly to the other fields (as
linear terms are canceled) and it is not obvious how one can compute simply Green functions with the background
gauge field as a source.
In the present work we determine the two loop contribution to the beta function for Yang Mills theories using the
background gauge field as the only source present in the functional formalism. Of course the beta function is known
up the fourth order [7] in the MS scheme so our main interest lies in the method that we introduce and the possibility
for that to be developed for higher orders. We rely on the wellknown result of the one loop effective potential (derived
either in the perturbative or in the instanton approach) and on the decomposition of the one loop operators into spin
dependent and spin independent operators corresponding to each field. Our derivation is entirely based on an analytic
functional approach (see the Appendix) that does not involve the computation of any two loop Feynman diagram.
II. THE METHOD
The Yang Mills Lagrangian in the background gauge field method (where the gauge field is separated into Ba
and Ba
µis the background gauge field) has the expression:
µ+Aa
µ
L = −
1
4g2[Fa
1
2g2(DµAa
µν+ DµAa
ν− DνAa
µ+ fabcAb
µAc
ν]2−
−
µ)2+ ¯ ca[(−D2)ac− DµfabcAb
µ]cc
(1)
This lagrangian contains quantum gauge fields Aa
bution and a higher order one. The procedure for extracting the quadratic terms in this lagrangian is standard and
after integration leads to the one loop effective potential [8].
µand ghosts ca, ¯ caand can be separated into a quadratic contri
∗Email: rjora@theory.nipne.ro
Page 2
2
After some simplification the quadratic part of the Lagrangian can be written as:
L2= −
1
2g2[Aa
µ(−(D2)acgµν− 2fabcFbµν)Ac
ν] + ¯ ca(−(D2)ac)cc
(2)
The trilinear and quadrilinear contributions are summarized below:
L3,4 = −
1
2g2(DµAa
1
4g2fabcfadeAb
ν− DνAa
µ)fabcAbµAcν−
−
µAc
νAdµAeν+ ¯ ca(−DµfabcAb
µ)cc
(3)
Then the effective action reduces to:
eiΓ[B]=
?
?
?
DADcexp[i
?
d4x(L + Lct] =
1
4g2(Fa
1
4g2(Fa
? ?
= D⌋exp[i
?
d4x[−
?
µν)2+ Lct+ L2+ L3,4]] =
=DADcexp[i
?
d4x[−
µν)2+ L2+ Lct]]
× [1 + id4xL3,4−1
2
d4xd4yL3,4(x)L3,4(y) + ....] (4)
where Lctis the counterterm Lagrangian.
The quadratic term alone leads to the one loop effective potential whereas using standard functional procedures one
can find higher order Feynman diagrams and solve for the higher order contributions to it. Since the first two orders
of the beta function are renormalization scheme independent one may wonder if it not possible to practically deduce
two loop contributions using only Eq(4) and an entirely functional approach without considering and calculating any
two loop Feynman diagram. In what follows we will show that this is indeed the case by considering the background
gauge field as a source in the functional approach. For that first we need to express in a suitable form the quadratic
operators:
−
1
2g2[−(D2)ac− 2fabcFbµν] = −
1
2g2[−∂2+ ∆1+ ∆2+ ∆J] = −
1
2g2∆, (5)
where ∆1+ ∆2is a spin independent operator and ∆Jis a spin dependent operator,
∆1= i[∂µBb
∆2= BaµtaBb
µfabc+ Bb
µtb
µfabc∂µ]
∆j= −2fabcFbµν. (6)
No spin dependent operator acts on ghosts.
The one loop effective potential for a Yang Mills theory is obtained by computing,
exp[iΓ[B]] = exp[i
?
d4x[−
1
4g2(Fa
µν)2][det(∆G,1)]−1/2det(∆G,0), (7)
where ∆G,1refers to the gauge fields and ∆G,0to the ghost fields. This leads to:
Γ[B] = −1
4(1g2
?
d4x(Fa
µν)2+1
2ln[det(∆G,1)] − ln[det(∆G,0)]).(8)
Each operator ∆ has the decomposition from Eq (6)with the following calculated contribution to the one loop
effective potential:
−
1
4g2
1
4g2
1
4g2
?
?
?
d4x(Fa
µν)2−→ −1
4[−4N lnM2
4[2
4[−1
k2]
?
d4x(Fa
µν)2zeromodescontributionforquantumgaugefields
−
d4x(Fa
µν)2−→ −1
3N lnM2
3N lnM2
k2]
?
d4x(Fa
µν)2positivemodescontributionforquantumgaugefields
−
d4x(Fa
µν)2−→ −1
k2]
?
d4x(Fa
µν)2positivemodescontributionforghostfields.(9)
Page 3
3
However it is more convenient for us to represent these results as,
ln[det(∆1+ ∆2)G,1] =1
3NX
?
1
12NX
?
d4x(Fa
µν)2
ln[det(∆J)G,1] = −2NXd4x(Fa
µν)2
ln[det(∆1+ ∆2)G,0] =
?
d4x(Fa
µν)2, (10)
to keep track of the proper regularization procedures. The exact role of X will be revealed in section VI.
Note that in the case of dimensional regularization each term in Eq(10) will be multiplied by a different X factor
such that the infinite parts are the same whereas the finite parts are different.
In the next section we will implement a procedure for computing the two loop beta function by considering the
background gauge field as a source. Knowing the one loop effective potential before and after the path integration we
use simple results as those in the Appendix to derive the relevant contributions. The key point in the whole approach
is the separation of the one loop operators in spin dependent and spin independent ones both in the integrand and in
the final results. In order to justify that let us consider the general form of the integrals with Dij and Bij the spin
independent and spin dependent operators respectively:
??
??
i
dξiexp[−ξi(Bij+ Dij)ξj] =
k
dxkexp[−xi(Ot(B + D)O)iixi] =√π[det[Ot(B + D)O]]−1/2
(11)
Here O is the orthogonal operator which realizes the diagonalization. Taking the logarithm of the expression we
observe that at first order at least there is no interference between the eigenmodes of D and B such that Ot(B+D)O =
Bd+Dd+... where Bdand Ddare the diagonalized operators. This means that at least in the first order the operators
B and D are diagonalized by the same orthogonal matrix. We will use this feature in our computation.
III.THE QUADRILINEAR TERM
We start with the simplest contribution to the effective potential, the quadrilinear term:
?
DA−i
4g2fabcfadeAb
µAc
νAdµAeνexp[i
?
d4x[−
1
4g2(Fa
µν)2+ L2]](12)
The correct structure can be obtained from
DAig2
?
?
4
?
d4xd4yδ(x − y)
δ2
δFb
µν(x)δFbµν(y)exp[
?
?
d4xi
g2fabcFbµνAa
µAc
ν] =
=DA−i
4g2fabcfadeAb
µAc
νAdµAeνexp[ d4xi
g2fabcFbµνAa
µAc
ν] (13)
so it is clear that this operator comes only from the spin dependent part in the one loop effective potential. Then
quite clearly using the results in the appendix one finds:
?
ig2[−1
?
DA−i
4g2fabcfadeAb
?
d4xd4yδ(x − y)3
δ
δFb
?
µAc
νAdµAeνexp[i
?
d4x[−
1
4g2(Fa
µν)2+ L2]] =
?
8
d4xd4yδ(x − y)
δ2
δFb
µν(x)Fbµν(y)exp[−2NXd4zF2(z)]exp[5
3NX
?
d4zF2(z)] +
+
16
δ
δFb
µν(x)exp[−2NX
?
d4zF2(z) × oneloopcontribution × exp[−1
?
d4zF2(z)] × exp[10
3NX
?
d4zF2(z)] ×
×
µν(x)exp[−2NXd4zF2(z)] × oneloopcontribution
= ig24X2N2
3NX
?
d4zF2(z)][1 + ...](14)
For each stage of the calculations we select only the term proportional to the background gauge field invariant
Fa
µνFaµνand drop conveniently terms proportional to higher order invariants.
Page 4
4
IV.THE TRILINEAR PURE GAUGE TERMS
This contribution corresponds to the term:
−
1
2g4[
?
d4x(AaνDρAc
νAmρfacm)(x)
?
d4y(AdµDσAe
µAnσfden)(y)]exp[i
?
d4x[−
1
4g2(Fa
µν)2+ L2]](15)
It is simpler in this case to work with the gauge tensor Fa
such that,
µνtawhere tais the generator in the adjoint representation
Tr(Fa
µνtaFaµνta) = NFa
µνFaµν
(16)
and Bµ= bµctc.
First we notice that the part of the term in Eq (15) that contains covariant derivatives can be easily obtained from
the spin independent quadratic operator in accordance to:
δ[?d4xexp[−
i
2g2Aa
δ(Bρ)ac
µ(∆1+ ∆2)acAc
ν]]
=
1
g2Aa
µDρAc
νgµνexp[−
i
2g2
?
d4xAa
µ(∆1+ ∆2)acAc
ν] (17)
Then,
δ2
acδBσ
δBρ
[−i
g2Aa
de
νδρσδadδce+1
exp[−
i
2g2
?
d4xAa
µ(∆1+ ∆2)acAc
ν] =
νAc
g4Aa
νDρAc
νAd
µDσAe
µ] ×
×exp[−
i
2g2
?
d4xAa
µ(∆1+ ∆2)acAc
ν].(18)
We need two more component gauge fields which can be simply obtained from the spin dependent operator. The
desired result is finally obtained from:
δ2
δBρ
= [i
ac(x)δBσ
de(y)exp[−
i
2g2
?
d4xAa
µ(∆1+ ∆2)Ac
ν]
δ
δFmn
ρσ(u)exp[1g2
?
d4xAa
µFµν
acAc
ν] =
g4AmρAnσAaνAc
i
2g2
νδρσδµνδadδceδ(x − y) +1
g6(AaνDρAc
ν)(x)Amρ(u)(AdµDσAe
µ)(y)Anσ(u)] ×
×exp[−
?
d4xAa
µ(∆1+ ∆2)Ac
ν+1
g2
?
d4xAa
µFµν
acAc
ν].(19)
In order to get the desired term we need to multiply by facmfdenwhich will lead to the cancelation by symmetry
of the unwanted first term in the last line of Eq (19).
Eq(19) has the correct structure except for the space time dependence. We will use a small artifice in order to
correct that. First we write:
1
g2
?
d4xAmρ(x)Fmn
ρσ(x)Anσ(x) =
1
g2
?
d4ud4vAmρ(u)Fmn
ρσ(u)Anσ(v)δ(u − v).(20)
Then,
δ2
δ(δ(w1− w2)δFmn
ρσ(w1)
1
g2
?
d4ud4vAmρ(u)Fmn
ρσ(u)Anσ(v)δ(u − v) =
1
g2Amρ(w1)Anσ(w2).(21)
The desired contribution can be computed from:
−
1
2g4[
?
d4x(AaνDρAc
νAmρfacm)(x)
?
d4yAdµDσAe
µAnσfden)(y)]exp[i
?
d4x[−
1
4g2(Fa
µν)2+ L2]] =
Page 5
5
−1
2g2
δ(Bρ)ac(x)δ(Bσ)de(y)exp[−i1
δ2
δFmn
?
d4xd4yd4ud4vδ(x − u)δ(y − v)facmfden×
δ2
×
2g2
?
d4xAa
ν(∆1+ ∆2)Ac
ν] ×
×
ρσ(u)δ(δ(u − v))exp[1
Using Eq(22) and the results from Appendix A we obtain:
g2
?
d4xAmρFmn
ρσAnσ] × oneloopghostterm. (22)
−
= −g2
−1
1
2g4[
?
d4x(AaνDρAc
νAmρfacm)(x)
?
d4y(AdµDσAe
µAnσfden)(y)]exp[i
?
d4x[−
1
4g2(Fa
µν)2+ L2]] =
2
?
d4xd4yd4ud4vδ(x − u)δ(y − v)facmfden×
δ2
δ(Bρ)ac(x)δ(Bσ)de(y)exp[N
δ2
δFmn
?
2g2
3X
?
d4xF2] ×
d4xF2] × exp[5×
ρσ(u)δ(δ(u − v))× exp[−2NX
= −4ig2N2X2
?
3NX
?
d4xF2] × ghostcontribution
d4xF2(x) × oneloopcontribution.(23)
V.TERMS THAT INCLUDE GHOSTS
There is one quadratic term which contains ghosts and two higher order contributions. We will need to determine
two terms, respectively:
−1
2(Dµ¯ cafabcAb
1
2g2Dµ¯ cafabcAb
µcc)2exp[i
?
d4x[−
1
4g2(Fa
µν)2+ L2]]
?
−
µcc(DρAd
σfdefAeρAfσ)exp[id4x[−
1
4g2(Fa
µν)2+ L2]].(24)
We start by analyzing the first term in Eq(24).
Both these expressions contain the ghost fields mixed with quantum gauge fields. For the sake of simplicity we
write:
¯ ca(−DµfabcAb
µ)cc≡ Dµ¯ cafabcAb
µcc
(25)
which is true up to a total derivative. Moreover the quadratic term must also be written in a similar manner as:
¯ ca(−D2)accc≡ (−D2¯ cacc)(26)
We can switch in all these terms the order of the ghost field without problem since we are dealing with the square
of the trilinear operator. Then the analogy with the previous case is obvious and with exactly the same derivation
we obtain:
−
= −g2
−1
1
2g4[
?
d4x(caDρccAmρfacm)(x)
?
d4ycdDσceAnσfden)(y)]exp[i
?
d4x[−
1
4g2(Fa
µν)2+ L2]] =
2
?
d4xd4yd4ud4vδ(x − u)δ(y − v)facmfden×
δ2
δ(Bρ)ac(x)δ(Bσ)de(y)exp[1
δ2
δFmn
×exp[(5
= ig21
6N2X2
2g2
12NX
?
?
d4xF2] ×
×
ρσ(u)δ(δ(u − v))exp[−2NX
3N −
?
d4xF2] ×
1
12N)X
d4xF2(x) × exp[−(1
?
d4F2] × oneloopspinindependentgaugecontribution
?
3NXd4F2] × oneloopcontribution.(27)