Gauge invariance, background fields and modified Ward identities

6 Pages·0.11 MB·English

by  F. Freire

DIAS-STP-01-01
CERN-TH-2001-010
FAU-TP3-01-01
GAUGE INVARIANCE, BACKGROUND FIELDS
AND MODIFIED WARD IDENTITIES
1 F. FREIRE
0 School of Theoretical Physics, Dublin Institute for Advanced Studies,
0 10 Burlington Road, Dublin 4, Ireland
2
D. F. LITIM
n
a Theory Division, CERN,
J CH-1211 Geneva 23, Switzerland

7
J. M. PAWLOWSKI
1
Institut fu¨r Theoretische Physik III, Universit¨at Erlangen-Nu¨rnberg,

2 Staudtstraße 7, D-91058 Erlangen, Germany
v
8 In this talk the gauge symmetryfor Wilsonianflows inpure Yang-Millstheories is dis-
0 cussed. Thebackgroundfieldformalismisusedfortheconstructionofagaugeinvariant
1 effectiveaction. Thesymmetriesoftheeffectiveactionundergaugetransformationsfor
1 boththe gauge fieldandthe auxiliarybackground fieldareseparately evaluated. Mod-
0 ified Ward-Takahashi and background field identities are used in my study. Finally it
1 isshownhowthesymmetrypropertiesofthefulltheoryarerestoredinthelimitwhere
0 thecut-offisremoved.
/
h
t Introduction and aim: Wilsonian or Exact renormalisation group (ERG) equations
-
p have successfully been applied to non-perturbative phenomena in quantum field
e theories. Hence an ERG formulation of gauge theories is a promising tool for re-
h
solving open questions concerning the non-perturbative regime of these theories,
:
v e.g. confinement, chiral symmetry breaking. A key hurdle in such a task concerns
i the consistent and practicable introduction of an infra-red cut-off in a theory with
X
a non-linear local symmetry.
r
a In this talka I will discuss this difficult matter in some detail. The quest can
be presented as follow: How can it be ensured that a Wilsonian effective action
shows the gauge symmetry of the underlying full theory? Let me first explain in
more detail why this question requires a surgicallook into. Implementations of the
ERG1 are most intricate when the symmetries of the theory are deformed by the
intrinsic infra-red cut-off to this approach. As the integration of the flow equation
is carried out it is necessary to guarantee that the information about the inherent
symmetries of the theory is not washed out. In general, much work has been
devotedtoovercomingthe involvedproblemsmainly withindifferentapproachesto
non-Abelian theories.3,4,6,7 Here, for the sake of clarity, I will discuss this problem
for pure Yang-Mills theory. Moreover I use an approach to these theories within
the background field formalism which permits the definition of a gauge invariant
effective action.
atalkpresentedbyFFattheSecondConferenceonthe
ExactRenormalizationGroup,Roma,Sep.18-22,2000.
1
The aim is to clarify how it can be ensured that after the complete integration
oftheERGequations,i.e. flowthecut-offscalektozero,agaugeinvariantsolution
is obtainedwithout the need ofanextra fine-tuning. Furthermore,I will show that
the resurgent symmetry does indeed correspond to the inherent gauge one. Most
technicaldetailswillbe bypassedin this presentationandmaybe foundina recent
letter.2
At the centre of my presentation is the quest of understanding how physi-
cal information is encrypted along the flow through an interplay between gauge
invariance, Ward-Takahashi identities and background field identities. A similar
programme has been pursued a few years ago in the context of Abelian theories.5
Background field identities were shown to contain the Ward identities under a re-
quirement of gauge invariance.
ThekeyingredientofthepresentapproachistheERGequationfortheeffective
action Γ . It describes the logarithmic rate of the change of Γ with respect to the
k k
scale k. Following the standard implementation of the background field formalism
I introduceanon-dynamicalauxiliaryfieldA¯,the so-calledbackgroundgaugefield.
Then formally the flow equation for pure Yang-Mills may be written as:
1 δ2Γ −1 δ2Γ −1
∂ Γ [A,c,c∗;A¯]= Tr k +R ∂ R − Tr k +R ∂ R . (1)
t k 2 (cid:18)δAδA A(cid:19) t A (cid:18)δcδc∗ C(cid:19) t C
I use the common notation where t = lnk and the trace Tr denotes a sum over
momenta, Lorentz and gauge group indices. The functions R and R implement
A C
the infra-red cut-off for the gauge field A and ghost fields c and c∗ respectively.
Theymayalsodependonthebackgroundfield,towhichInowturnyourattention.
Background field formalism: I briefly summarise some important points about the
backgroundfieldformalism,inparticulartherˆoleofdifferentgaugetransformations.
The formalism is settled on the use of a background field dependent gauge-fixing
condition that is invariant under a simultaneous gauge transformation of A¯ and of
the fields A,c and c∗. This can be used for a definition of an effective action which
is invariant under this combined gauge transformation. As A¯ is involved in this
transformation, the invariance of the effective action is, a priori, only an auxiliary
symmetry. However,forthe choiceA¯=A itbecomesthe inherentgaugesymmetry
of the theory.
For a pure Yang-Mills theory including the ghost term,
S =S +S +S . (2)
cl gf gh
The classical action S = 1 Fa Fa contains the field strength tensor F (A)=
cl 4 x µν µν µν
∂µAµ−∂νAµ+g[Aµ, Aν], wRhere Fµν ≡Fµaνta and Aµ =Aaµta with the generators
ta satisfying[ta, tb]=fabctc andtrtatb =−1δab. Ialsousetheshorthandnotation
2
≡ ddx. In the adjoint representation, the covariant derivative is
x
R R
Dab(A)=δab∂ +gfacbAc . (3)
µ µ µ
Thenaturalchoiceforthe gaugefixingisthe so-calledbackgroundfieldgauge. The
corresponding gauge-fixing and ghost actions are respectively,
S =− 1 (A−A¯)aD−abD−bc(A−A¯)c , S =− c∗D−acDcdc , (4)
gf 2ξ Z µ µ ν ν gh Z a µ µ d
x x
2
−
which involves the covariant derivative D ≡ D(A¯). The symmetries of the action
in(2) canbe inspectedby two differentgaugetransformations. The firstonegauge
transforms the dynamic fields A, c, andc∗– it represents the underlying symmetry
of the theory. Its infinitesimal generator Ga in a natural representation is defined
as
δ δ δ
Ga =Dab −gfabc c +c∗ . (5)
µ δAb (cid:18) c δc c δc∗(cid:19)
µ b b
From the action of Ga on the fields it can be shown that A transforms inhomo-
geneously, the ghosts transform as tensors and A¯ is invariant. It follows that the
covariant derivative transforms as a tensor.
The second transformation, denoted by the generated G¯a, acts only on the
background field,
G¯a =D−ab δ , (6)
µ δA¯b
µ
and under its action A¯transforms inhomogeneously like A under Ga, and therefore
−
the covariant derivative D also transforms as a tensor. Note that the auxiliary
transformation G¯a as it stands, does not carry any physical information.
I now turn your attention to the manner in which Ga and G¯a operate on the
action S. The classicalaction is invariantunder both transformations since it does
not depend on the background field, while neither S nor S are invariant under
gf gh
(5) or (6). For (5) it follows,
Ga(x)S = 1DabD−bcD−cd(A−A¯)d(x) , Ga(x)S =fbdcD−ad c∗Dcec . (7)
gf ξ µ µ ν ν gh µ b µ e
(cid:0) (cid:1)
Now fromthe explicitexpressionfor S andS , asgivenin(4), it followsthat(7)
gf gh
justdisplays−G¯aS and−G¯aS respectively. Thus,inthebackgroundfieldgauge,
gf gh
each term in the Yang-Mills action (2) is separately invariant under the combined
transformation G + G¯. A key point of the background field formalism has been
reached: the action resulting from setting the background field equal to the gauge
field, i.e. Sˆ[A,c,c∗]≡ S[A,c,c∗;A¯=A] is invariant under the physical symmetry
generatedby(5),GaSˆ[A,c,c∗]=0,withS[A,c,c∗;A¯]satisfyingtheclassical‘Ward-
Takahashi identity’, GaS =Ga(S +S ).
gf gh
Atquantumlevelthissymmetryturnsintothe gaugeinvarianceoftheeffective
action Γ[A,c,c∗;A¯=A], which in turn satisfies the Ward-Takahashi identity for a
non-Abeliangaugetheory. HoweverI remindyouthatit isonly the combinationof
both statements that gives a physical meaning to this gauge invariance. Note that
heuristicallythisresultstemsfromtheobservationthatinthequantisedtheorythe
source only couple to the fluctuation field
aa =Aa −A¯a , (8)
µ µ µ
and the gauge fixing condition (4) only constrains aa.
µ
3
Background field dependent Wilsonian flows: In order to effectively implement the
background field formalism for the coarse-grained effective Yang-Mills action I
choose the regulator terms
∆S =∆S +∆S , (9)
k k,A k,C
for the gauge and the ghost fields, respectively, to be3
1
∆S = (A−A¯)aR ab(P2)(A−A¯)b (10)
k,A 2Z µ Aµν A ν
x
∆S = c∗Rab(P2)c . (11)
k,C a C C b
Z
x
The arguments P2 and P2 of the regulator functions are appropriately defined
A C
backgroundfielddependent Laplaceans,andtheir choicemightdetermine the sym-
metries ofthe resultingtheory. Forinstance, inorderto haveanactionS whichis
k
gauge invariantunder the combinedtransformationit is only necessaryto required
that both P2 and P2 transform as tensors under G +G¯. Thus, in the following, I
A C
shall assume that such a choice has been made.
Up to this point, I have restricted the presentation to the classical action with
the regulator terms (9) added. The computation of the coarse-grained effective
action Γ follows the usual procedure. Consider the Schwinger functional W ≡
k k
W [J , η, η∗; A¯ ],
k µ µ
expW = DAaDc Dc∗ exp −S + (Ja(A−A¯)a +η∗c −c∗η ) .(12)
k Z µ a a (cid:20) k Z µ µ a a a a (cid:21)
Ya (cid:8) (cid:9)
where (J,η,η∗) are the respective sources. Then the effective action Γ is given by
k
Γ [A,c,c∗;A¯]=−W [J,η,η∗;A¯]−∆S [A,c,c∗;A¯]
k k k
+ Ja(A−A¯)a +η∗c¯ −c∗η . (13)
µ µ a a a a
Z
x(cid:0) (cid:1)
The flow equation for Γ has already been given at the beginning, (1). Now I
k
introduce the effective action Γˆ ,
k
Γˆ [A,c,c∗]≡Γ [A,c,c∗; A¯=A]. (14)
k k
AsIshallarguelater,thisnewactionisgauge-invariant. Itsflowequation,ofcourse,
is given by the flow of Γ in (1), but evaluated at A¯=A. It is important to stress
k
that ∂ Γˆ , since it depends on the second functional derivatives of Γ with respect
t k k
to the dynamicalfields (atA¯=A), is a functionalofΓ andnota functionalofΓˆ .
k k
This means that is not sufficient to study the symmetries of Γˆ but also necessary
k
to study those of Γ .
k
Modified and background field Ward-Takahashi identities: I will now discuss the
Ward-Takahashi identities that are related to the gauge transformations (5) and
(6). In the Wilsonian formalism, due to the presence of a coarse-graining, these
identities receive a contribution from the regulator term. The identity that follows
4
from considering GaΓ is called the modified Ward-Takahashi identities (mWI).
k
A second identity follows from G¯aΓ and I shall denote it as the background field
k
Ward-Takahashi identities (bWI).
As S is invariant under the action of Ga + G¯a it can be read off from the
k
Schwinger functional (12) and the effective action (13), that Ga+ G¯a leaves the
functional Γ invariant for generic A and A¯ configurations,
k
Ga+ G¯a Γ =0. (15)
k
(cid:0) (cid:1)
Therefore by requiring A=A¯, Γˆ is also invariant, GaΓˆ = 0, which for k = 0
k k
expresses the desired physical gauge invariance. Consequently, for k 6= 0, physical
gauge invariance is encoded in the behaviour of Γ under the transformation Ga. I
k
emphasise again that, in order to attain this crucial result, I had to keep track of
the effects from the transformations Ga and G¯a on Γ separately.
k
For pure Yang-Mills theories the mWI is given by
Ga(x)Γ =Ga(x) (S +S )+La(x)+La (x) . (16)
k gf gh k R,k
Both L and L display loop terms. The first term L stands for the well-known
k R,k k
loopcontributionstoWard-Takahashiidentitiesinnon-Abeliangaugetheoriesorig-
inating fromhGa(S +S )i , whilst the secondterm is due to the regulatorterms
gf gh J
and clearly vanishes when k → 0. It follows that the mWI (16) turns into the
standard WI for k =0,
GaΓ=La . (17)
0
As for the bWI, by applying G¯a to W [J,η,η¯;A¯] it follows from (12) and (13) after
k
some manipulations that the effective action Γ obeys the equation
k
G¯aΓ =G¯a(S +S )−(La+La ) . (18)
k gf gh k R,k
ThecombinedgaugeinvarianceofΓ ,Eq.(15),followsimmediatelyfromthisiden-
k
tity and the mWI given by (16).
Symmetriesoftheflowandphysicalgaugeinvariance: Theimplementationofacoarse-
graining modifies the gauge symmetry of the theory as I mentioned before. At the
formal level it is clear that the original symmetry is restored when the coarse-
graining scale is removed. A more delicate problem is to guarantee that this also
happens at the level of the solution to the flow equation.
Tounderstandhowgaugeinvarianceisencodedthroughouttheflow,itispivotal
to also study the action of the symmetry transformations on ∂ Γ , Eq,(1). Under
t k
the combined gauge transformationthe flow of Γ transforms as
k
(Ga+ G¯a)∂ Γ =0 . (19)
t k
An immediate consequence is that Ga∂ Γˆ = 0. Note that the only input for (19)
t k
wastheinvarianceofΓ . Thus,whentheinitialeffectiveactionΓ isinvariantunder
k Λ
Ga+G¯a, it follows that the full effective action Γ is also invariant, (Ga+G¯a)Γ =
0 0
0. In other words, (15) and (19) are the proof that the flow and the combined
5
transformation commute. Moreover, Γ satisfies the usual WI (17). This means
0
that the line of arguments for the backgroundfield formalism can be followed here
as well. Hence the equation GaΓˆ =0 displays physical gauge invariance.
0
Now I wish to make a final remark on the consistency of the mWI (16) with
the flow. Here,as in other formulationsof Wilsonianflows in gaugetheories,4,7 the
flow of the mWI is proportional to itself. Such common property states that if the
effectiveactionΓ satisfiesthemWIatsomescalek,e.g theinitialonek =Λ,then
k
Γ automatically satisfiesthemWI atanyscalek,provideditisobtainedfrominte-
k
grating the flow equation, and in particular, Γ satisfies the usual Ward-Takahashi
0
identity.
SummaryoftheTalk: Ihaveestablishedacompletesetofequationsrelevantforthe
control of gauge invariance in the ERG approach to pure Yang-Mills theories. In
particular,itcanbe inferredthat there is no requirementfor additionalfine-tuning
conditions,despitethepresenceofabackgroundfield. Moreover,Ihaveshownthat
invarianceoftheeffectiveactionunderacombinedgaugetransformationofallfields
follows from mWI and bWI. More generally, two of these three properties of the
effective action (invariance, mWI and bWI) lead to the third one.
Consequently, a key property of the usual background field formalism is main-
tained in the ERG approach: by virtue of the auxiliary background identity it
follows that physical gauge invariance is reflected in the invariance of the effective
action under the combined gauge transformation of all fields.
The formalism is not only suitable for formal or analytic analysis, but also,
it elucidates the problem of how to filter the contribution of spurious unphysical
modes in a numerical computation of approximate solutions to flow equations in
gauge theories.
Acknowledgements
FF thanks the organisers for their financial support.
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