A space-time approach to multigluon computations in QCD: An application to effective action terms

10 Pages·0.13 MB·English

by  A. I. Karanikas

UA/NPPS-1-03
3 A Space-time approach to multi-gluon loop computations in QCD:
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An application to effective action terms
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7 A. I. Karanikas 1 and C. N. Ktorides2
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University of Athens, Physics Department

1 Nuclear & Particle Physics Section
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Panepistimiopolis, Ilisia GR 157–71, Athens, Greece
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v Abstract
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The applicability of the space-time formulation of the gluonic sector of QCD in
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a terms of the Polyakov worldline path integral, via the use of the background field
gauge fixing method, is extended to multi-gluon loop configurations. Relevant master
formulas are derived for the computation of effective action terms.
[e-mail]
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Space-time based approaches to QCD, employing first quantization methods, have estab-
lished a viable alternative for expediting perturbative computations in the theory. The first
attempts in this direction have used strings as underlying agents [1-3] through which a set of
prescriptions, known as Bern-Kosower rules, were successfully applied at the one gluon loop
level. Subsequently, Strassler [4] established the possibility to reproduce the whole approach
by basing the formalism on worldline paths instead of strings, see also Refs [5,6]. More
recently, Sato, Schmidt and collaborators [7,8] have extended the worldline considerations
to the two loop level. In this letter we apply Polyakov’s version [9] of the worldline cast-
ing of QCD, which we have been consistently pursuing for some time [10-12], to multi-gluon
loopconfigurations and formulate specific rules for constructing the relevant expressions that
represent them. More than just as a strategy pertaining to perturbative calculations this
approach provides a framework within which a systematic treatment of the propagation of
particle modes accommodating Wilson lines (loops) can be achieved.
As in a recent paper [12], in which we presented one gluon loop master formulas for
effective action terms, we shall formulate the worldline path integral for the gluon by em-
ploying the background gauge fixing method [13], which is based on a splitting of the gauge
field potentials Aa by setting Aa = aa +Ba and treating the aa modes as fully fluctuating
µ µ µ µ µ
in a background represented by the fields Ba. For the present purposes the latter will be
µ
considered as classical. The idea, however, that in a quantized context they could be used as
‘carriers’ of non-perturbative physics in QCD presents interesting possibilities on which we
shall devote some comments in the end. In any case, they facilitate the gauge fixing through
the condition
(δab∂ +gfabcBc)αb ≡ Dab(B)αb = 0. (1)
µ µ µ µ µ
Our starting point is the generating functional (partition function in Euclidean space-
time) given, for the gluonic sector of the theory, by
Z(B) = N Dc¯DcDαexp(−S ) = NZ Det−21[−(D2)abδ −2gfabcFc (B)]Det[−R( δ )]
A 0 µν µν δJ
Z
δ 1
×exp −S exp d4xd4x′Ja(x)iG[1](x,x′)abJb , (2)
" I δJ!# (cid:20)2 Z µ µν ν(cid:21)J=0
where c,c¯ denote ghost fields, S is the interaction part of the Yang-Mills action for the α
I
fields, Z is the classical contribution, iG[1] is the gluon propagator and
0
←−ac
Rab(α) = (D2)ab +gD fcbdαd. (3)
µ µ
Propagators of particle modes (field quanta) according to the Polyakov worldline path
integral have the following generic form, see Refs [10-12],
∞ T
−1
iG[s](x,x′)ab = dT Dx(t)exp dtx˙2 Φ[s] [Cxx′]U[Cxx′]ab, (4)
mn  4  mn
Z0 x(0)=x′Z,x(T)=x Z0
 
where s = 0,1/2,1,... stands for the spin of the propagating particle entity and m,n are
‘Lorentz generator’ indices for each given value of s, e.g., mn → µν for spin 1. Furthermore,
2
Φ[s] [Cxx′] is the so-called spin-factor for the particle mode whose generic form is (P stands
mn
for path ordering)
T
i
Φ[s] [Cxx′] = Pexp dtJ ·ω ; x(0) = x, x(T) = x′ (5)
mn 2 
Z
0 mn
 
with ω the tensor T[x¨ (t)x˙ (t)−x¨ (t)x˙ (t)] whose indices are contracted with those of the
ρσ 2 ρ σ σ ρ
generators, namely Jρσ and where U[Cxx′]ab is the Wilson line element, specified in terms
mn
of the background gauge fields, i.e.
ab
T
U[Cxx′]ab = Pexp ig dtx˙(t)·B . (6)
 
Z
0
 
In the particular case of the pure gluonic sector, to which we shall exclusively devote our
attention in this paper, we have (J )ρσ → (δ δ − δ δ ), while for the propagation of
µν ρµ σν ρν σµ
the ghost modes the spin factor is unity. For completeness, we mention that in the case of
spin-1/2 modes one has (J )ρσ → 1σρσ, where the α, β are Dirac spinor indices.
mn 2 αβ
The above set of formulas suffice to conduct any perturbative, at least, computation
in the theory. To be concrete, we can produce prescriptions for combining the worldline
carriers of gluons and ghosts thereby arriving at the worldline version of the Feynman rules
for the corresponding vertices. Given that the gist of any such computation are the 1PI
configurations, we shall, in the present paper, restrict our considerations to effective action
functionals, whose one-loop contribution term has been confronted in Ref [12]. For two-
loops we must separately deal with the configuration having a single 4-gluon vertex and the
one with two 3-gluon vertices. For the first case we obtain, upon inserting the Polyakov
worldline path integral expression for the propagators involved and designating by C and
1
C the respective loops that compose the ‘figure-8’ formation,
2
1 2 ∞ 2 1 Ti
Γ(2) = d4x dT Dx (t )exp − dt x˙2
4 8  i i i  4 i i
Z iY=1Z0 iY=1xi(0)=xZi(Ti)=x Z0 
×Φ[1][Cxx]Φ[1][Cxx](V )abcd U[Cxx]abU[Cxx]cd,   (7)
µν 2 ρσ 1 4 µνρσ 2 1
with
(V )abcd = g2[feabfecd(δ δ −δ δ )+feacfebd(δ δ −δ δ )
4 µνρσ µρ νσ µσ νρ µν ρσ µσ νρ
+feadfebc(δ δ )−δ δ )] (8)
µν ρσ µρ νσ
the well known 4-gluon vertex.
In order to deal with the second configuration, which involves a derivative operation at
each vertex, we first observe that
∞ T
1 1
Dab iG[s](x,x′)bc = dT Dx(t)x˙ (T)exp − dtx˙2 Φ[1][Cxx′]U[Cxx′]ac
µ,x νρ 2 µ  4  νρ
Z0 x(0)=x′Z,x(T)=x Z0
  (9)
3
and similarly, with a relative minus sign, when the covariant derivative acts with respect to
′
x. We now face a situation where the relevant configuration has three, open line, branches.
′
The following organization is adopted. Place x andx on diametrically opposite points on a
′
circle. Go, always from x to x, in three different ways. Once along one semicrcle, contour
C , once along the diameter, contour C , and once along the other semicircle, contour C .
1 2 3
Following the convention that velocities are positive at the end point we determine, after
some algebraic manipulations,
1 1 3 ∞ 3 1 Ti
Γ(2) = d4xd4x′ dT Dx (t )exp − dt x˙2
3 23!  i i i  4 i i
Z iY=1Z0 iY=1xi(0)=x′Z,xi(Ti)=x Z0 
×Φ[1][Cxx′]Φ[1][Cxx′]Φ[1][Cxx′](Vx)abc (Vx′)defU[Cxx′]adU[Cxx′]beU[Cxx′]cf,(10)
µρ 3 νσ 2 κλ 1 3 µνκ 3 ρσλ 3 2 1
where the first vertex reads
i
(Vx)abc = gfabc{δ [x˙ (T )−x˙ (T )] +δ [x˙ (T )−x˙ (T )]
3 µνκ 2 µν 2 2 1 1 κ µκ 1 1 3 3 ν
+δ [x˙ (T )−x˙ (T )] }, (11)
νκ 3 3 2 2 µ
while the second has a relative minus sign -accounting for opposite velocity orientations- and
makes the replacement T → 0.
i
The last specification needed is for the the two loop configuration which involves vertices
(2)
with ghosts. Choosing our paths the same as for Γ , with C and C traversed by ghosts
3 1 2
and C by gluons and once determining the action of the corresponding covariant derivative
3
operatorsontheghostfieldpropagator,inanalogytoEq. (9),thefollowing result isobtained
1 1 3 ∞ 3 1 Ti
Γ(2) = d4xd4x′ dT Dx (t )exp − dt x˙2
gh 23!  i i i  4 i i
Z iY=1Z0 iY=1xi(0)=x′Z,xi(Ti)=x Z0 
×Φ[1][Cxx′]Φ[0][Cxx′]Φ[0][Cxx′](Vx)abc(Vx′)defU[Cxx′]adU[Cxx′]beU[Cxx′]cf, (12)
µν 3 2 1 g µ g ν 3 2 1
where
i
(Vx)abc = − fabcx˙ (T ) (13)
g µ 2 1µ 1
with 1 → 2 and T → 0 for the second vertex.
1
One observes that the expressions for the vertices are related to those of the second
quantized (field theoretical) formulation of QCD via the correspondence p ↔ ix˙. This
2
occurence explicitly illustrates the space-time character of our approach to the theory.
Fromwhat we have so farpresented the following rules follow forthe space-time approach
to perturbation theory:
•Draw all diagrams (1PI for our present purposes) in terms of worldlines for the particle
modes which employ the allowed vertices. The analytical expressions for the latter are
furnished by Eqs. (8), (11) and (13).
• For each line segment between adjacent vertices assign the appropriate spin factor and
Wilson line(loop).
• Each diagram is assigned a combinatorial factor which coincides with that of the cor-
responding conventional Feynman diagram.
4
The perturbative computation of the effective action, at two loop level, proceeds once we
expand the Wilson exponential factors entering the corresponding expressions. Keeping the
Mth order, in g, of the expansion means that one determines up to M+2-order contributions
to the effective action. Before taking this step we find it convenient to recast our two loop
expressions in a form wherein the various worldlines entering them are described in terms
of parameters running along the traversed (Wilson) paths as opposed to being characterized
′
through the points of reference x,x. In this way one circumvents zero mode problems due
to translational invariance while at the same time achieves a form which, in the Feynman di-
agrammatic language, amounts to directly reaching expressions at the stage where Feynman
parameters have been introduced.
(2)
The new form of Γ becomes, once making the redefinition t → T +T −t and the
4 2 1 2 2
replacements T = s , T +T = s,
1 1 1 2
∞ s s
1 1
Γ(2) = ds ds Dx(t)δ[x(s)−x(0)]δ[x(s)−x(s )]exp − dtx˙2
4 8 1 1  4 
Z Z Z Z
0 0 0
×Φ[1][Css1]Φ[1][Cs10](V )abcd U[Css1]abU[Cs10]cd,   (14)
µν ρσ 4 µνρσ
s′
wherewehaveusedthenotationΦ[1][Css′] ≡ Pexp i dtJ ·ω andsimilarlyforU[Css′]ab.
µν 2
s !
µν
R
Also, we have taken into account the fact that the contour segment C of Eq (7) is now being
2
reversely traversed since, on account of the uniform parametrization, one branch is given a
clockwise and the other a counterclockwise orientation.
Fortheconfigurationswiththepairof3-verticesthecalledforchangesaret → T +T +t
3 1 2 3
and t → T +T −t while the corresponding replacements are T = s , T +T = s and
2 1 2 2 1 1 1 2 2
T +T +T = s. Once taking into account direction reversals since, for the sake of achieving
1 2 3
a single parametrization, the contour is to be traced in a continuous manner, we obtain
∞ s s2 s
1 1
Γ(2) = ds ds ds Dx(t)δ[x(s)−x(s )]δ[x(0)−x(s )]exp − dtx˙2
3 8 2 1 1 2  4 
Z Z Z Z Z
0 0 0 0
×V [C,x˙]fabcfdefU[Css2]adU[Cs1s2]beU[Cs10]cf,  (15)
3
where
V [C,x˙] ≡ g2({Φ[1][Css2]Tr Φ[1][Cs20]−Φ[1][Css2]Φ[1][Cs20]}x˙ (s −0)
3 µν L µρ νρ µ 1
×[x˙ (0)+x˙ (s −0)]+{Φ[1][Css1]Φ[1][Cs10]+Φ[1][Cs0]}x˙ (s −0)x˙ (0)) (16)
ν ν 2 ρµ ρν µν µ 1 ν
(2)
and similarly for Γ with
gh
V [C,x˙] → V [C,x˙] ≡ −g2Φ[1][Css2]x˙ (s −0)x˙ (s +0)). (17)
3 g µν µ 1 ν 1
We now proceed with the expansion of the Wilson exponential. To Mth order we have
s′
U[Css′]ab = δab +ig(tc1)ab dt x˙(t )·Bc1[x(t )]+···
G 1 1 1
Zs
1 ab 1 s′
+ ig(tcGn) θ(tn −tn−1)x˙(tn)·Bcn[x(tn)]+O(gM+1). (18)
" #
nY=M nY=MZs
5
Upon writing
dDq
Ba = eiq·x˜ǫa(q) (19)
µ (2π)D µ
Z
and using the equation of motion DabFb (B) = 0 obeyed by the (classical) background field,
µ µν
one determines
ǫb ·p ǫc
˜ǫa(q) = (2π)Dδ(q −p )ǫa +g(2π)Dδ(q −p −p )(ta)bc2 c µ +O(g2), (20)
µ a µ b c G (p +p )2
b c
where the ǫa are polarization vectors and the p four momenta for the background fields.
µ aµ
In a generic computation of an n-point Green’s function the above expansion serves as the
meansbywhichagivenworldline gluon(multi)loopconfigurationis‘punctured’byanetwork
of gluonic tree diagrams whose contribution to the overall computation is determined by
classicalfieldperturbationtheory. Forthepurposeofcomputingcontributionstotheeffective
action we are exclusively interested in 1PI terms, hence it suffices to set {Ban[x(t )]} →
µn n
{ǫn exp[ip ·x(t )]}. Upon employing the representation [4]
µn n n
ǫn ·x˙(t )] = dξ dξ¯ exp[iξ ξ¯ ǫn ·x˙(t )], (21)
n n n n n n
Z
¯
where the ξ ,ξ are Grassmann variables, we determine the Mth order term in Eq. (18) to
n n
take the form
1 ab 1 s′ M
ig(tcGn)  dξndξ¯n dtnθ(tn −tn−1)exp i kˆ(tn)·x(tn) +permutations.
" # " #
nY=M nY=MZ Zs nX=1
 
  (22)
In the above expression we have set
∂
kˆ (t ) ≡ p +ξ ξ¯ ǫn , (23)
µ n n,µ n n µ∂t
n
while the term ‘permutations’ refers to all possible rearrangements of the t .
n
As shown in Refs [11,12], via the employment of the Migdal-Polyakov area derivative
[14,15], for a number of M background gluon field punctures of a given loop configuration
one obtains the following result for the spin factor
i M i M
Φ[1][C] → Φ[1][M] = Pexp J ·φ(n) = δ + (J ) φ (n)
µν µν "2 # µν 2 ρσ µν ρσ
nX=1 µν nX=1
i 2 M n2−1
+ (J ) (J ) φ (n )φ (n )+... (24)
2 ρ2σ2 µλ ρ1σ1 λν ρ2σ2 2 ρ1σ1 1
(cid:18) (cid:19) nX2=1nX1=1
with
4
φ (n) = 2ξ ξ¯ (εnp −εnp )+ ξ ξ¯ ξ ξ¯ (εn+1εn −εn+1εn)δ(t −t ) (25)
µν n n µ n,ν ν n,µ T n+1 n+1 n n µ ν ν µ n+1 n
¯
and where we have set ξ = ξ = 0.
M+1 M+1
6
For the generic partitioning, wherein we keep the first N terms (0 ≤ N ≤ M) from the
Wilson loop expansion of the contour Cs10 and the first M−N terms from the contour Css1,
we obtain the following result for the two loop configuration with the four-vertex
1 M N+1 bc 1 de 1
Γ(2) = gM (tan) (V )bcde (tan) dξ dξ¯
4,(M+2) 8 " G # 4 µνρσ" G # " n n#
N=0 n=M n=N n=MZ
X Y Y Y
∞ s N+1 s 1 s1 1
× ds ds1 dtn dtn θ(tn −tn−1)
  " #
Z0 Z0 nY=MsZ1 nY=NZ0 nY=M
  
×Φ[1][M −N]Φ[1][N]Q(1)[{ǫ},{p}]+permutations, (26)
µν ρσ
where we have denoted (the indication ‘(1)’ signifies the presence of a single vertex)
s
1 M
Q(1) = Dx(t)δ[x(0)−x(s)]δ[x(s )−x(s)]exp − dt x˙2 + kˆ(t )·x(t ) . (27)
1 i n n
 4 
Z Z n=1
0 X
 
For the three-vertex, two-loop gluon configuration and keeping M −N terms from Css2,
2
N −N from Cs2s1 and the remaining N −N from Cs10 we get
2 1 2 1
ab
1 M M N2+1 N1+1 1
Γ3gluons = − gM tan tb tan ta tan
2,M+2) 8  G  G G  G G 
NX1=0NX2=0 nY=M nY=N2 nY=N1
1 ∞ s s2 N2+1 s N1+1 s2 1s1
¯
× dξ dξ ds ds ds dt dt dt
n n 2 1 n n n
" #    
nY=MZ Z0 Z0 Z0 nY=MsZ2 nY=N2sZ1 nY=N1Z0
   
1 δ
× θ(tn −tn−1) V3 N1,N2,−i R(2)[{ǫ},{p},j]j=0+permutations, (28)
" # " δj#
n=M
Y
where the current source j has been introduced for the purpose of facilitating the compu-
′
tation of correlators of the type < x˙ (t)x˙ (t) > which invariably enter the three-vertex
µ ν
computation. Finally,we have set
s s
1 M
R(2)(j) = Dx(t)δ[x(s)−x(0)]δ[x(s )−x(s )]exp − dtx˙2 +i kˆ(t )·x(t )+i dtj ·x˙(t )
1 2 n n n
 4 
Z Z n=1 Z
0 X 0
 (29) 
with the ‘(2)’ serving to note that the configuration being considered has two vertex points.
The ghost-containing configuration results from Eq (28) via the replacement V → V .
3 g
We are now at a point where we must execute the path integrals. Their gaussian nature
calls for determining the appropriate Green’s functions on each given line contour. It is
possibletogeneralizetheGreen’sfunctionscorrespondingtopropagationofbosonic‘particle’
modes on unobstructed closed, or open, contours [4-6,12] in (Euclidean) space-time to our
two-loop situation where one or two intervening vertex points are present. For loops with n
vertices the Green’s function we need can be defined as follows
1
G(n)(t,t′) = G(n−1)(t,t′)− [G(n−1)(t,0)−G(n−1)(t,s )−G(n−1)(t′,0)+G(n)(t′,s )]2,
G(n−1)(s ,0) n n
n
(30)
7
with G(0)(t,t′) = 1|t−t′|(s−|t−t′|).
s
The corresponding expression for contours with non-coinciding initial and final points
(open contours), needed for the three-vertex configuration, reads
G˜(n)(t,t′) = G(n)(t,t′)+ 1 (t(n) −t′(n))2 (31)
(n) R R
s
R
with G˜(o)(t,t′) = |t−t′|.
(n)
In the above equation s is the reduced total time given by
R
1 1 1 1
= + +··· (32)
s(n) s1 s2 −s1 s−sn
R
(n)
and t is the reduced partial time
R
t s −t t−s
(n) (n) 2 n
t = s θ(s −t) +[θ(s −t)−θ(s −t)] +···+θ(t −s) . (33)
R R 1 s 2 1 s −s n s−s
(cid:26) 1 2 1 n(cid:27)
Using the above expressions one arrives at the following results:
M 1
Q(1) = (2π)4δ p exp kˆ(t )·kˆ(t )G(1)(t ,t ) (34)
n=1 n! (4π)D[s(s−s1)]D/2 "m>n n m n m #
X X
and
M 1
R(2)(0) = (2π)4δ p
n
n=1 ! (4π)D[s2(s−s2)+s1(s2 −s1)]D/2
X
×exp kˆ(t )·kˆ(t )G˜(2)(t ,t ) . (35)
n m n m
" #
m>n
X
For the three vertex computation we need velocity correlators. The result obtains from
the operation involving the current source j entering Eq (28). One obtains, generically,
< x˙µ(t)x˙ν(t′) > = −δµν∂t∂t′G˜(2)(t,t′)− kˆ(tn)kˆ(tm)∂t[G˜(2)(t,0)−G˜(2)(t,tn)]
n,m
X
×∂t′[G˜(2)(t′,0)−G˜(2)(t′,tm)]. (36)
Substituting Q(1), R(2) and the corresponding correlators in Eqs (26) and (28), respec-
tively (the latter in the ‘ghost’ term as well) a set of master expressions is produced which
call for the execution of integrations over two sets of parameters. We have been able to
reproduce known results [16] for the M = 0 case and M = 2 cases (forthcoming paper).
For our closing comments let us begin by saying that the ability to reformulate QCD in
termsofspace-timepathshasledtotheresultofproducingmasterexpressions, giveninterms
of two sets of parameters (Grassmann and Feynman) which correspond to multi-gluon loop
configurations that enter perturbative expansions. Once the overall strategy is established
the only real determination that needs to be made is the computation of Green’s functions
describing bosonic propagation on paths with a number of ‘marked’ points satisfying two
8
types of boundary conditions. Even though the examples we have discussed pertain to
paths that can be continuously traced nothing, in principle, prevents one from treating
more complex, higher loop, configurations which call for more than a single continuous
parametrization. The bottom line is that the relevant path integral, whose execution calls
for determining the Green’s functions, is Gaussian, therefore manageable.
Aside from the execution of multi-gluon loop computations, which call for numerical
approaches to confront the relevant master expressions as the configurations get increasingly
complicated, there is another aspect of the Polyakov path integral approach to QCD that we
find more intriguing. This has to do with the fact that each worldline, in addition to being
geometrically characterized by the spin of the propagating particle mode through the spin
factor, is accompanied by a ‘cloud’ of background gauge potentials on account of the Wilson
line(loop). Consider a context within which the background fields are quantized in a way
that they carry non-perturbative physics. This, e.g., can be achieved in a lattice context or,
alternatively, in lattice inspired continuum schemes, as in Refs [17-19]. One would then be in
possession ofamethodologyformulatedstrictlywithin theboundsofQCD thatwillfacilitate
the study of the interplay between the perturbative and the non-perturbative regime of the
theory. It is to the confrontation of this problem that we intend to direct our future efforts.
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