Relation between Worldline Green Functions for Scalar Two-Loop Diagrams

10 Pages·0.12 MB·English

by  H-T. Sato

NBI-HE-95-40
hep-th/9511158
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Relation between worldline Green functions
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for scalar two-loop diagrams
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v Haru-Tada Sato 1
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The Niels Bohr Institute, University of Copenhagen
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1 Blegdamsvej 17, DK-2100 Copenhagen, Denmark
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Abstract
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v Wediscussarelation betweentwo-loop bosonicworldlineGreenfunctionswhich
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are obtained by Schmidt and Schubert in two different parametrizations of a two-
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loop worldline. These Green functions are transformed into each other by some
transformation rules based on reparametrizations of the proper time and worldline
modular parameters.
1 Fellow of the Danish Research Academy, [email protected]
One of interesting and important theoretical aspects of quantum field theories, in
particular of gauge theories, is the Bern-Kosower formalism [1] which provides a refor-
mulation of one-loop Feynman amplitudes. Their idea is very natural that amplitudes in
ordinary field theory may be reproduced by the infinite string tension limit of superstring
amplitudes. It results in a new set of rules instead of Feynman rules and enables several
calculations: five point gluon [2], four point graviton amplitudes [3] and so on [4]. The
advantage of this formalism is to get rid of handling a vast number of Feynman diagrams
and of taking care of gauge cancelation of diagrams in gauge invariant theories. It might
clarify underlying structure, which we have not been recognized yet in gauge theories; for
example, various useful informations have been developed in worldline approaches [5]-[7].
Extension ofthis formalismto multi-loopdiagramshas alsobeeninvestigated recently
[11],[12],[13] generalizing Strassler’s approach [8] to the Bern-Kosower rules. Strassler de-
rived the worldline Green functions [9] of spinor, scalar and gauge fields for one-loop dia-
gramsrewriting one-loopeffective actions as pathintegrals of (supersymmetric) worldline
actions. In one-loop case, the modular parametrization of a loop is unique and we can
define the worldline Green functions in a unique way. However, this situation changes
in multi-loop cases because we have a variety of choices of the parametrization due to
node points, i.e., internal vertices. Schmidt and Schubert have actually obtained two ex-
pressions for the two-loop worldline Green function in the φ3 theory [11]. They explicitly
checked that amplitudes derived from these respective Green functions coincide with at
least three or four point functions from the Feynman rule calculations.
It isapparent that thiskind ofequivalence check between worldline Greenfunctions in
different forms becomes difficult in more general situations such as higher loop diagrams
anda largenumber of external legs. Inaddition, we can not recognize which parametriza-
tion is natural or convenient one in a complicated case, and hence it is useful to know
how to convert the Green functions to a differently parametrized form. It is therefore
important to find a clear connection between multi-loop Green functions defined on a
differently parametrized worldline.
In this report, we discuss a relationship between two-loop bosonic worldline Green
1
functions proposed by Schmidt and Schubert in the scalar φ3 theory. First let us survey
the vacuum amplitude at one-loop level in two different ways which are instructive to get
an insight into two-loop case. One is given by the path integral for a worldline action
with cyclic boundary condition and the other may be given by sewing two propagators
of path integral expression [8],[9];
∞ dT T 1 dT
I(1) = N(T) Dxexp[− dτ x˙2(τ)] = (4πT)−D/2 (1)
0 Z T Z Z 4 Z T
0 x(0)=x(T) 0
and
2 ∞ Ti 1
J(1) = dDx dDx dT N˜ (T ) Dx exp[− dτ x˙2(τ)] (2)
0 Z a bYZ0 i i i Zxi(0)=xa i Z0 4 i
i=1 xi(Ti)=xb
= dDx dT dT [4π(T +T )]−D/2,
Z bZ 1 2 1 2
where the path integral normalizations N and N˜ are determined as the second equality
in each case. We omit the mass term e−m2T in (1) for simplicity. Comparing these
equations, we think of the transformation from J(1) to I(1) in the following form
0 0
T = T(1−u), T = Tu (3)
1 2
which gives
∞ ∞ 1
dT dT = dTT du. (4)
Z 1 2 Z Z
0 0 0
RHS of the latter expression (2) becomes
∞ dT 1 ∞ dT T T
dDx · T2 du(4πT)−D/2 = (2π)DδD(0)· dτ dτ (4πT)−D/2, (5)
Z b Z T Z Z T Z aZ b
0 0 0 0 0
where we have used the fact that the integrand is independent of any τ variables. If we
consider N-point functions as discussed later, we see that (2π)DδD(0) corresponds to the
conservation law of external momenta, which is implicit in (1). The integrals with respect
to τ and τ correspond to node point integrals which appear inmulti-loop integrals. Here
a b
we inevitably encounter the node point integrals because we need an additional leg at
a glue point of two propagators regardless of external or internal leg. Strictly speaking,
eq.(5) is not so much a one-loop vacuum amplitude as a two-point function with external
zero momenta or a kind of two-loop vacuum amplitude of one internal line of infinite
2
length (this will become clear later). We hence pick up pure one-loop contribution from
(5) through dropping node point integrals by hand. In this way, we understand that (3)
relates (1) with (2) which is symmetrized in T and T .
1 2
Wefurtherproceed toN-pointfunctions which concerntheworldline Greenfunctions.
Evaluating path integrals with insertion of N scalar vertex operators in both (1) and (2),
we have the following two expressions which are seemingly different from each other
∞ dT N T 1 N
I(1) = (4πT)−D/2 · dτ exp[ p p G (τ ,τ )] (6)
N Z0 T Y Z0 n 2 X j k B j k
n=1 j,k
and
J(1) = (2π)DδD( p(i)) ∞dT dT [4π(T +T )]−D/2 · N1 T1 dτ(1) N2 T2 dτ(2)
N n Z 1 2 1 2 Z n Z n
X 0 Y 0 Y 0
n,i n=1 n=1
1 2 Na N1 N2
×exp[ p(a)p(a)G (τ(a),τ(a))+ p(1)p(2)G (τ(1),τ(2))] (7)
2 j k aa j k j k 12 j k
XX XX
a=1 j,k j k
where G is the one-loop bosonic Green function [8],[9]
B
(x−y)2
G (x,y) = |x−y|− (8)
B T
and G are
ij
(x−y)2
G (x,y) = G (x,y) = |x−y|− (9)
11 22 T +T
1 2
(x+y)2
G (x,y) = x+y − . (10)
12 T +T
1 2
In (9) and (10), G are rearranged from original path integral results with the use
ij
of momentum conservation in accordance with two-loop results [11]. Note that τ(2)
is defined to run in the opposite direction to τ and τ(1) because we have chosen the
boundary condition of path integral (2) as x (0) = x and x (T ) = x for i = 1,2. If
i a i i b
we choose x (0) = x (T ) = x and x (T ) = x (0) = x , we will obtain an expression
1 2 2 a 1 1 2 b
with reversed sign for τ(2) variable. Since we have seen a correspondence between T-
parameters’ integral measures in (5), we do not have to repeat that here. Changing the
sign of τ(2) and renaming the variables {τ(1),τ(2)} and {p(1),p(2)} as τ and p , we obtain
n n n n n n
the same equation as (6) after omitting the delta function and node point integrals as
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a result of the transformation (3). Here we have assumed that the transformation of
τ-integration measures follow a naive replacement
N T Ni Ti
dτ → dτ(i). (11)
Z n Z n
Y 0 Y Y 0
n=1 i=1,2n=1
Let us promote the above one-loop idea to the two-loop case. Two-loop N-point
functions with different parametrizations from each other are obtained through insertion
of one propagator parametrized by length T
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∞ T3 1
dT N˜ (T ) Dx exp[− dτ x˙2(τ)]. (12)
Z0 3 3 3 Z x3(0)=xa 3 Z0 4 3
x3(T3)=xb
The insertion of this propagator and scalar vertex operators into (1) and (2) leads to [11]
∞ dT ∞ T T N T
I(2) = dT (4π)−D dτ dτ [TT +TG (τ ,τ )]−D/2 dτ
N Z T Z 3 Z aZ b 3 B a b Z n
0 0 0 0 Y 0
n=1
1 N
×exp[ p p G(1)(τ ,τ )] (13)
2 X j k B j k
j,k
and
J(2) = 3 ∞dT (4π)−D(T T +T T +T T )−D/2 N1 T1dτ(1) N2 T2dτ(2)
N Z a 1 2 2 3 3 1 Z n Z n
Y 0 Y 0 Y 0
a=1 n=1 n=1
1 2 Na N1 N2
×exp[ p(a)p(a)Gsym(τ(a),τ(a))+ p(1)p(2)Gsym(τ(1),τ(2))], (14)
2 XX j k aa j k XX j k 12 j k
a=1 j,k j k
where our G(1) and Gsym are same ones in [11] denoted by G˜(1) and G˜sym; namely,
B ij B ij
1(G (x,τ )−G (x,τ )−G (y,τ )+G (y,τ ))2
G(1)(x,y) = G (x,y)− B a B b B a B b (15)
B B 4 T +G (τ ,τ )
3 B a b
and
T +T
Gsym(x,y) = |x−y|− a+1 a+2 (x−y)2 (16)
aa T T +T T +T T
1 2 2 3 3 1
x2T +y2T +(x+y)2T
Gsym (x,y) = x+y − a+1 a a+2. (17)
aa+1 T T +T T +T T
1 2 2 3 3 1
The subscripts on G mean the superscripts of first and second arguments as easily un-
derstood in (14) and T ≡ T . It is worth noticing that the integrand of (14) coincides
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with that of (7) in the limit T → ∞. This implies the previous note that J(1) may be
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regarded as a two-loop contribution from an infinite internal line.
Similarly to the one-loop case, we begin with the vacuum (N = 0) case. According
to the change of variables (3) and (4), eq.(14) becomes
∞ 1
J(2) = dT dTT du(4π)−D[TT +T2u(1−u)]−D/2. (18)
0 Z 3 Z 3
0 0
We then transform the variable u in order to reproduce the Green function G (τ ,τ )
B a b
which can be seen in (13)
|τ −τ |
u = a b . (19)
T
Taking account of this transformation, eq.(18) coincides with the vacuum amplitude I(2)
0
∞ dT T T
J(2) = I(2) = dT dτ dτ (4π)−D[TT +TG (τ ,τ )]−D/2. (20)
0 0 Z 3 T Z aZ b 3 B a b
0 0 0
We here observe that (19) is needed in addition to (3) differently from one-loop case (5)
where u disappears.
In the case of N-point functions, we need a further consideration. We use the fact
that to deal only with τ ≤ τ is guaranteed by the symmetry of (13) under τ ↔ τ .
a b a b
The reversal of τ(2) considered in the one-loop case should be modified taking account of
dependence on the proper time position of a node point;
τ(1) = x −τ , τ(2) = τ −y . (21)
n n b n b n
This is the τ(2) reversal transformation with respect to τ which is regarded as the origin
b
of τ(1) coordinate shifted by τ accordingly. In one-loop case, we can put the origin τ = 0
b b
in safety (because of no u dependence). Notice also that to adopt τ as the origin of the
a
reversal transformation instead of τ corresponds to consider τ ≤ τ exchanging τ ↔ τ .
b b a a b
From (3) and 0 ≤ τ(i) ≤ T , we see also
i
τ ≤ y ≤ τ ≤ x . (22)
a n b n
It is convenient to write down G(1) explicitly according to this sequence (22),
B
|τ −τ |2 (x −x )2
G(1)(x ,x ) = G (x ,x )− a b n m
B n m B n m T +G (τ ,τ ) T2
3 B a b
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(T2 −|τ −τ |2)2(y −y )2
G(1)(y ,y ) = G (y ,y )− a b n m (23)
B n m B n m T +G (τ ,τ ) T2
3 B a b
1 x−y
G(1)(x,y) = G (x,y)− (τ −y −|τ −τ | )2.
B B T +G (τ ,τ ) b a b T
3 B a b
With substitution of (3),(19),(21) in (16), we verify that Gsym for i,j = 1,2 are trans-
ij
formed into G(1) as follows:
B
Gsym(τ(1),τ(1)) = G(1)(x ,x ),
11 n m B n m
Gsym(τ(2),τ(2)) = G(1)(y ,y ), (24)
22 n m B n m
Gsym(τ(1),τ(2)) = G(1)(x ,y ).
12 n m B n m
Finally, weapplyourtransformationtoanothersetofworldlineGreenfunctionswhich
can be obtained from inserting vertex operators also into the T propagator part (12).
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The exponential parts including Green functions of (13) and (14) are generalized to the
following respectively
1 N′ 1 N3 N′ N3
exp[ p p G(1)(τ ,τ )+ p(3)p(3)G(1)(τ(3),τ(3))+ p p(3)G(1)(τ ,τ(3))] (25)
2 j k 11 j k 2 j k 33 j k j k 13 j k
X X XX
jk jk j k
and
1 3 Na 3 Na Na+1
exp[ p(a)p(a)Gsym(τ(a),τ(a))+ p(a)p(a+1)Gsym (τ(a),τ(a+1))], (26)
2 j k aa j k j k aa+1 j k
XX XX X
a=1 j,k a=1 j k
where N′ = N +N , N = N etc., and G(1) are
1 2 4 1 ij
G(1)(x,y) = G(1)(x,y), (27)
11 B
(z −z )2
G(1)(z ,z ) = |z −z |− 1 2 , (28)
33 1 2 1 2 T +G (τ ,τ )
3 B a b
1
G(1)(x,z) = G(1)(x,τ )+ T z −z2 +z[G (x,τ )−G (x,τ )] (29)
13 B a T +G (τ ,τ ) (cid:16) 3 B b B a (cid:17)
3 B a b
or
1
G(1)(x,z) = G(1)(x,τ )+ T z −z2 +z[G (x,τ )−G (x,τ )] . (30)
13 B b T +G (τ ,τ ) (cid:16) 3 B a B b (cid:17)
3 B a b
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Eqs.(29) and (30) are different from each other, however they are essentially same under
the exchange of integral variables τ ↔ τ . If we follow the rule of transformations
a b
(3),(21) and (22) for τ ≤ τ , we have to choose (30) which can be derived from boundary
a b
conditions x(0) = x(τ ) and x(T ) = x(τ ) in the path integral (12), where τ is regarded
b 3 a b
astheoriginofτ parameter. Ifwewanttodiscuss(29),wehavetousethetransformations
(obtained from τ ↔ τ ) for the reverse case τ ≤ τ as mentioned above. This situation is
a b b a
completely same as the third equation in (23). As for (27) and (28), we have no attention
to such choice as whether (29) or (30) because they are already symmetric form in τ and
a
τ .
b
Eq.(27) hasbeenchecked in(24)to betransformed intoGsym, Gsym andGsym. Eq.(28)
11 22 12
also coincides with Gsym under our transformation rules discussed above
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G(1)(τ(3),τ(3)) = Gsym(τ(3),τ(3)). (31)
33 n m 33 n m
Now, it is enough to show the correspondence of (30) to the rest Gsym and Gsym. First,
23 31
writing down G(1)(x ,z) and G(1)(y ,z) according to (22), and then using (3) and (19),
13 n 13 n
we can rearrange them into similar forms as Gsym or Gsym; for example,
31 23
1
G(1)(y ,z) = b−y +z− T (b−y )2 +Tuz2 +T(1−u)(b−y +z)2 .
13 n n T(T +G (τ ,τ )) h 3 n n i
3 B a b
(32)
With the identification (21) and z = τ(3), we get the following relations
G(1)(x,z) = Gsym(τ(3),τ(1)), (33)
13 31
G(1)(y,z) = Gsym(τ(2),τ(3)). (34)
13 23
In this paper, we have discussed the relation between different worldline Green func-
tions G(1) and Gsym. It is shown that both sets of these Green functions are transformed
ij ij
into each other under the relations (3), (19) and (21). Also the integral measures on the
modular parameters T , T and those on node points τ , τ and T are in correspondence
1 2 a b
to each other. From (3), τ is related to τ(i) as
τ = (Tu−τ(2))θ(Tu−τ)+(τ(1) +Tu)θ(τ −Tu) (35)
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and this yields
T Tu T(1−u)
dτ = dτ(2) + dτ(1), (36)
Z Z Z
0 0 0
which produces combinatorial factors in (11), however such factors should be removed
taking account of a constraint on proper times similarly to the analysis in [10].
Our transformation rules are found through generalization of simple one-loop case.
Our discussion on two-loop Green functions might be useful for other theories; for exam-
ple, scalar and spinor QED diagrams to which super Green functions are applied [12] (see
also [13] where the same Green function as (15) and multi-loop diagrams are discussed
in scalar QED). We finally anticipate that our analysis may be a help to clarification of
multi-loop constructions in the worldline path integral approach.
Acknowledgments
TheauthorwouldliketothankJ.W.vanHolten,L.Magnea,K.Roland,M.G.Schmidt
for useful suggestions and T. Onogi for discussions in the early stage of this work.
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