Patterns of gauge symmetry in the background field method

The correlation functions of Yang–Mills theories formulated in the background field method satisfy linear Slavnov–Taylor identities, which are naive generalizations of simple tree level relations, with no deformations originating from the ghost-sector of the theory. In recent years, a stronger version of these identities has been found to hold at the level of the background gluon self-energy, whose transversality is enforced separately for each special block of diagrams contributing to the gluon Schwinger–Dyson equation. In the present work we demonstrate by means of explicit calculations that the same distinct realization of the Slavnov–Taylor identity persists in the case of the background three-gluon vertex. The analysis is carried out at the level of the exact Schwinger–Dyson equation for this vertex, with no truncations or simplifying assumptions. The demonstration entails the contraction of individual vertex diagrams by the relevant momentum, which activates Slavnov–Taylor identities of vertices and multi-particle kernels nested inside these graphs; the final result emerges by virtue of a multitude of extensive cancellations, without the need of performing explicit integrations. In addition, we point out that background Ward identities amount to replacing derivatives of propagators by zero-momentum background-gluon insertions, in exact analogy to standard properties of Abelian gauge theories. Finally, certain potential applications of these results are briefly discussed.

The BFM is a powerful framework that enables the implementation of the gauge-fixing procedure necessary for quantizing gauge theories without losing explicit gauge invariance, in contradistinction to the conventional quantization schemes [54][55][56][57][58][59][60][61][62][63].The starting point of the BFM is the splitting of the gauge field A a µ appearing in the classical action of the theory according to A a µ = B a µ + Q a µ , where B a µ and Q a µ are the background and quantum (fluctuating) fields, respectively.The quantum field is the variable of integration in the generating functional Z(J), and external sources are coupled only to it, as J • Q.The background field does not enter in loops; it couples externally to Feynman diagrams, connecting them with the asymptotic states to form S-matrix elements.Then, by virtue of a special gauge-fixing condition, the resulting action (with the corresponding ghost terms included) is no longer invariant under transformations of the quantum field, but retains its invariance intact with respect to the background field [5,61].
A key consequence of the background gauge invariance of the action is that Green's functions involving the B a µ field satisfy ghost-free Slavnov-Taylor identities (STIs), akin to the Takahashi identity known from QED: the STIs are straightforward generalization of tree level relations, receiving no ghost-related contributions after the inclusion of quantum corrections.Instead, in the standard STIs [66,67] of the R ξ gauges [68], starting already at one-loop, the ghost sector modifies these tree level relations non-trivially.To fix the ideas with a simple example, the quark-gluon vertex with either a B or a Q gluon satisfies at tree level the simple identity (suppressing color) q µ γ µ = / q = (− / p − m) − (/ r − m) = S −1 0 (− / p) − S −1 0 (/ r), where S 0 is the tree level version of the quark propagator S. In the case of a B gluon, the all-order STI is obtained from the above tree level identity by simply substituting S 0 → S, namely q µ Γ µ (q, r, p) = S −1 (− / p) − S −1 (/ r).Instead, in the case of a Q gluon (R ξ gauges), the STI gets modified by quantum corrections [66,67], which induce a dependence on the ghost dressing function and the quark-ghost kernel [69][70][71][72].
As was pointed in earlier studies, the STI satisfied by the background-gluon self-energy, Π µν (q), namely the standard transversality condition q µ Π µν (q) = 0, is implemented in a very special way, which has been denominated "block-wise" [8,64,65].Specifically, the diagrammatic representation of the SDE governing Π µν (q) is composed by four distinct blocks, namely one-and two-loop diagrams containing only gluons, and one-and two-loop diagrams containing ghost fields, as shown in Fig. 1.The block-wise realization of the STI in this case is the simple statement that the transversality of Π µν (q) is enforced independently for each of the four blocks.This is in sharp contrast to what happens in the R ξ gauges, where, already at the one-loop perturbative level, it is only the sum of gluon and ghost diagrams that is transverse [73,74].The proof of this property is particularly simple; it proceeds by contracting the various diagrams by q µ from the side of the fully dressed vertices, thus triggering the corresponding naive STIs of the BFM.It is important to emphasize that the proof holds for any value of the gauge-fixing parameter ξ Q used to define the propagators of the Q-type gluons entering in the various loops.
The basic question that arises naturally in this context is whether the special blockwise realization of the STI described above is particular to the two-point function, or if it is a common feature of all Green's functions containing only B gluons.In the present work we take a first step in the exploration of this issue, and demonstrate that the same pattern persists in the STI of the BFM vertex with three incoming background gluons, to be denoted by Γ αµν (q, r, p).This Abelian STI relates the contraction q µ Γ αµν (q, r, p) to the difference Π µν (r) − Π µν (p) [5,64,75].It turns out that the diagrams comprising the SDE of Γ αµν (q, r, p) may also be classified into four subsets in a way completely analogous to the case of Π µν (q).Then, the contraction of each subset by q µ generates the difference of the corresponding subsets of Π µν , confirming the block-wise realization of this STI.We emphasize that, as in the case of the gluon self-energy, this property is completely ξ Q -independent.
An additional noteworthy aspect of the BFM Green's functions is the Ward identities (WIs) they satisfy, namely the relations that emerge when the momentum that triggers the STIs is taken to vanish.For example, in the case of the Γ µ (q, r, p) mentioned above, a Taylor expansion of the STI around q = 0 and subsequent matching of terms linear in q yields the relation Γ µ (0, −p, p) = ∂S −1 ( / p)/∂p µ , which is the precise equivalent of the text-book WI known from QED, relating the photon-electron vertex with the electron propagator [73,74].
In fact, exactly as happens in QED, this WI admits a simple diagrammatic interpretation: the derivative of the inverse quark propagator may be depicted as the insertion of a background gluon carrying zero momentum.These observations may be straightforwardly extended to the case of the three-gluon vertex Γ αµν (q, r, p), allowing for a completely analogous pictorial representation of the corresponding WI.In fact, the block-wise realization of the STI leads to a corresponding pattern for the WIs that emerges from it: the derivative acting on any of the blocks of Π µν (q) is identical to the diagrams comprising the associated block of Γ αµν (q, r, p), when the corresponding momentum is set to zero.To the best of our knowledge, the notions described above appear for the first time in the literature.
The article is organized as follows.In Sec.II we review certain pivotal properties of the BFM, and explain the notion of the block-wise transversality at the level of the SDE that governs the background gluon propagator.Sec.III contains the main result of this work, namely the demonstration of the block-wise realization of the STI for the case of the background three-gluon vertex.Then, in Sec.IV we focus on the WIs of the BFM, their graphical representation in terms of zero-momentum gluon insertions, and demonstrate the block-wise realization of the three-gluon WI, for the operationally simplest subset of graphs.In Sec.V we summarize our findings and discuss future directions.Finally, in four Appendices we present complementary material that facilitates the perusal of the article.

II. GENERAL THEORETICAL FRAMEWORK
In this section we highlight some of the significant features of the BFM formalism that are relevant for the demonstrations that follow; for further details the reader is referred to the extensive literature on the subject, see, e.g., [5,59,63].
(i) The initial decomposition of the gauge field into B a µ and Q a µ components increases considerably the number of Green's functions that can be defined, which may be classified into three broad subsets: those with B a µ fields only, those with Q a µ fields only (corresponding to the standard Green's functions of the R ξ gauges), and mixed ones, with both B a µ and Q a µ fields.We will occasionally denote Green's functions according to the type of incoming fields, such as "BB" for the case of the propagator connecting two background fields, or "BBB" for the case of the three-gluon vertex connecting three such fields.
In what follows we will use extensively a number of three-and four-particle vertices, which we list here.In particular, the relevant three-particle vertices are while the four-particle vertices are where g denotes the gauge coupling constant, and f abc are the SU(3) structure constants.
(v) In order to elucidate with a simple example how this special transversality is enforced at the diagrammatic level, we consider the case of Π (2) µν (q), whose diagrams are enclosed by the orange box of Fig. 1.
The diagrams (a 3 ) and (a 4 ) are given by where we have used the Feynman rules given in Eq. (B2) and Eq.(B6) of the Appendix B, and factored out the trivial color structure δ ab from both expressions.In addition, we have defined where C A is the Casimir eigenvalue of the adjoint representation [N for SU(N )].Furthermore, we have introduced where the use of a symmetry-preserving regularization scheme is implicitly assumed.
We next contract graph (a 3 ) µν (q) by q ν , thus triggering the STI satisfied by Γ ν (q − k, k, −q), given in Eq. (C2), to obtain which is exactly the negative of the contraction q ν (a 4 ) µν (q).Hence, We emphasize that the above strategy of contracting directly individual diagrams and triggering the corresponding STIs will be followed unaltered in the more complicated case of the three-gluon vertex treated in the next section.Note finally that the entire demonstration leading to Eq. (2.8) is carried out for a general value of the gauge-fixing parameter ξ Q [64,76,77].

III. BLOCK-WISE STI OF THE THREE-GLUON VERTEX
In this section we demonstrate the block-wise realization of the STI satisfied by the BBB three-gluon vertex.

A. General considerations
(i) The one-particle irreducible three-gluon vertex, Γ abc αµν (q, r, p), is defined from the vacuum expectation value of the time ordered product of three background gluons (in momentum space), as The three-gluon vertex Γ abc αµν (q, r, p) is naturally cast in the form where the tree level component Γ αµν (q, r, p) coincides with that of the conventional three-gluon vertex (QQQ), i.e., while Γ abc αµν (q, r, p) captures all quantum corrections, both perturbative and nonperturbative.(ii) The SDE that defines Γ abc αµν (q, r, p) is shown diagrammatically in Fig. 2, written with respect to the gluon that carries momentum q; therefore, the corresponding vertices to which this leg is attached are kept at tree level.The corresponding Feynman diagrams have been FIG. 2. The block-wise structure of the SDE which describes the three-gluon vertex Γ abc αµν (q, r, p) (BBB).The blue circles represent full one-particle irreducible vertices, while the purple ones the four-and five-point scattering kernels.
classified in four blocks, applying the exact same criterion as in the case of Π µν (q), and employing the same color code for the individual boxes as in Fig. 1.Thus, where, as shown in Fig. 2, the four blocks are comprised by the diagrams (suppressing indices) (iii) It is well-known that Γ abc αµν (q, r, p) satisfies the Abelian STI [5,64,75] and cyclic permutations thereof.The STI of Eq. (3.6) may be obtained by means of formal manipulations of the BFM generating functional, or simply from the STI of the conventional QQQ vertex [69,78,79], by setting all ghost-related contributions to their tree level values.
From Eq. (3.3) it is elementary to show that p ν Γ (0) αµν (q, r, p) = r 2 P αµ (r) − q 2 P αµ (q) .(3.7) Then, from Eqs. (2.5), (3.2) and (3.6) follows that (iv) The central observation of the present study is that, as happens in the case of Eq. (2.6), the STI in Eq. (3.8) admits a block-wise realization.Specifically, as we will demonstrate in this section, In diagrammatic terms, Eq. (3.9) states that the contraction by p ν of the diagrams within a given block (color) in Fig. 2 generates the difference between diagrams within the corresponding block (color) of Fig. 1.In fact, the validity of Eq. (3.9) will be demonstrated by acting with p ν on vertex diagrams, and exploiting the STIs triggered by this contraction in order to cast the result in the form of self-energy contributions.
(v) Note that the diagrams shown in Fig. 2 have a factor g removed from them, which cancels against the g appearing in the definition of the three-gluon vertex in Eq. (3.1).In addition a factor of i will be factored out, which will cancel against the explicit i appearing on the r.h.s. of Eq. (3.9).Thus, on the r.h.s. of all intermediate formulas will appear directly the self-energy diagrams (a i ) of Fig. 1.Furthermore, with the exception of Sec.IV, we will suppress the argument (q, r, p) in all vertex graphs.
(vi) We introduce the definitions (vii) We introduce the short-hand notation The special relations will be employed in the analysis that follows.Their validity may be established by appealing to the Bose symmetry of the BQQ vertex with respect to its two quantum legs or the ghostantighost symmetry of the Bcc vertex, and the change of integration variable r − k → k.

An alternative demonstration proceeds by noting that
with where the STI of Eq. (C2) was used.The first relation in Eq. (3.12) may be proved in the exact same way, employing the STI of Eq. (C1).
(viii) Lastly, from now on we adopt the convention that 1PI vertices containing at least one background gluon will be represented diagrammatically by a blue circle (see, e.g., the BBQQ vertex in diagram (b 3,2 ) of Fig. 3).

B. One-loop gluonic sector (first block)
We begin by considering the one-loop gluonic vertex graphs, namely the set {(b 1 ), (b 2 ), (b 3 )}, enclosed in the blue box of Fig. 2.
As a first step, we recognize that, by virtue of Eq. (3.12), diagrams (b 1 ) and (b 2 ) vanish, given in Fig. 3, with The contraction p ν (b 3 ) abc αµν may be evaluated using the STIs in Eqs.(C1) and (C4) and a moderate amount of algebra, yielding The first term in Eq. (3.18) is exactly −f bce (a 1 ) ae αµ (q).The second term, after k → −k − p, generates f cae (a 1 ) be αµ (r); note, in particular, that where, due to Eq. (3.12), the last three terms give vanishing contributions.Thus, one arrives at At this point, we add and subtract on the r.h.s. of Eq. (3.20) the momentum-independent seagull graph (a 2 ), to obtain which, in view of Eq. (3.15), is precisely Eq. (3.9) for i = 1.
C. One-loop ghost sector (second block) We next focus on the one-loop ghost graphs, forming the set {(c 1 ), (c 2 ), (c 3 )}, enclosed in the orange box of Fig. 2. The demonstration that follows is completely analogous to that of the previous subsection.Due to Eq. (3.12), diagrams (c 1 ) and (c 2 ) vanish, and we only need to consider the contraction of graph (c 3 ).This diagram contains three contributions, depicted in Fig. 4, with The contraction of diagram (c 3 ) by the momentum p ν activates the STIs of Eqs.(C2) and (C6), and one obtains.
The second term of Eq. (3.25) is simply −f bce (a 3 ) ae αµ (q), while the first term, after the shift k → −k − p and use of Eq. (3.12), furnishes f cae (a 3 ) be αµ (r).Thus, we conclude that which, after adding and subtracting the momentum-independent (a 4 ), is tantamount to the validity of Eq. (3.9) for i = 2.

D. Two-loop gluonic sector (third block)
We turn to the two-loop dressed gluonic contributions, contained within the diagram (d), enclosed by the purple box in Fig. 2. To that end, in Fig. 5 we show the individual diagrams that emerge upon implementing the skeleton expansion of the five-gluon kernel (purple circle) inside (d).Note that all vertices appearing in these graphs are one-particle irreducible.The diagrams in Fig. 5 where and This particular separation is motivated by the observation that the contraction by p ν of each subset generates a concrete term of the STI, namely where the self-energy diagrams (a 5 ) and (a 6 ) (purple box in Fig. 1) are given by with s = q − k − l.In passing from the first to the second expression for (a 5 ), the explicit form of Γ (0)adme αβσρ , given in Eq. (B3), has been used.The contraction of the momentum p ν with the above diagrams will activate a series of STIs, which will furnish the desired structures, together with a considerable number of terms that will cancel exactly among each other.In what follows, we briefly outline how this calculation may be best organized.
We begin with the diagrams comprising (d 1 ) abc αµν .The action of p ν on (d 1,1 ) abc αµν leads to the contraction p ν R ρλ ν (l, −p − l), triggering the STI of Eq. (C1), and yielding The second term in the above expression will give where the ellipsis denotes terms that do not contribute to the r.h.s. of Eq. (3.31).We now change the integration variables as k → −k and l → −l − p, and exploit Lorentz invariance to replace Γ bnmd µλητ (r, −l, −t, −k) → Γ bnmd µλητ (−r, l, t, k).Then, substituting the Feynman rule of Eq. (B3) and using the Bose symmetry of the vertices, we arrive at which is precisely f cae (a 5 ) be µν (r), in the form given in the second line of Eq. (3.32).Similarly, the contraction of (d 1,2 ) abc αµν by p ν activates the STI for the five-point function Γ cbemd νµλητ (p, r, l, s, k), given in Eq. (C7), namely Note that only the first term on the r.h.s. of Eq. (3.36) contains the four-point function with a q entry, since r + p = −q.Thus, one gets which is precisely f bce (a 5 ) ae (q).
After appropriate changes in the integration variables and judicious use of Bose symmetry, one may show that all remaining terms, denoted by the ellipses in Eqs.(3.35) and (3.37), cancel against each other.Thus, one is left with the first equation in Eq. (3.31).
A similar line of reasoning reveals that the term f cae (a 6 ) be αµ (r) originates from the contraction of p ν with the diagrams (d 2,2 ) and (d 2,4 ) of the second group.Specifically, one triggers the STI of Eq. (C1) to obtain Using the Feynman rule given by Eq. (B3) for the tree level vertex one gets that the substitution of the first term into Eq.(3.38) gives precisely f cae (a 6 ) be αµ (r), while the ellipsis contains the terms that will cancel.
A close inspection of these expressions reveals that FIG. 6. Contributions from the diagram (e) in Fig. 2 after expanding the five-point kernel represented by the purple blob.This group is organized in the three subsets (e 1 ), (e 2 ), and (e 3 ).
It is then a matter of straightforward algebra to demonstrate that all terms contained in the ellipses of Eqs.
leaving Eq. (3.43) as the final result.Thus, the validity of Eq. (3.9) for i = 4 is confirmed.
The final conclusion drawn from the analysis presented in subsections III B, III C, III D, and III E is that the block-wise realization of the STI announced in subsection III A, holds.
Notice, in fact, that the validity of Eq. (3.9) has been demonstrated for an arbitrary value of the gauge-fixing parameter ξ Q .

IV. ABELIAN WARD IDENTITIES WITH BACKGROUND GLUONS
In this section we derive Abelian WIs from the STIs satisfied by the BFM vertices, and apply to them the text-book diagrammatic representation for the WIs known from QED [73].
In addition, we demonstrate the block-wise realization of the WI that connects the vertex Γ αµν (0, r, −r) with the derivative of ∆(r).Here we define the notation of a perforated circle as being the derivative acting on the Green's function.
As is well-known in the context of Abelian gauge theories, such as spinor or scalar QED, the implementation of the limit q → 0 of the Takahashi identity gives rise to the corresponding WI.In order to fix the ideas consider the latter theory, describing the interaction of a photon with a complex scalar, where the full photon-scalar vertex Γ µ (r, p, q) satisfies the Abelian STI (Takahashi identity) with D(p) denoting the fully dressed propagator of the scalar field.Then, the standard WI is determined by expanding both sides of Eq. (4.1) around q = 0, and equating the linear terms.Specifically, this procedure yields or, equivalently, ∂ D(r) The version of the WI given in Eq. (4.3) admits the text-book diagrammatic interpretation: the derivative of the propagator D(r) is equivalent to the insertion of a zero-momentum photon in it [73].
It turns out that the Abelian STIs satisfied by the BFM three-point functions give rise to WIs completely analogous to that of Eq. (4.3), which admit the same diagrammatic interpretation given above, but now in terms of zero-momentum insertions of a background gluon.The simplest case is that of the ghost-gluon vertex Γ µ (r, p, q), whose WI is identical to that of Eqs.αµν (q, r, p) given in Eq. (B1), capturing correctly its dependence on the gauge-fixing parameter ξ Q .
We next focus our attention on the WIs satisfied by BFM vertices with more than three incoming fields.As a concrete example, consider the vertex BBcc; when contracted with respect to the momentum carried by one of the background gluons, it satisfies the STI given by Eq. (C6).Expanding both sides of Eq. (C6) around q = 0, and using the Jacobi identity to eliminate the zeroth order term on the r.h.s., we obtain the WI Exactly analogous expressions may be deduced for higher point Green's functions; for a formal derivation of the STI satisfied by a general vertex of the form BQ n , see Appendix A.
To prove Eq. (4.8), we first symmetrize the process of differentiation of (a 3 ) αµ by shifting the loop momentum (k → u − k, with u = q/2), to get with (a 3,1 ) αµν (q) = −2λ the last three contributions are depicted graphically in the first line of Fig. 9.

V. DISCUSSION AND CONCLUSIONS
It has been known for some time [64] that the transversality of the background self-energy is enforced in a special way, namely independently for each one of the four subsets (blocks) of diagrams comprising the corresponding SDE.In the present work we have shown that the Abelian STI of the background three-gluon vertex is also realized according to the exact same pattern, at the level of the corresponding SDE: the momentum contraction of each subset of vertex diagrams generates the difference of the corresponding self-energy subsets.
The demonstration of this property has been carried out at the level of the fully dressed Feynman diagrams that comprise the relevant SDEs.In particular, the contraction of all three-gluon vertex diagrams by the appropriate momentum triggers STIs satisfied by the vertices and the kernels embedded in them, giving rise to crucial rearrangements and cancellations, which are implemented algebraically, with no need to resort to any integrations.
Note that the extensive reorganization of diagrams observed here has been first identified in the context of the pinch technique [5,[80][81][82][83], where the "gauge-invariant" three-gluon vertex was first studied at the one-loop level [75,84,85].Evidently, it would be particularly interesting to explore the origin of the block-wise STIs at a formal level, and establish its validity by means of the Batalin-Vilkovisky functional machinery [5,[86][87][88][89][90] .
It is natural to conjecture that the STI of the background four-gluon vertex, B 4 , given by [84,91] q µ Γ mnrs µαβγ (q, r, p, t) = f mse f ern Γ αβγ (r, p, q + t) + f mne f esr Γ βγα (p, t, q + r) + f mre f ens Γ γαβ (t, r, q + p) , is realized according to the same block-wise pattern described above.A diagrammatic demonstration along the lines presented in this work appears to be quite feasible, and would give further support to the notion that the STI of any B n -type of vertex is enforced in this characteristic manner.
Some of the results presented in Sec.IV may be used in order to explore the numerical impact of certain truncations or approximations, in the spirit of the recent study presented in [92].For example, the equality shown in Fig. 9 will be distorted if the vertex BBcc were to be replaced by its tree level value, given by Eq. (B6).The amount of discrepancy induced between the two sides of this equation is a quantitative indicator of the veracity of such an approximation.
Throughout the present analysis we have assumed that the BBB vertex does not contain irregularities in the form of massless poles.However, as has been shown in detail in a series of studies, the emergence of a dynamical gluon mass [80] through the operation of the Schwinger mechanism [93,94]  (q, p) (q − r) ν g αµ + (r − p) α g µν + (p − q) µ g αν (B1)

FIG. 1 .
FIG.1.Diagrammatic representation of the self-energy Π µν (q).The small gray circles at the end of gluon legs indicate a background field.The orange and green circles represent conventional fully dressed propagators and vertices, respectively, while the blue circles represent fully dressed vertices with one background gluon.

FIG. 3 .
FIG. 3.The two contributions which arise from diagram (b 3 ) of Fig.2after performing the skeleton expansion of the four-gluon scattering kernel (purple blob).

FIG. 4 .
FIG. 4. The three contributions which emerge from the diagram (c 3 ) shown in the Fig. 2, after implementing the skeleton expansion of the four-point scattering kernel (purple blob) formed by two background gluons with a ghost-antighost pair.

FIG. 7 .
FIG. 7. Diagrammatic representation of the WI for the Bcc vertex, where the derivative of the ghost propagator can be identified as the insertion of a zero-momentum background gluon leg.

FIG. 8 .
FIG.8.representation of the WI for the BQQ vertex, where the derivative of the gluon propagator can be identified as the insertion of a zero-momentum background gluon leg.