In this work we calculate two two-loop massless Feynman integrals pertaining
to self-energy diagrams using NDIM (Negative Dimensional Integration Method).
We show that the answer we get is 36-fold degenerate. We then consider special
cases of exponents for propagators and the outcoming results compared with
known ones obtained via traditional methods.Comment: LaTeX, 10 pages, 2 figures, styles include

Schmidt, Alexandre G. M.

Suzuki, Alfredo T.

English

arXiv

A.T. Suzuki

A.G.M. Schmidt

Crossref

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Two-loop self-energy diagrams worked out with
NDIM
Alfredo T. Suzuki‡ and Alexandre G. M. Schmidt§
Instituto de F́isica Teórica, Universidade Estadual Paulista, R.Pamplona 145, São
Paulo SP, CEP 01405-900, Brazil.
Abstract. In this work we calculate two two-loop massless Feynman integrals
pertaining to self-energy diagrams using NDIM (Negative Dimensional Integration
Method). We show that the answer we get is 36-fold degenerate. We then consider
special cases of exponents for propagators and the outcoming results compared with
known ones obtained via traditional methods.
PACS numbers: 02.90+p, 03.70+k, 12.38.Bx
Short title: Two-loop self-energy diagrams worked out with NDIM
August 17, 2018
‡ E-mail: suzuki@axp.ift.unesp.br
§ E-mail: schmidt@power.ift.unesp.br
2
1. Introduction.
The dimensionality of space-time plays a key role in all branches of Physics. The
quantities we calculate, as theoreticians, depend very much on the number of dimensions
we are considering. Theories in higher and lower dimensions than four have been put
forth by many researchers and plentiful of good insights have been gained through
this exercise. Zooming in the arena of quantum field theory, we discover that the
dimensionality of space-time gained a more sophisticated status, being promoted from
a mere integer number to that of a complex variable, with the advent and development
of dimensional regularization by ’t Hooft et al and several other pioneers in the field [1].
In other words, we could say that quantum field theory (QFT), besides other great
ideas it inspired, physical and mathematical alike, did reveal this amazing possibility:
the analytic continuation of the space-time dimension D.
The union between the theory of analytic functions and QFT is very profitable.
Dimensional regularization (DREG), the technique that bears the concept of analytically
continued D, is one of its profits. As a step further in this direction Halliday et al [2, 3]
developed the idea of analytically continued D to negative values. Of course, the seminal
idea of negative values for D is already contained in the work of ’t Hooft and others. But
what is novel in Halliday’s insight is the amazing possibility of letting field propagators
be raised to positive powers, so that the integrand becomes polynomial. The thrust
behind the idea is that solving a polynomial integral should be — in principle at least
— easier to perform than rational ones elicited in the usual Feynman integrals. This very
simple argument, which we call negative dimensional integration method (NDIM), can
simplify the calculation of Feynman integrals in an astounding way [4, 5, 6, 7, 8, 9, 10].
In the usual DREG [11, 12, 13] the only quantities that preserve their meaning are
the Green’s functions [14]. We will not try to discuss whether they still have (or have
not) their meaning preserved within the context of NDIM nor speculate what are the
features, if any, of this “new world” of negative dimensions [8]. What we do is simply to
allow for it just for calculational purposes. The reader must have this important point
in mind.
In our previous works [7, 8] we calculated massive one-loop four point functions
(former reference) and a massless two-loop three-point vertex (latter reference) with the
NDIM approach. In the first, NDIM provided not only the well-known hypergeometric
functions but six other new results in a very straightforward manner [9]; while for the
two-loop vertex, we considered the particular case where two of its external momenta
were set on-shell, and NDIM responded with as many as twelve times — surprisingly
enough, all of them yielding the same correct result — that is, a twelvefold degeneracy.
This led us to conjecture that when the power series had unit argument and they were
all summable, then the result would be degenerate. That is, if this conjecture is correct,
3
we need only to carry out one sum — the most convenient one, of course. The conjecture
remains to be proven or disproved.
Here we put our NDIM to another “lab-test” [8] by considering two two-loop self-
energy diagrams which we call by the funny name “flying saucer” diagrams — side view
(Fig. 1) and front view (Fig. 2), just to make it easier for us to refer to them. The
outline for this article is as follows: In Section 2 we solve the two-loop Feynman integral
relative to these two graphs, i.e., the space-time dimension and the exponents of the
pertinent propagators are left arbitrary. Then, in Section 3, we particularize to suit
either the “flying saucer, side view” or the “flying saucer, front view” diagram cases.
And finally, Section 4 is devoted to our concluding remarks.
p− k
k − q
pp
q
k
Figure 1. Two-loop massless Feynman diagram: the “flying saucer”, side view.
2. Feynman Graphs with Four Massless Propagators.
The NDIM approach to solve Feynman integrals is beautiful in its simplicity: First,
we take the propagators of the integral we want to work out, multiply each one of
them by a specific parameter and then solve the D-dimensional gaussian integral whose
argument is that very expression. Let us see how it works in practice. Consider the
gaussian integral,
I(p2;D) =
∫
dDk dDq exp
[
−αq2− βk2 −γ(p− k)2 −ω(k − q)2
]
, (1)
which clearly is relevant to the diagrams we want to work out. It is not difficult to
integrate it; the result is,
I(p2;D) =
(
π2
λ
)D/2
exp
[
−
γp2
λ
(βω + αβ + αω)
]
, (2)
4
p− k
k − q
pp
q
kk
Figure 2. Two-loop massless Feynman diagram: the “flying saucer”, front view.
where λ = αβ+βω+αγ+ γω+αω. Expanding (2) in Taylor series and also expanding
the multinomial expression in λ, we get an eightfold summation,
I(p2;D) =
∞
∑
ni=0
(−p2)n1+n2+n3(−n1 − n2 − n3 −
1
2
D)!
n1!n2!n3!n4!n5!n6!n7!n8!
× αn1+n2+n4+n6+n8βn1+n3+n4+n5γn1+n2+n3+n6+n7
× ωn2+n3+n5+n7+n8, (3)
with the constraint −n1 − n2 − n3 −
1
2
D = n4 + n5 + n6 + n7 + n8 coming from the
multinomial expansion.
The second step is simpler and faster: expand the exponential (1) in Taylor series
first to get,
I(p2;D) =
∞
∑
i,j,l,m=0
(−1)i+j+l+m
αiβjγlωm
i!j!l!m!
J (i, j, l,m;D), (4)
where we define,
J (i, j, l,m;D) =
∫
dDk dDq (q2)i(k2)j
[
(p− k)2
]l [
(k − q)2
]m
, (5)
which is our negative dimensional integral.
Comparing (3) and (4) we get an expression for the negative-D integral,
J (i, j, l,m;D) = (−π)D(p2)σ G
∞
∑
ni=0
1
n1!n2!n3!n4!n5!n6!n7!n8!
, (6)
where we define the product of gamma functions,
G = Γ(1 + i)Γ(1 + j)Γ(1 + l)Γ(1 +m)Γ(1 − σ −
1
2
D),
5
and since the two expressions must equal, sum indices in the former and exponents of
propagators in the latter, must satisfy the system,















n1 + n2 + n4 + n6 + n8 = i
n1 + n3 + n4 + n5 = j
n1 + n2 + n3 + n6 + n7 = l
n2 + n3 + n5 + n7 + n8 = m
n1 + n2 + n3 = σ
(7)
where σ = i+j+l+m+D and the last equation comes from the multinomial expansion.
Observe that the equations above are linear, but because we have eight unknowns and
only five equations, in order to solve this system we must choose three of the unknowns
and solve it in terms of them. There are many ways in which this choice can be done;
in fact, there are C83 = 8!/(5!3!) = 56 possibilities altogether. However, 20 out of the 56
lead us to trivial solutions, which present no interest at all. The remaining 36 give us
the results for the Feynman integral when we plug their solutions in equation (6).
We will solve the non-trivial systems and write down the general results, but before
doing that, let us see what we can do to lessen our task. Looking at the Feynman
diagram we can spot symmetry properties that help us in this. Thus, we expect the
outcoming result to be symmetric under the exchange i ↔ m, which in turn will reduce
by half the number of distinct systems that we need to deal with, since the symmetry
will account for the remaining half.
Let us then first consider the solution that leaves n5, n6, n8 as free indices in the
summation; let us call it Ja. It yields,
Ja = (−π)
D(p2)σP1
∞
∑
n5,n6,n8=0
(−1)n6(−i−m− 1
2
D|n8)(−l + σ|n6)
n5!n6!n8!
×
(−j −m− 1
2
D|n5 − n6)(
1
2
D + l|n5 + n8)
(1 + l −m|n5 − n6 + n8)
, (8)
where
P1 =
Γ(1 + i)Γ(1 + j)Γ(1 + l)Γ(1 +m)
Γ(1 + l −m)Γ(1 + i+m+ 1
2
D)Γ(1 + j +m+ 1
2
D)Γ(1 + l − σ)
×
Γ(1− σ − 1
2
D)
Γ(1− l − 1
2
D)
,
with the Pochhammer symbol [15] denoted by,
(a|m) ≡ (a)m =
Γ(a+m)
Γ(a)
.
Using one of the properties of the Pochhammer symbol [15], i.e.,
(a| −m) =
(−1)m
(1− a|m)
, (9)
6
one can identify these series as hypergeometric [16]; in fact, we can rewrite them in a
convenient manner using another property,
(a|b+ c) = (a|b)(a + b|c), (10)
and sum, for example, the n8 series using the well-known formula [15],
2F1(a, b; c|1) =
Γ(c)Γ(c− a− b)
Γ(c− a)Γ(c− b)
, (11)
yielding,
Ja = (−π)
D(p2)σP1P2
∞
∑
n5,n6=0
(1 + i| − n6)(−j −m−
1
2
D|n5 − n6)
n5!n6!(1−m−
1
2
D| − n6)
×
(1
2
D + l|n5)(−l + σ|n6)
(1 + i+ l + 1
2
D|n5 − n6)
, (12)
where
P2 =
Γ(1 + i)Γ(1 + l −m)
Γ(1−m− 1
2
D)Γ(1 + i+ l + 1
2
D)
. (13)
In a similar manner we can sum the two remaining series, getting as a result,
Ja = (−π)
D(p2)σ
Γ(1 + i)Γ(1 + l)Γ(1 +m)Γ(1− σ − 1
2
D)
Γ(1 + σ)Γ(1− i− 1
2
D)Γ(1−m− 1
2
D)Γ(1− l − 1
2
D)
×
Γ(1 + i+ j +m+ 1
2
D)Γ(1− i−m−D)
Γ(1 + i+m+ 1
2
D)Γ(1 + l − σ)
. (14)
The last and final step that need to be taken is to bring this result back to our
real physical world with positive D. Grouping the gamma functions in the numerator
with the ones in the denominator in convenient Pochhammer symbols and using (9), we
arrive at
J ACa = π
D(p2)σ(−i|i+m+
1
2
D)(−m|i+m+
1
2
D)(σ +
1
2
D| − 2σ −
1
2
D)
× (−l|σ)(−i − j −m−
1
2
D|j)(i+m+D| − i+ l −m−
1
2
D). (15)
This very simple operation allows us to analytically continue the result back into
our real physical world, D > 0. Equation (15) is the general result, and we note
that it is symmetric in i ↔ m as it should be, and it is correct [17]. The reader will
ask immediately: What is(are) the result(s) that the other solution(s) provide? NDIM
answers in as brief and surprising a manner as it could be: the same. Indeed just to
make sure we went through all of them, and verified that it is possible to sum all the
emerging series and they provide the same result, namely, equation (14) which leads to
the correct expression, equation (15), i.e., we have a thirty-six-fold degeneracy!
7
A word of caution here: Not all the sums can be so easily dealt with. Yet, just
to convince the reader that it is possible to sum them all, we shall carry out one more
summation, the hardest one. The degeneracy above mentioned can be classified into
two sets: 32 solutions are like Ja with relatively easy sums to carry out, while 4 of them
are like the following one which we call Jb. Consider then the solution with indices
n1, n4, n5,
Jb = (−π)
D(p2)σP3
∞
∑
n1,n4,n5=0
(−1)n4(1
2
D + l|n4 + n5)(
1
2
D +m|n1 + n4)
n1!n4!n5!(1− j + σ|n4 + n5)
×
(−j|n1 + n4 + n5)
(1− i− j − 1
2
D|n1 + n4)
, (16)
where
P3 =
Γ(1 + i)Γ(1 + l)Γ(1 +m)Γ(1− σ − 1
2
D)
Γ(1− j + σ)Γ(1−m− 1
2
D)Γ(1− i− j − 1
2
D)Γ(1− l − 1
2
D)
. (17)
Using the same procedure we used for Ja, we can carry out the n5 summation,
yielding,
Jb = (−π)
D(p2)σP3P4
∞
∑
n1,n4=0
(1
2
D + l|n4)(
1
2
D +m|n1 + n4)(−σ|n1)
n1!n4!(
1
2
D + l − σ|n1 + n4)
×
(−j|n1 + n4)
(1− i− j − 1
2
D|n1 + n4)
, (18)
where
P4 =
Γ(1− j + σ)Γ(1− l − 1
2
D + σ)
Γ(1 + σ)Γ(1− j − l − 1
2
D + σ)
.
However, this time, neither of the remaining sums (in n1 or n4) can be written in
terms of 2F1, but rather in terms of 3F2 which are not summable for arbitrary values of
its parameters. Here comes the trick that will do the required job: Put the n4 series in
terms of a 3F2 function and use the property [16]
3F2(a, b, c; e, f |1) = Q 3F2(e− a, f − a, s; s+ b, s+ c|1), (19)
where s = e+ f − a− b− c and
Q =
Γ(e)Γ(f)Γ(s)
Γ(a)Γ(s+ b)Γ(s + c)
.
A good choice is to take
a = −j + n1
b = 1
2
D + l
c = 1
2
D +m+ n1
e = 1
2
D + l − σ + n1
f = 1− i− j − 1
2
D + n1
(20)
8
so that the gamma functions in Q simplify several factors in the series and some of them
can be grouped by the property (10) giving,
Jb = (−π)
D(p2)σP3P4P5
×
∞
∑
n1,n4=0
(1
2
D + j + l − σ|n4)(
1
2
D +m|n1)(−σ|n1)
n1!n4!(1− i− σ −
1
2
D|n1 + n4)
×
(1− i− 1
2
D|n4)(1− i−m− σ −D|n4)
(1− i+ l −m− σ − 1
2
D|n4)
, (21)
where
P5 =
Γ(1
2
D + l − σ)Γ(1− i− j − 1
2
D)Γ(1− i−m− σ −D)
Γ(−j)Γ(1− i− σ − 1
2
D)Γ(1− i+ l −m− σ − 1
2
D)
.
Now the sum in n1 is a 2F1 function and we can sum it using the Gauss summation
formula (11). The gamma functions that arise from this summation simplify the
remaining series in n4 from a 3F2 into a 2F1 function that can be summed again using
the usual Gauss summation formula. The end result is,
Jb = (−π)
D(p2)σP3P4P5P6P7, (22)
where P6 came from the n1 sum,
P6 =
Γ(1− i− σ − 1
2
D)Γ(1− i−m−D)
Γ(1− i− 1
2
D)Γ(1− i−m− σ −D)
,
and P7 came from the last one,
P7 =
Γ(−j)Γ(1− i+ l −m− σ − 1
2
D)
Γ(1
2
D + l − σ)Γ(1− σ + l)
.
Multiplying all the gamma factors (P3, ..., P7) we get, exactly, the expression (14),
that leads to the correct result (15).
3. Special Cases.
The scalar integral we calculated in the previous section has particular cases of interest,
namely, the “flying saucer” diagrams, side view (Fig.1) and front view (Fig.2). For the
side view diagram the exponents of the propagators are all equal to minus one, while
for the front view diagram the exponents are minus one except for j = −2.
Let us denote by J AC the general result for the “flying saucer” diagram; when we
take i = j = l = m = −1 we have the result for the side view diagram,
J ACSV = π
D(p2)D−4
Γ3(1
2
D − 1)Γ(D − 3)Γ(2− 1
2
D)Γ(4−D)
Γ(3− 1
2
D)Γ(D − 2)Γ(3
2
D − 4)
, (23)
9
while when we take i = −1, j = −2, l = m = −1 we have the result for the front view
diagram,
J ACFV = π
D(p2)D−5
Γ3(1
2
D − 1)Γ(5−D)Γ(D − 4)Γ(2− 1
2
D)
Γ(4− 1
2
D)Γ(D − 2)Γ(3
2
D − 5)
, (24)
which reproduce well-known results [17, 18].
4. Conclusion.
The analytic continuation of the space-time dimension D into negative values has shown
us advantages never dreamed of before: we interpret the analytic continuation like
in usual DREG but solve the Feynman integrals in a much simpler way because the
integrands we have to deal with are polynomial. The way back road, via another
analytic continuation is straightforward and the whole procedure is quite simple and
elegant. There are no cumbersome parametric integrals to solve; on the contrary, the
only things one needs to know are how to solve gaussian integrals, and systems of linear
algebraic equations! Furthermore, we have surprising manifold degenerate solutions for
a single integral. Our previous conjecture on this topic seems to hold, though further
research is needed to prove it.
Acknowledgments
AGMS wishes to thank CNPq (Conselho Nacional de Desenvolvimento Cient́ifico e
Tecnológico) for financial support.
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(Ph.D. Thesis, Imperial College, 1987).
[3] G.V.Dunne, I.G.Halliday, Phys.Lett. B193,2(1987)247.
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10
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Two-loop self-energy diagrams worked out with NDIM

In this work we calculate two two-loop massless Feynman integrals pertaining to self-energy diagrams using NDIM (Negative Dimensional Integration Method). We show that the answer we get is 36-fold degenerate. We then consider special cases of exponents for propagators and the outcoming results compared with known ones obtained via traditional methods

Suzuki, A. T.

Schmidt, A. G M

Acervo Digital da Universidade Estadual Paulista

The European Physical Journal C

In this work we calculate two two-loop massless Feynman integrals pertaining to self-energy diagrams using NDIM (Negative Dimensional Integration Method). We show that the answer we get is 36-fold degenerate. We then consider special cases of exponents for propagators and the outcoming results compared with known ones obtained via traditional methods.Inst. de Fis. Teórica Universidade Estadual Paulista, R.Pamplona 145, São Paulo SP, CEP 01405-900Inst. de Fis. Teórica Universidade Estadual Paulista, R.Pamplona 145, São Paulo SP, CEP 01405-90

Two-loop self-energy diagrams worked out with NDIM

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