Page 1

arXiv:1202.1368v1 [hep-ph] 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 MG-6, Bucharest-Magurele, 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 well-known 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)

Page 6

6

In this approach the second term in Eq (24) will give no contribution since it will appear as a product of three

functional derivatives corresponding to the spin dependent, spin independent and ghost terms in the one loop potential

and this would lead to a result proportional to a gauge invariant (in the background gauge field) of order higher than

two.

VI.CONNECTING THE DOTS

We add the results from Eq(14), Eq (23) and Eq(27) to obtain for the second order correction:

− ig217

6N2X2

?

d4x(Fa

µν)2

(28)

Here X is just the result of the regularization at one loop. After taking into account all gauge and internal indices

X amounts to a one loop scalar integral so one can write schematically for the proper loop result:

≈

?

d4xd4yFa

µνU(x − y)Faµν(y) (29)

The two loop expression then corresponds to:

≈

?

d4xd4yFa

µν(UU)(x − y)Faµν(y)(30)

where (UU)(x − y) is the result of the scalar two loop diagram with two bubbles and two external legs. But this

regularized is just the square of U regularized at one loop so practically we do not need it. So finally we will take for

X the expression:

X = i

1

(4π)2

?1

0

ln(

xΛ2

−x(1 − x)k2) = i

1

(4π)2(1 + lnΛ2/k2)(31)

We multiply by a loop factor1

2to obtain the second order contribution to the coupling constant:

ig2(k)

4

?

d4x(Fa

µν)2= ig2

4[1 −11N

3

1

(4π)2lnM2/k2−34

3N2 g2

(4π)4lnM2/k2+ ...]

?

d4x(Fa

µν)2

(32)

From that the known result for the two loop beta function is obtained:

β(g2) =

g4

(4π)2[−11

3N −34

3N2g2

16π2].(33)

Here we defined β(g) =

dg2

dln(µ2).

VII.DISCUSSION

It is important to know the beta function for non-abelian gauge theories for several reasons. First the one loop

coefficient of beta function was the main clue that these theories are endowed with asymptotic freedom. Second higher

order coefficients can reveal information about the phase structure of these type of models. And it is always useful to

learn more about the mathematical structures that lie at the basis of contemporary particle physics.

In the present work we do not aim to obtain higher order correction to the beta function but rather to introduce

a new method that can ease their calculation. All previous computations of order higher than two of beta function

rely heavily on computer technique, rightly so because for example the fourth order involves about 50.000 Feynman

diagrams. We determine the two loop coefficient of the beta function as a check of the method. In the process we do

not compute any two loop diagram and the computation in entirely analytic.

Our procedure can be easily extended to higher orders and be adjusted to work with dimensional regularization

and in MS scheme. This would amount to small changes in the general set-up. We leave all these for future work.

Acknowledgments

I am happy to thank J. Schechter for support and encouragement and for useful comments on the manuscript. This

work has been supported by PN 09370102/2009.

Page 7

7

Appendix A

First let us review some basic steps regarding integration in the functional approach. We start with the simple

formula:

?

k

?

dξkexp[−ξiBijξj] =

?

k

?

dxkexp[−bix2

i] =

?

i

?π

bi

= const[detB]−1/2

(A1)

In what follows we will drop the constant factors.

We extend this to a slightly more complicated case; assume the following:

?

k

?

dξkexp[ξiBijξj]exp[ξiDijξj] =

?

i

?

π

det(B + D)

(A2)

Note that B and D correspond in our case to the spin dependent and spin independent operators respectively in

the quadratic part of the Lagrangian. Let us now differentiate one of the above factors with respect to a quantity

Hmwhere the index includes any type of subscript (Note that in the end this Hmwill be the background gauge field

tensor or any component of it).

?

k

−1

?

ξkexp[−ξiBijξj]

δ

δHmexp[−ξiDijξj] =

2[det(B + D)]−1/2?

i

δdi

δHm

1

di+ bi

(A3)

Here diand biare the eigenmodes of D and B. The operators B and D are diagonalized together in the one loop

effective potential such that,

−ξiBijξj⇒ −bx2

−ξiDijξj⇒ −dix2

i+ xiKijxj

i− xiK′

ijxj

(A4)

where diand biare proportional to the square of the background gauge field tensor (Fa

contain higher powers of the background gauge field. It is perfectly safe to ignore corrections of order Kijor K′

We need to compute higher order derivatives of various types:

µν)2whereas Kij and K′

ij

ij.

?

k

[−1

×[det(B + D)]−1/2.

?

dξkexp[−ξiBijξj]

δ2

δHmδHnexp[−ξiDijξj] =

2

δ2di

δHnδHm

1

bi+ di

+

δdi

δHm

δdj

δHn(1

4

1

(bi+ di)(bj+ dj)+12

1

(bi+ di)2δij)]

(A5)

Furthermore,

?

k

[δbk

δHp

δbk

δHp

−1

4

?

dξk

δ

δHp

exp[−ξiBijξj]

δ2

δHmδHnexp[−ξiDijξj] =

δ2di

δHmδHn(1

δdi

δHm

(bi+ di)2(bk+ dk)δij+1

4

1

(bi+ di)(bj+ dj)+1

δHn(−1

8

1

2

1

(bi+ di)2δik) +

δdj

1

(bi+ di)(bj+ dj)(bk+ dk)−1

1

(bi+ di)3δijδik)][

4

1

(bi+ di)2(bj+ dj)δik−1

4

1

(bi+ di)2(bj+ dj)δjk

2

?

l

(bl+ dl)]−1/2.(A6)

We treat separately the ghost terms. Thus,

?

k

?

dθkdθ∗

kexp[θ∗

iCijθj] =

?

k

?

dzidz∗

iexp[−ci|zi|2] =

?

i

ci= det[C](A7)

Page 8

8

from which we can deduce,

?

k

?

dθkdθ∗

k

δ

δHmexp[θ∗

iCijθi] =

?

k

?

dzkdz∗

k

δ

δHmexp[−z∗

icizi] =

δci

δHm

1

ci[

?

k

ck](A8)

and further,

?

k

?

dθkdθ∗

k

δ2

δHmδHnexp[θ∗

iCijθi] =

?

k

?

dzkdz∗

k

δ2

δHmδHnexp[−z∗

icizi] =

δ2ci

δHmδHn

1

ci[

?

k

ck] +

δci

δHm

δcj

δHn[−1

c2

i

δij+

1

cicj][

?

k

ck]. (A9)

It is useful to give in what follows some results regarding the functional derivatives of the square of the gauge tensor

(Here Fa

mνta= Fµν, where tais the generator in the adjoint representation).

δ2

δBρ

ac(x)δBσ

de(y)[

?

d4zTr(Fµν)2(z)] =

?

8i[(Fρσ)ae(x)δcdδ(x − y) − (Fρσ)dcδaeδ(x − y)].

d4z2(Fµν)gh(z)

δ2(Fµν)hg

δBρ

ac(x)δBσ

de(y)+δ(Fµν)gh(z)

δBρ

ac(x)

δ(Fmuν

hg

δBσ

(z)

de(y)

=

(A10)

Furthermore from this one can deduce:

?

d4xd4yd4ud4vδ(x − u)δ(y − v)facmfden×

δ2

δ(Bρ)ac(x)δ(Bσ)de(y)exp[k1

δ2

δFmn

?

?

?

d4z(Fa

µνFaµν)(z)] ×

×

ρσ(u)δ(δ(u − v))exp[k2

= −16ik1k2

d4z(Fa

µνFaµν)(z)] =

d4x(Fa

µν)2× exp[(k1+ k2)

?

d4z(Fa

µνFaµν)(z)](A11)

[1] G ’t Hooft, Phys. Rev. D 14, 3432, 1976.

[2] C. G. Callan, R. Dashen and D. J. Gross, Phys. Rev. D 17, 2717, 1978;19, 1826, 1979.

[3] A. Vainshtein, V. Zakharov, V. Novikov and M. Shifman, Sov. Phys. Usp. 25, 195, 1982.

[4] R. Jora, Phys. Rev. D 82, 056005, 2010; arXiv:1004.3660.

[5] R. Jora, arXiv:1101.1395, 2011.

[6] L. F. Abbott, Acta Phys. Polonica, B13, 33, 1982.

[7] W. E. Caswell, Phys. Rev. Lett. 33, 244, 1974; D. R. T. Jones, Nucl. Phys. B 75, 531, 1974; E. S. Egorian, O. V. Tarasov,

Theor. Mat. Fiz. 41, 26, 1979; O. V. Tarasov, A. A. Vladimirov, A. Yu Zharkov, Phys. Lett B 93, 429, 1980; S. A. Larin,

J. A. M. Vermaseren, Phys. Lett. B 303, 334, 1993; T.van Ritbergen, J. A. M. Vermaseren and S. A. Larin, Phys. Lett B

400, 379, 1997.

[8] M. E. Peskin and D.V. Schroeder, ” Quantum Field Theory”, Perseus Books Publishing, L. L. C., 1995(pg 533-543).