arXiv:hep-ph/9406271v1 10 Jun 1994 lentv a fDrvn h ic Technique’s Pinch the Deriving of Way Alternative ∗ § ‡ † upre npr yteMnuh rn-nAdfrScientifi for [email protected] Grant-in-Aid Monbusho address: Scientifi the e-mail for by Grant-in-Aid part Monbusho in Scientifi the Supported for by Grant-in-Aid part Monbusho in the Supported by part in Supported by h aeWr dniyta a on ihPT. thi with that found show was that and identity BFM Ward in same vertex the v four-gluon obeys derived were the which results calculate same leve also the one-loop get at we gluons that for demonstrate vertices gauge- three-point obta construct and we are energy illustration which For results (PT). gauge-invariant technique pinch same the deriving of H AKRUDFEDMETHOD: FIELD BACKGROUND THE hj Hashimoto Shoji eso httebcgon edmto BM sasml way simple a is (BFM) method field background the that show We et fPyis ooaaNtoa University National Yokohama Physics, of Dept. et fPyis iohm University Hiroshima Physics, of Dept. iah-iohm 2,JAPAN 724, Higashi-Hiroshima ∗ ioKODAIRA Jiro , ooaa20 JAPAN 240, Yokohama e SASAKI Ken Results Abstract † § n ohaiYASUI Yoshiaki and eerhN.050076. No. Research c C-05640351. No. Research c 040011. No. Research c a 1994 May YNU-HEPTh-94-104 HUPD-9408 yBMand BFM by l nain self- invariant ndb the by ined aP.We PT. ia vertex s ‡ 1 Introduction

Formulation of a gauge theory begins with a gauge invariant Lagrangian. However, except for lattice gauge theory, when we quantize the theory in the continuum we are under compulsion to fix a gauge. Consequently, the corresponding Green’s functions, in general, will not be gauge invariant. These Green’s functions in the standard formulation do not directly reflect the underlying gauge invariance of the theory but rather obey complicated Ward identities. If there is a method in which we can construct systematically gauge-invariant Green’s functions, then it will make the computations much simpler and may have many applications. Along this line exist two approaches: One is the pinch technique and the other, the background field method. The pinch technique (PT) was proposed some time ago by Cornwall [1] [2] for a well-defined algorithm to form new gauge-independent proper vertices and new propagators with gauge-invariant self-energies. Using this technique Cornwall and Papavassiliou obtained the one-loop gauge-invariant self- energy and vertex parts in QCD [3] [4]. Later it was shown [5] that PT works also in spontaneously broken gauge theories, and since then it has been applied to the standard model to obtain a gauge-invariant electromagnetic form factor of the neutrino [5], one-loop gauge-invariant W W and ZZ self-energies [6], and γW W and ZW W vertices [7]. On the other hand, the background field method (BFM) was first introduced by DeWitt [8] as a technique for quantizing gauge field theories while retaining ex- plicit gauge invariance. In its original formulation, DeWitt worked only for one-loop calculations. The multi-loop extension of the method was given by ’t Hooft [10], De- Witt [9], Boulware [11], and Abbott [12]. Using these extensions of the background field method, explicit two-loop calculations of the β function for pure Yang-Mills theory was made first in the Feynman gauge [12]- [13], and later in the general gauge [14]. Both PT and BFM have the same interesting feature. The Green’s functions

1 (gluon self-energies and proper gluon-vertices, etc) constructed by the two methods retain the explicit gauge invariance, thus obey the naive Ward identities. As a result, for example, a computation of the QCD β-function coefficient is much simplified. Only thing we need to do is to construct the gauge-invariant gluon self-energy in either method and to examine its ultraviolet-divergent part. Either method gives the same correct answer [3] [12]. Thus it may be plausible to anticipate that PT and BFM are equivalent and that they produce exactly the same results. In this paper we show that BFM is an alternative and simple way of deriving the same gauge-invariant results which are obtained by PT. Although the final results obtained by both methods are gauge invariant, we have found, in particular, that the BFM in the Feynman gauge corresponds to the intrinsic PT. In fact we explicitly demonstrate, for the cases of the gauge-invariant gluon self-energy and three-point vertex, both methods with the Feynman gauge produce the same results which are equal term by term. We also give the gauge-invariant four-gluon vertex calculated in BFM and show explicitly that this vertex satisfies the same simple Ward identity that was found with PT. The paper is organized as follows. In Sec.2 we review the intrinsic PT and explain how the gauge-invariant proper self-energy and three-point vertex for gluon were derived in PT. In Sec.3 we write down the Feynman rule for QCD in BFM and compute the gauge-invariant gluon self-energy at one-loop level in BFM with the Feynman gauge. The result is shown to be the same, term by term, as the one obtained by the intrinsic PT. The BFM is applied, in Sec.4, to the calculation of the three-gluon vertex. The result is shown to coincide, again term by term, with the one derived by the intrinsic PT. In Sec.5 we compute the gauge-invariant four-gluon vertex at one-loop level in BFM. We present each contribution to the vertex from the individual Feynman diagram. Then we show that the acquired four-gluon vertex satisfies the same naive Ward identity that was found with PT and is related to the gauge-invariant three-gluon vertex obtained previously by PT and BFM.

2 2 The intrinsic pinch technique

There are three equivalent versions of the pinch technique: the S matrix PT [1] - [3], the intrinsic PT [3] and the Degrassi-Sirlin alternative formulation of the PT [6]. To prepare for the later discussions and to establish the notations, we briefly review, in this section, the intrinsic PT and explain the way how the gauge-invariant proper self-energy and three-point vertex for gluons at one-loop level were obtained in Ref. [3]. In the S-matrix pinch technique we obtain the gauge-invariant effective gluon propagator by adding the pinch graphs in Fig. 1(b) and 1(c) to the ordinary prop- agator graphs [Fig. 1(a)]. The gauge dependence of the ordinary graphs is canceled by the contributions from the pinch graphs. Since the pinch graphs are always missing one or more propagators corresponding to the external legs, the gauge- dependent parts of the ordinary graphs must also be missing one or more external propagator legs. So if we extract systematically from the proper graphs the parts which are missing external propagator legs and simply throw them away, we obtain the gauge-invariant results. This is the intrinsic PT introduced by Cornwall and Papavassiliou [3]. We will now derive the gauge-invariant proper self-energy for gluons of gauge group SU(N) using the intrinsic PT. Since we know that PT successfully gives gauge-invariant quantities, we use the Feynman gauge. Then the ordinary proper self-energy whose corresponding graphs are shown in Fig. 2 is given by 2 D 0 iNg d k 1 Πµν = D 2 2 2 Z (2π) k (k + q) ×[Γαµλ(k, q)Γλνα(k + q, −q) − kµ(k + q)ν − kν(k + q)µ], (1) where we have symmetrized the ghost loop in Fig. 2(b) and omitted a trivial group- theoretic factor δab. We assume the dimensional regularization in D = 4 − 2ε dimensions. The three-gluon vertex Γαµλ(k, q) has the following expression [15]:

Γαµλ(k, q) ≡ Γαµλ(k,q, −k − q)

3 = (k − q)λgαµ +(k +2q)αgµλ − (2k + q)µgλα. (2)

Here and in the following we make it a rule that whenever the external momentum appears in the three-gluon vertex, we put it in the middle of the expression, that is, F like qµ in Eq. (2). Now we decompose the vertices into two pieces: a piece Γ which has the terms with the external momentum q and a piece ΓP (P for pinch) carries the internal momenta only.

F P Γαµλ(k, q) = Γαµλ +Γαµλ, F Γαµλ(k, q) = −(2k + q)µgλα +2qαgµλ − 2qλgαµ, (3) P Γαµλ(k, q) = kαgµλ +(k + q)λgαµ.

The full vertex Γαµλ(k, q) satisfies the following Ward identities:

α −1 −1 k Γαµλ(k, q) = Pµλ(q)d (q) − Pµλ(k + q)d (k + q)

λ −1 −1 (k + q) Γαµλ(k, q) = Pαµ(q)d (q) − Pαµ(k)d (k) (4) where we have defined

−2 −1 2 Pµν (q)= −gµν + qµqν q , d (q)= q . (5)

The rules of the intrinsic PT are to let the pinch vertex ΓP act on the full vertex and to throw out the d−1(q) terms thereby generated. We rewrite the product of two full vertices ΓαµλΓλνα as

F F P P P P ΓαµλΓλνα =ΓαµλΓλνα +ΓαµλΓλνα +ΓαµλΓλνα − ΓαµλΓλνα (6)

Using the Ward identities in Eq. (4) we find that the sum of the second and third terms of Eq. (6) turns out to be

P P −1 ΓαµλΓλνα +ΓαµλΓλνα = 4Pµν(q)d (q) −1 −2Pµν(k)d (k)

−1 −2Pµν(k + q)d (k + q). (7)

4 We drop the first term on the RHS of (7) following the intrinsic PT rule. Now we use the dimensional regularization rule (which we adhere to throughout this paper)

dDkk−2 =0, (8) Z and discard the parts which disappear after integration, then the second and third terms can be written as

−1 −1 − 2Pµν(k)d (k) − 2Pµν(k + q)d (k + q)= −2kµkν − 2(k + q)µ(k + q)ν. (9)

Also applying the dimensional regularization rule Eq.(8) to the fourth term on the RHS of Eq.(6), we find

P P − ΓαµλΓλνα = −2kµkν − (kµqν + qµkν). (10)

Now combining the first term on the RHS of Eq.(6) with Eqs.(9) and (10), and inserting them into Eq.(1), we arrive at the following expression for the gauge- invariant self-energy [3]

iNg2 dDk 1 Πµν = D 2 2 2 Z (2π) k (k + q) b F F ×[Γαµλ(k, q)Γλνα(k + q, −q) − 2(2k + q)µ(2k + q)ν]. (11)

The same rules are applied to obtain the gauge-invariant three-gluon vertex at one-loop level. The contributions of the graphs depicted in Fig.3 to the ordinary proper three-gluon vertex are summarized as [15]

2 D 0 iNg d k 1 Γµνα = − D 2 2 2 Nµνα (12) 2 Z (2π) k1k2k3 Nµνα = Γσµλ(k2, q1)Γλνρ(k3, q2)Γρασ(k1, q3)

+k1νk2αk3µ + k1αk2µk3ν. (13) where the momenta and Lorentz indices are defined in Fig.3(a) and the overall group-theoretic factor gf abc is omitted.

5 F P −1 Decomposing Γ into Γ +Γ and dropping the terms involving d (qi) which are generated by application of ΓP to the full vertices, Cornwall and Papavassiliou obtained the following expression for gauge-invariant proper three-gluon vertex:

Γµνα(q1, q2, q3)= iNg2 dDk 1 b F F F − D 2 2 2 Γσµλ(k2, q1)Γλνρ(k3, q2)Γρασ(k1, q3) 2 Z (2π) k1k2k3 

+2(k2 + k3)µ(k3 + k1)ν(k1 + k2)α  −8(q1αgµν − q1ν gµα)A(q1) − 8(q2µgαν − q2αgµν )A(q2)

e −8(q3νgµα − q3µgνα)A(eq3) , (14)  e where A(qi) is defined by

e dDk 1 A q . ( i)= D 2 2 (15) Z (2π) k (k + qi) e

3 The background field method and the gauge- invariant gluon self-energy

In this section we write down the Feynman rules for QCD in the background field calculations and compute the gluon self-energy in the Feynman gauge. Then we will see that the result coincides, term by term, with the gauge-invariant gluon self-energy derived via the intrinsic PT. In BFM, the field in the classical lagrangian is written as A + Q, where A (Q) denotes the background (quantum) field. The Feynman diagrams with A on external legs and Q inside loops need to be calculated. The relevant Feynman rules [12] are given in Fig.4. It is noted that the Feynman rule for the ghost-A vertex is similar to the one which appears in the scalar QED. Now let us write a three-point vertex

6 b with one Aµ field as

abc bac Γλµν (p,q,r) = gf Γλµν (p,q,r) (16) 1 1 Γe (p,q,r) = (p − qe + r) g +(q − r − p) g λµν ξ ν λµ ξ λ µν e +(r − p)µgνλ. (17)

Then we find that in the Feynman gauge ξ = 1, Γαµλ(k,q, −k − q) turns out to be

e Γαµλ(k,q, −k − q)|ξ=1 = −2qλgαµ +2qαgµλ − (2k + q)µgλα, (18)

e F which coincides with the expression of Γαµλ in Eq.(3). This fact gives us a hint that BFM may reproduce the same results which are obtained by the intrinsic PT (and we find later that it is true in fact). Now we calculate the gluon self-energy in BFM with the Feynman gauge. The relevant diagrams are depicted in Fig.5. The diagram 5(a) gives a contribution iNg2 dDk 1 (a) F k, q F k q, q , Πµν = D 2 2 Γαµλ( )Γλνα( + − ) (19) 2 Z (2π) k (k + q) b F F where we have used the fact Γαµλ|ξ=1 = Γαµλ and Γλνα|ξ=1 = Γλνα. On the other hand, through the scalar QED-likee coupling for thee background field and ghost vertices, the diagram 5(b) gives iNg2 dDk 1 (b) k q k q . Πµν = D 2 2 [−2(2 + )µ(2 + )ν] (20) 2 Z (2π) k (k + q) It is interestingb to note that the contributions of diagrams 5(a) and 5(b) , respec- tively, correspond to the first and second terms in the parentheses of Eq.(11) and the sum of the two contributions coincides with the expression of the gauge-invariant self-energy Πµν which was derived in section 2 by the method of intrinsic PT.

b 4 The gauge-invariant three-gluon vertex

The success in deriving the gauge-invariant PT result for the gluon self-energy by BFM gives us momentum to study, for the next step, the gauge-invariant three-gluon

7 vertex at one-loop level. The relevant diagrams are shown in Fig.6, where momenta and Lorentz and color indices are displayed. With the fact that an AQQ vertex in F the Feynman gauge, Γξ=1, is equivalent to Γ in Eq.(3), it is easy to show that the contribution of the diagrame 6(a) is 3 abc D (a) abc iNg f d k 1 Γµνα(q1, q2, q3) = − D 2 2 2 2 Z (2π) k1k2k3 F F F × Γσµλ(k2, q1)Γλνρ(k3, q2)Γρασ(k1, q3) . (21)   The contribution of the diagram 6(b) (and the similar one with the ghost running the other way) is

3 abc D (b) abc iNg f d k 1 Γµνα(q1, q2, q3) = − D 2 2 2 2 Z (2π) k1k2k3

× 2(k2 + k3)µ(k3 + k1)ν(k1 + k2)α . (22)   When we calculate the diagram 6(c), again we use the Feynman gauge (ξ = 1) for the four-point vertex with two background fields. Remembering that the diagram 1 6(c) has a symmetric factor 2 and adding the two other similar diagrams, we find 3 abc (c) abc iNg f Γµνα(q1, q2, q3) = 8(q1αgµν − q1νgµα)A(q1) 2  e +8(q2µgαν − q2αgµν)A(q2)+8(q3ν gµα − q3µgνα)A(q3) .(23)  Finally, the contribution of the diagram 6(d) (ande two other similar diagrams)e turns out to be null because of the group-theoretical identity for the structure constants f abc such as f ead(f dbxf xce + f dcxf xbe)=0. (24)

Now adding the contributions from the diagrams (a)-(c) in Fig. 6 and omitting the overall group-theoretic factor gf abc, we find that the result coincides with the expression of Eq.(14) which was obtained by the intrinsic PT. Also we note that each contribution from the diagrams (a)-(c), respectively, corresponds to a particular term in Eq.(14).

8 Finally we close this section with a mention that the constructed Γµνα(q1, q2, q3) is related to the gauge-invariant self-energy Πµν of Eq.(11) through a Wardb identity [3]

µ b q1 Γµνα(q1, q2, q3)= −Πνα(q2)+ Πνα(q3), (25) which is indeed a naive extensionb of the tree-levelb one.b

5 The gauge-invariant four-gluon vertex and its Ward identity

The gauge-invariant four-gluon vertex has been constructed by Papavassiliou [4] using the S-matrix PT. As he stated in Ref. [4], the construction was a nontrivial task because of the large number of graphs and certain subtleties of PT. Although he did not report the exact closed form of the gauge-invariant four-gluon vertex, he showed that the new four-gluon vertex is related to the previously constructed Γµνα in Eq.(14) through a simple Ward identity. In this section we apply BFM withb the Feynman gauge to obtain the gauge-invariant four-gluon vertex at one-loop level. We give the closed form of this vertex and show that it satisfies the same Ward identity which was proved by Papavassiliou. 2 abcd The bare four-gluon vertex in Fig.7(a) is expressed as −ig Γµναβ with

abcd abx cdx Γµναβ = f f (gµαgνβ − gµβgνα) acx dbx + f f (gµβgαν − gµνgαβ)

adx bcx + f f (gµνgβα − gµαgβν), (26)

abc while the bare three-gluon vertex in Fig.7(b) is expressed as gΓµνλ(k1,k2,k3) with

abc abc Γµνλ(k1,k2,k3)= f Γµνλ(k1,k2,k3) (27)

µ abcd and Γµνλ(k1,k2,k3) is given by Eq.(2). Now acting with q1 on Γµναβ, we get

µ abcd abx cdx q1 Γµναβ = f f (q1αgνβ − q1βgνα)

9 acx dbx + f f (q1βgαν − q1νgαβ) adx bcx + f f (q1ν gβα − q1αgβν). (28)

Next with a help of the Jacobi identity

f abxf cdx + f acxf dbx + f adxf bcx =0, (29) we add

0 = f abxf cdx + f acxf dbx + f adxf bcx  

× (q4 − q3)νgαβ +(q2 − q4)αgβν +(q3 − q2)βgνα (30)   to the RHS of Eq.(28) and we obtain the tree-level Ward identity [15]

µ abcd abx cdx q1 Γµναβ = −f Γαβν (q3, q4, q1 + q2) acx dbx −f Γβνα(q4, q2, q1 + q3) adx bcx −f Γναβ(q2, q3, q1 + q4). (31)

The bare three- and four-gluon vertices are manifestly gauge independent. However, if we consider the usual one-loop corrections to these vertices, they become gauge dependent and do not satisfy Eq.(31) any more. We now apply BFM to the case of the four-gluon vertex and show that the con- structed gauge-invariant vertex satisfies the generalized version of the Ward identity in Eq.(31). The diagrams for the four-gluon vertex at one-loop level are shown in 1 Fig.8. It is noted that the two-gluon loop diagrams 8(e) have a symmetric factor 2 . For later convenience let us introduce the following group-theoretic quantities:

f(abcd) ≡ f almf bmnf cnef del, (32) which satisfy the relations

f(abcd)= f(bcda)= f(badc) (33) N f(abcd) − f(abdc)= − f abxf cdx. (34) 2

10 The last relation is derived from an identity for the structure constants f abc N f almf bmnf cnl = f abc (35) 2 and the Jacobi identity Eq.(29). It is straightforward to evaluate the diagrams in Fig.8, and the relevant momenta, color and Lorentz indices are indicated in the graphs. The relations for color factors in Eqs.(33)-(34) are extensively used. We present each contribution to the vertex from the individual Feynman diagram, expecting that each contribution corresponds to the particular term of the intrinsic PT result once the calculation is made in the future. The results are the following.

[a]: The diagrams 8(a) give

dDk 1 (a) abcd q , q , q , q ig2f abcd Γµναβ( 1 2 3 4)= ( ) D 2 2 2 2 Z (2π) k (k + q2) (k − q1 − q4) (k − q1) b F F F F × Γλµρ(k − q1, q1)Γρντ (k, q2)Γτακ(k + q2, q3)Γκβλ(k − q1 − q4, q4)  

+ (q2, b, ν) ←→ (q3,c,α) + (q3,c,α) ←→ (q4,d,β) , (36)     where the notation (q2, b, ν) ←→ (q3,c,α) represents a term obtained from the   first one by the substitution (q2, b, ν) ←→ (q3,c,α). The same notation applies to the third term, and also to the expressions below. [b]: The diagrams 8(b) give

dDk 1 (b) abcd q , q , q , q ig2f abcd Γµναβ( 1 2 3 4)= − ( ) D 2 2 2 2 Z (2π) k (k + q2) (k − q1 − q4) (k − q1) b × 2(2k − q1)µ(2k + q2)ν(2k − q1 − q4 + q2)α(2k − 2q1 − q4)β  

+ (q2, b, ν) ←→ (q3,c,α) + (q3,c,α) ←→ (q4,d,β) . (37)     [c]: The diagrams 8(c) give

dDk 1 (c) abcd q , q , q , q ig2 Γµναβ( 1 2 3 4)= D 2 2 2 Z (2π) k (k + q2) (k − q1) b 11 F F × − f(abcd)+ f(abdc) gαβΓλµρ(k − q1, q1)Γρνλ(k, q2)    abx cdx F F F F +Nf f Γβµλ(k − q1, q1)Γλνα(k, q2) − Γαµλ(k − q1, q1)Γλνβ(k, q2)  

+ (q2, b, ν) ←→ (q3,c,α) + (q2, b, ν) ←→ (q4,d,β)    

+ (q1,a,µ) ←→ (q3,c,α) + (q1,a,µ) ←→ (q4,d,β)    

+ (q1,a,µ) ←→ (q3,c,α), (q2, b, ν) ←→ (q4,d,β) . (38)   [d]: The diagrams 8(d) give dDk 1 (d) abcd q , q , q , q ig2 Γµναβ( 1 2 3 4)= D 2 2 2 Z (2π) k (k + q2) (k − q1) b × f(abcd)+ f(abdc) 2gαβ(2k − q1)µ(2k + q2)ν  

+ (q2, b, ν) ←→ (q3,c,α) + (q2, b, ν) ←→ (q4,d,β)    

+ (q1,a,µ) ←→ (q3,c,α) + (q1,a,µ) ←→ (q4,d,β)    

+ (q1,a,µ) ←→ (q3,c,α), (q2, b, ν) ←→ (q4,d,β) . (39)   [e]: The diagrams 8(e) give

(e) abcd 2 Γµναβ(q1, q2, q3, q4)= ig A(q1 + q2) abx cdx b × f(abcd)+ f(abdce ) Dgµνgαβ +4Nf f (gµαgνβ − gµβgνα)   

+ (q2, b, ν) ←→ (q3,c,α) + (q2, b, ν) ←→ (q4,d,β) . (40)     where A(qi) is defined in Eq.(15).

[f]: Thee diagrams 8(f) give

(f) abcd 2 Γµναβ(q1, q2, q3, q4)= −ig A(q1 + q2) f(abcd)+ f(abdc) 2gµνgαβ   b e + (q2, b, ν) ←→ (q3,c,α) + (q2, b, ν) ←→ (q4,d,β) . (41)     abcd Then the final form of the gauge-invariant four-gluon vertex Γµναβ at one-loop level is given by the sum b

abcd (a) abcd (b) abcd (c) abcd Γµναβ = Γµναβ + Γµναβ + Γµναβ

b b b b 12 (d) abcd (e) abcd (f) abcd + Γµναβ + Γµναβ + Γµναβ (42)

b b b abcd Our next task is to show explicitly that the above four-gluon vertex Γµναβ satisfies the following generalized version of the Ward Identity in Eq.(31): b

µ abcd abx cdx q1 Γµναβ = −f Γαβν(q3, q4, q1 + q2) acx dbx b −f Γbβνα(q4, q2, q1 + q3) adx bcx −f Γb ναβ(q2, q3, q1 + q4), (43) where b abc abc Γµνλ(q1, q2, q3)= f Γµνλ(q1, q2, q3) (44) and Γµνλ(q1, q2, q3) is theb gauge-invariant three-gluonb vertex given in Eq.(14). The

Wardb identity Eq.(43), first found by Papavassiliou with PT [4], is naturally expected to hold in BFM formalism. µ abcd We act with q1 on the individual contributions to Γµναβ which are expressed in

Eqs.(36)-(41). Before going through the evaluation we makeb some preparations. Let us introduce the following integrals for the three-point vertex with the constraint q1 + q2 + q3 = 0: dDk 1 B q , q , q µνα( 1 2 3) ≡ D 2 2 2 Z (2π) k (k + q2) (k − q1) F F F × Γλµρ(k − q1, q1)Γρντ (k, q2)Γταλ(k + q2, q3) ,   dDk 1 C q , q , q µνα( 1 2 3) ≡ D 2 2 2 Z (2π) k (k + q2) (k − q1) ×(2k − q1)µ(2k + q2)ν(2k − q1 + q2)α. (45)

In terms of Bµνα and Cµνα, the gauge-invariant three-gluon vertex Γµνλ in Eq.(14) is expressed as b iNg2 Γµνλ(q1, q2, q3) = − Bµνα(q1, q2, q3)+2Cµνα(q1, q2, q3) 2  b −8(q1αgµν − q1ν gµα)A(q1) − 8(q2µgαν − q2αgµν)A(q2)

e−8(q3ν gµα − q3µgνα)A(q3) e. (46)  e 13 These Bµνα and Cµνα satisfy the following relations:

Bµνα(q1, q2, q3) = Bναµ(q2, q3, q1)= −Bνµα(q2, q1, q3), (47)

Cµνα(q1, q2, q3) = Cναµ(q2, q3, q1)= −Cνµα(q2, q1, q3), (48) which can be proved by changing integration variables under the constraint q1 +q2 + q3 = 0 and using the fact

F F Γλµρ(k, q)= −Γρµλ(−k − q, q). (49)

Throughout the algebraic manipulations, we often take the means of changing the integration variables under the constraint q1 + q2 + q3 + q4 = 0 and make use of identities

µ F 2 2 q1 Γλµρ(k − q1, q1) = (k − q1) − k gλρ,   µ 2 2 q1 (2k − q1)µ = k − (k − q1) (50) to reduce the number of propagators by one, the relations of Eqs.(47)-(49) for Bµνα, F Cµνα and Γλµρ to classify terms into groups with the same color factors. We also use the identity of Eq.(34). The results are

[a]:

µ (a) abcd q1 Γµναβ(q1, q2, q3, q4)= dDk ΓF (k − q , q )ΓF (k, q ) igb2 f abcd f abdc q λαρ 3 3 ρβλ 4 ( )+ ( ) 1ν D 2 2 2   Z (2π) (k − 3) k (k + q4) abx cdx 1 +Nf f Bαβν (q3, q4, q1 + q2) 2 dDk ΓF (k − q , q )ΓF (k, q ) − (ν ←→ λ) qλ λαρ 3 3 ρβν 4 + 1 D 2 2 2 Z (2π) (k − 3) k (k + q4)  + cyclic permutations (51)   where {cyclic permutations} represents two terms which are obtained from the first term by the substitution {(q2, b, ν) → (q3,c,α) → (q4,d,β) → (q2, b, ν)} and the

14 substitution {(q2, b, ν) → (q4,d,β) → (q3,c,α) → (q2, b, ν)}. The same notation applies to the expressions below. [b]:

µ (b) abcd q1 Γµναβ(q1, q2, q3, q4)= dDk (2k − q ) (2k + q ) igb 2 f abcd f abdc q 3 α 4 β ( )+ ( ) (−2 1ν ) D 2 2 2   Z (2π) (k − 3) k (k + q4) abx cdx +Nf f Cαβν(q3, q4, q1 + q2)  + cyclic permutations (52)   [c]:

µ (c) abcd q1 Γµναβ(q1, q2, q3, q4)=

b 2 ig f(abcd)+ f(abdc) −Dq1ν gαβA(q1 + q2)   dDek ΓF (k − q , q )ΓF (k, q ) q λαρ 3 3 ρβλ 4 − 1ν D 2 2 2 Z (2π) (k − 3) k (k + q4)  abx cdx +Nf f 4(q2αgβν − q2βgνα) A(q1 + q2) − A(q2)    dDk ΓF (k − qe, q )ΓF (k, qe) − (ν ←→ λ) qλ λαρ 3 3 ρβν 4 − 1 D 2 2 2 Z (2π) (k − 3) k (k + q4) ! + cyclic permutations (53)   [d]:

µ (d) abcd q1 Γµναβ(q1, q2, q3, q4)= 2 igb f(abcd)+ f(abdc) 2q1νgαβA(q1 + q2)   e dDk (2k − q ) (2k + q ) q 3 α 4 β +2 1ν D 2 2 2 Z (2π) (k − 3) k (k + q4)  + cyclic permutations . (54)   [e]:

µ (e) abcd 2 q1 Γµναβ(q1, q2, q3, q4) = ig f(abcd)+ f(abdc) Dq1ν gαβA(q1 + q2)   b e 15 abx cdx +Nf f 4(q1αgβν − q1βgνα)A(q1 + q2)  + cyclic permutations e (55)   [f]:

µ (f) abcd 2 q1 Γµναβ(q1, q2, q3, q4) = ig f(abcd)+ f(abdc) (−2q1ν )gαβA(q1 + q2)   b + cyclic permutations . e (56)  

Adding together all the contributions [Eqs.(51)-(56)], we find

2 µ abcd ig N abx cdx q1 Γµναβ(q1, q2, q3, q4) = f f Bαβν(q3, q4, q1 + q2)+2Cαβν(q3, q4, q1 + q2) 2  b −8 (q1 + q2)βgνα − (q1 + q2)αgνβ A(q1 + q2)   e +8 q2βgνα − q2αgνβ A(q2)    + cyclic permutations e (57)   It is noted that all the terms which are proportional to factors [f(abcd)+ f(abdc)], [f(acdb)+ f(acbd)], and [f(adbc)+ f(adcb)] cancel out and only terms with factors Nf abxf cdx, Nf acxf dbx, and Nf adxf bcx remain in the final result. The last step is to add ig2N 0 = f abxf cdx + f acxf dbx + f adxf bcx 2  

× −8(q2βgνα − q2αgνβ)A(q2)  −8(q3ν gαβ − q3βgαν)eA(q3)

−8(q4αgβν − q4ν gβα)Ae(q4) (58)  e to the RHS of Eq.(57) and to use Eq.(46), and we arrive at the desired result of Eq.(43).

16 6 Conclusions

In this paper we demonstrated that the background field method is an alternative and simple way of deriving the same gauge-invariant results which are obtained by the pinch technique. We have found, in particular, in the cases of gauge-invariant gluon self-energy and three-gluon vertex, that both BFM in the Feynman gauge and the intrinsic PT produce the same results which are equal term by term. We also calculated the gauge-invariant four-gluon vertex in BFM and presented its exact form. Finally we explicitly showed that this four-gluon vertex satisfies the same simple Ward identity that was found with PT. We already know that PT works in spontaneously broken gauge theories, es- pecially, in the standard model. It will be very interesting to investigate whether BFM may reproduce the same results for the one-loop gauge-invariant W W and ZZ self-energies and γW W and ZW W vertices which were constructed by PT.

Acknowledgements

One of the authors (K.S) would like to thank Professor D. Zwanziger and Joannis Papavassiliou for the hospitality extended to him in the spring of 1994 at New York University where part of this work was done, and Professor T. Muta for the hospitality extended to him at Hiroshima University. He is also very grateful to Joannis Papavassiliou for informative and helpful discussions on the pinch technique.

17 References

[1] J. M. Cornwall, in Proceedings of the French-American Seminar on Theoretical Aspects of Quantum Chromodynamics, Marseille, France, 1981, edited by J. W. Dash (Centre de Physique Th´eorique, Marseille, 1982).

[2] J. M. Cornwall, Phys. Rev. D26, 1453 (1982).

[3] J. M. Cornwall and J. Papavassiliou, Phys. Rev. D40, 3474 (1989).

[4] J. Papavassiliou, Phys. Rev. D47, 4728 (1993).

[5] J. Papavassiliou, Phys. Rev. D41, 3179 (1990).

[6] G. Degrassi and A. Sirlin, Phys. Rev. D46, 3104 (1992).

[7] J. Papavassiliou and K. Philippides, Phys. Rev. D48, 4255 (1993).

[8] B. S. DeWitt, Phys. Rev. 162, 1195 (1967); also in Dynamical Theory of Groups and Fields (Gordon and Breach, New York, 1963).

[9] B. S. DeWitt, in Quantum Gravity 2, edited by C. J. Isham, R. Penrose, and D. W. Sciama (Oxford University Press, New York, 1981).

[10] G. ’t Hooft, in Acta Universitatus Wratislavensis No. 368, XIIth Winter School of Theoretical Physics in Karpacz, February-March, 1975; also in Functional and Probabilistic Methods in Quantum Field Theory, Vol. I.

[11] D. G. Boulware, Phys. Rev. D23, 389 (1981).

[12] L. F. Abbott, Nucl. Phys. B185, 189 (1981); Acta Physica Polonica B13, 33 (1982).

[13] S. Ichinose and M. Omote, Nucl. Phys. B203, 221 (1982).

[14] D. M. Capper and A. MacLean, Nucl. Phys. B203, 413 (1982).

18 F P [15] The definition of three-gluon vertex Γαµλ, and therefore, of Γαµλ and Γαµλ in this paper is different from the corresponding ones in Ref. [3] by the overall factor −1.

19 Figure captions Fig.1 The S-matrix pinch technique applied for the elastic scattering of two fermions. Graphs (b) and (c) are pinch parts, which, when added to the ordinary propagator graphs (a), yield the gauge-invariant effective gluon propagator. Fig.2 0 Graphs for the ordinary proper self-energy Πµν. (a): Gluon-loop. (b): Ghost-loop. Momenta and Lorentz indices are indicated. Fig.3 0 Graphs for the ordinary proper three-gluon vertex Γµνλ. (a): Gluon-loop. (b)(c): Ghost-loops. Momenta and Lorentz indices are indicated. Fig.4 Feynman rules for background field calculations in QCD. The wavy lines terminating in an A represent external gauge fields. The other wavy lines and dashed lines represent Q fields and ghost fields, respectively. Only shown are rules which are requisite for calculations in this paper. Fig.5

Graphs for a calculation of the gauge-invariant self-energy Πµν in BFM. (a):Gluon- loop. (b) Ghost-loop. Momenta and Lorentz indices are indicated. b Fig.6

Graphs for a calculation of the gauge-invariant three-gluon vertex Γµνλ in BFM. (a)(c): Gluon-loops. (b)(d): Ghost-loops. Momenta and Lorentz indices are indi- b cated. Fig.7 The bare four-gluon vertex (a) and the bare three-gluon vertex (b). Momenta, and color and Lorentz indices are indicated. Fig.8

Graphs for a calculation of the gauge-invariant four-gluon vertex Γµναβ in BFM. (a)(c)(e): Gluon-loops. (b)(d)(f): Ghost-loops. Momenta, and color and Lorentz b indices are indicated. (a)

(b)

(c) Fig.1 k+q

qµ qν

k

(a)

k+q

qµ qν

k

(b)

Fig.2 q2

k3=k

k1 q1 k2

q3

(a)

q2 q2

k3 k3 k1 k1 q1 q1 k2 k2

q3 q3 (b) (c) Fig.3 a b −iδab kµkν g − (1 − ξ) µ ν k2 + iε µν k2 k " #

iδ a b ab k2 iε k +

a,λ p b A abc 1 1 µ gf (p − q + r)νgλµ +(q − r − p)λgµν +(r − p)µgνλ q " ξ ξ # r c,ν

a p b A abc µ −gf (p + q)µ q c a,µ d,ρ A

2 abx xcd 1 −ig [f f (gµλgνρ − gµρgνλ + ξ gµνgλρ) adx xbc 1 +f f (gµν gλρ − gµλgνρ − ξ gµρgνλ) acx xbd +f f (gµν gλρ − gµρgνλ)]

A b,ν c,λ

a b

2 acx xdb adx xcb −ig gµν(f f + f f )

A A c,µ d,ν k+q

A A qµ qν

k

(a)

k+q

A A qµ qν

k

(b)

Fig.5 b,ν A

q2

k3 a,µ k1 A q1

k2

q3 A c,α

(a)

A

A

A

(b) b,ν A q k+q 2 a,µ A + 2 permutations q1 q3 k

A c,α

(c) b,ν A

e a,µ A + 2 permutations

d

A c,α

(d)

Fig.6 a,µ

q1 d,β q4 b,ν

2 abcd q2 −ig Γµναβ

q3

c,α

(a) b,ν

q2 a,µ

abc q1 gΓµνλ

q3 c,λ

(b) Fig.7 A a,µ

q1

k-q1 k d,β b,ν A A q4 q2 + 2 permutations

k-q1-q4 k+q2

q3 A c,α

(a)

A a,µ

q1

k k-q1 + 5 permutations d,β b,ν A A q4 k+q2 q2 q3

A c,α

(c) A

A A + 2 permutations

A

A

+ A A + 2 permutations

A

(b) A a,µ

q1

k-q1 k d,β b,ν + 5 permutations A A q4 k+q2 q2 q3

A c,α

A a,µ

q1

k-q1 k + d,β b,ν + 5 permutations A A q4 k+q2 q2 q3

A c,α

(d) A a,µ

q1

k A q2 b,ν d,β + 2 permutations q4 A k+q1+q2

q3

A c,α

(e)

A a,µ

q1

k A q2 b,ν + 5 permutations d,β q4 A k+q1+q2

q3

A c,α

(f) Fig.8 A

q1

k-q1 k

A A q4 q2 k+q k-q1-q4 2

q3 A A a,µ

q1

k-q1 k d,β b,ν A A q4 k+q2 q2 q3

A c,α