arXiv:1006.1806v5 [physics.gen-ph] 17 Aug 2010
Operator Regularization of Feynman diagrams at one loop A.Y. Shiekh∗ Department of Physics Mesa State College, Grand Junction, CO U.S.A.
Abstract It may be possible to use operator regularization with Feynman diagrams.
1
Introduction
Operator regularization of quantum field theory [1], [2] has not in general been used in conjunction with Feynman diagrams, although such use has been implied [4], [5]. It also has the distinct advantage that it can be used with formally non-renormalizable theories [3], [4] since the divergences are not reabsorbed, but each is removed and replaced by an arbitrary factor; so it might seem well worth the effort of having it work with Feynman diagrams. For use with Feynman diagrams, operator regularization needs to give results equivalent to other methods of regularization; the results will not ∗
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1
be identical however, since operator regularization, unlike say dimensional regularization, removes divergences. We begin with a description of operator regularization, how it works, and why it should give equivalent results to dimensional regularization. This is followed by two examples (one in scalar theory, one in QED) to show how operator regularization in practice is no harder than dimensional regularization.
2
Operator regularization
Using the generalized operator regularization scheme, namely εn dn (1) Ω = lim n (1 + α1 ε + ... + αn εn ) Ω−ε−m ε→0 dε n! where the αn s are arbitrary, and for the one loop case it is enough to use
−m
n = 1, namely d ε(1 + αε)Ω−ε−m ε→0 dε
Ω−m = lim
2.1
What operator regularization achieves
Starting with the generalized operator regularization for a divergent Ω−m εn dn Ω = lim n (1 + α1 ε + ... + αn εn ) Ω−ε−m ε→0 dε n! where the αn s are arbitrary.
−m
Noting that Ω−ε−m has the Laurent expansion c−1 c−n + ... + + c0 + O(ε) εn ε one can now see that the effect of operator regularization is to replace the Ω−ε−m =
divergent poles by arbitrary constants
2
1 → αn εn to yield the finite interpretation Ω−m = αn c−n + ... + α1 c−1 + c0 2.1.1
A cautionary tale
The operator regularization of the logarithm (as used in the Schwinger approach) dn ln Ω = − lim n ε→0 dε
εn−1 −ε Ω n!
!
(2)
can be transformed into the form given above (as used in the Feynman approach) using Ω−m =
(−1)m−1 dm ln Ω (m − 1)! dΩm
to yield −m
Ω
dn = lim n ε→0 dε
(1 + ε)...(m − 1 + ε) εn −ε−m Ω (m − 1)! n!
!
which simplifies to dn ε ε εn −ε−m Ω = lim n 1+ ... 1 + Ω (3) ε→0 dε 1 m − 1 n! and can be compared directly to operator regularization as might be used in −m
the Feynman diagram context dn εn −ε−m Ω (4) ε→0 dεn n! and they are seen to differ. However, so long as one includes all the arbitrary
Ω−m = lim
factors 3
resolutions of zero z }| {
εn dn (1 + α1 ε + ... + αn εn ) Ω−ε−m n ε→0 dε n!
Ω−m = lim
then they are equivalent.
3
Equivalence at one-loop
Starting from the basic integral that carries the divergences in one-loop Feynman diagrams Z
d2w l 1 Γ(A − w) 1 1 = 2w 2 2 A w 2 (2π) (l + M + 2l.p) (4π) Γ(A) (M − p2 )A−w
the other versions follow by differentiating with respect to pµ , so we can concentrate on this one alone. One needs to show that operator regularization and dimensional regularization treat the result 1 Γ(A − w) 1 (5) (4π)w Γ(A) uA−w in equivalent ways when divergent, namely in the limit A − w = 0, −1, .... In operator regularization this is !
d ε(ε − 1) Γ(ε − n) d lim (4π) 2 ε→0 dε Γ(A + ε) uε−n where n ≡ w − A, while for dimensional regularization it is 1
1 (4π) having put w =
d 2
d −ε 2
Γ(ε − n) 1 Γ(A) uε−n
(6)
(7)
− ε, where d is the number of space-time dimensions.
The operator regularized expression evaluates to (using the help of Mathematica [8]) 4
(−1)n
un α − ψ(A) + ψ(n + 1) − ln u d (4π) 2 (A − 1)!n!
(8)
and the dimensionally regularized expression has the Laurent expansion (−1)n
un 1 + ln(4π) + ψ(n + 1) − ln u + O(ε) d (4π) 2 (A − 1)!n! ε
(9)
where ψ is the polygamma function. These agree in form, and so all one loop results should be equivalent when using operator regularization or dimensional regularization.
Example 1: φ3 in 6 dimensions
4
This investigation began with the suggestion1 to look at the simpler divergent integral associated with scalar particles at one loop in 6 dimensions (rotated to Euclidean space) Z
d6 l 1 1 6 2 2 (2π) l + m (l + p)2 + m2
(10)
This integral can be prepared by using the Feynman parameter ‘trick’ xa1 −1 (1 − x)a2 −1 1 Γ(a1 + a2 ) Z 1 dx a1 a2 = D1 Dk Γ(a1 )Γ(a2 ) 0 (D1 x + D2 (1 − x))a1 +a2 to yield =
Z 1 0
dx
Z
d6 l 1 6 2 2 2 (2π) [l + m + p (1 − x) + 2l.p(1 − x)]2
(11)
One can now proceed by comparing the use of operator regularization for this divergent integral to that of dimensional regularization. 1
private communication with Professor D. G. C. McKeon
5
4.1
Operator regularization
Using the generalized operator regularization scheme Z 1 0
d Z d6 l dx lim ε→0 dε (2π)6
ε(1 + αε) [l2 + m2 + p2 (1 − x) + 2l.p(1 − x)]ε+2
!
then performing the momentum integrals using the identity Z
1 1 Γ(A − w) d2w l = 2w 2 2 A w (2π) (l + M + 2l.p) (4π) Γ(A) (M 2 − p2 )A−w
(12)
leads to 1 Z1 d = dx lim ε→0 dε (4π)3 0
ε(1 + αε) Γ(ε − 1) Γ(ε + 2) [m2 + p2 x(1 − x)]ε−1
!
where one can now perform the operator regularization limit, using d lim ε→0 dε
ε(1 + αε) Γ(ε − 1) Γ(ε + 2) uε−1
!
= u(−α + ln u)
to yield the finite result
=
1 Z1 2 2 2 2 dx m + p x(1 − x) −α + ln(m + p x(1 − x)) (4π)3 0
This can be evaluated further, but this is not necessary for comparison with the result from dimensional regularization, but µ2 should be included in the logarithm, and taken from the arbitrary (α) part, to yield the fixed part ! 1 Z1 m2 + p2 x(1 − x) 2 2 dx m + p x(1 − x) ln (4π)3 0 µ2
(13)
and the arbitrary part α p2 2 − m + (4π)3 6
6
!
(14)
4.2
Dimensional regularization
Regularization might now be done using the dimensional approach; starting from the same point (equation 11) with dimensional extension Z 1
dx
0
Z
1 d6−2ε l 6−2ε 2 2 2 (2π) [l + m + p (1 − x) + 2l.p(1 − x)]2
and again using the identity of equation 12 leads to Γ(−1 + ε) Z 1 1 = dx 3−ε 2 2 (4π) [m + p x(1 − x)]−1+ε 0 Using Mathematica [8] to help expand about ε = 0 1 Γ(−1 + ε) 1−ε u =− 3−ε (4π) (4π)3
u ln u 1 − γ + 1 + ln 4π u + + O(ε) ε (4π)3
where γ denotes the Euler-Mascheroni constant (0.577...); things become, for the finite part (again including the µ2 ) ! 1 Z1 m2 + p2 x(1 − x) 2 2 dx m + p x(1 − x) ln (4π)3 0 µ2
(15)
leaving the divergent part 1 − (4π)3
1 − γ + 1 + ln 4π ε
p2 m + 6 2
!
(16)
which are seen to agree in form with the result from operator regularization (equations 13 and 14).
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Example 2: QED
Applying operator regularization to the three divergent one loop Feynman diagrams in QED, following Ramond [6] with the Feynman gauge and in Euclidean space. 7
5.1
One loop correction to the fermion line
Starting with the Feynman diagram for the one loop correction to the fermion line (Σ(p)) (diagrams drawn using JaxoDraw [7])
l p
p!l
Σ(p) = −e2
Z
d4 l −i δµν γ γ µ ν (2π)4 p/ − /l + m l2
(17)
Following Ramond [6] this simplifies to γµ [p/(1 − x) − m]γµ d4 l Σ(p) = −ie dx (18) (2π)4 [l2 + m2 x + p2 x(1 − x)]2 0 which is taken as the common starting point for both dimensional and oper2
Z 1
Z
ator regularization. 5.1.1
Operator regularization (one loop fermion correction)
Now proceeding with operator regularization, following the same general route that would be taken with dimensional regularization
Σ(p) = −ie2
Z 1
d Z d4 l ε(1 + αε)γµ [p/(1 − x) − m]γµ ε→0 dε (2π)4 [l2 + m2 x + p2 x(1 − x)]ε+2
dx lim
0
using equation 35 to perform the momentum integrals e2 Z 1 d = −i dx γµ [p/(1 − x) − m]γµ lim 2 ε→0 dε (4π) 0
ε(1 + αε) Γ(ε) 2 Γ(ε + 2) [m x + p2 x(1 − x)]ε
Mathematica [8] for the limits d lim ε→0 dε
ε(1 + αε) Γ(ε) Γ(ε + 2) uε 8
!
= α − 1 − ln u
!
and the fact that in 4 dimensions γµ γµ = −4 and γµ γρ γµ = 2γρ (from {γµ , γν } = −2δµν ) yields
= −2i
e2 Z 1 2 2 dx [ p (1 − x) + 2m] α − 1 − ln(m x + p x(1 − x)) / (4π)2 0
to deliver the fixed part (µ2 taken from the arbitrary α) ! m2 x + p2 x(1 − x) e2 Z 1 dx [p/(1 − x) + 2m] ln 2i (4π)2 0 µ2
(19)
and the arbitrary part −i
e2 (α − 1)(p/ + 4m) (4π)2
(20)
Compare this to the result from dimensional regularization [6], the finite part ! e2 Z 1 m2 x + p2 x(1 − x) 2i dx [p/(1 − x) + 2m] ln (4π)2 0 4πµ2
(21)
and the divergent part e2 −i (4π)2
e2 1 − γ − 1 (p/ + 4m) −i 2m ε (4π)2
|
{z
(22)
}
extra term where γ denotes the Euler-Mascheroni constant (0.577...).
There is a constant difference between these two methods that stems from dimensionally continuing the gamma matrices (in dimensional regularization alone), but this will be absorbed by the counter terms.
5.2
One loop correction to the photon line
Continuing with the Feynman diagram for the one loop correction to the photon line (Πµν (p))
9
l+p
l
p
Πµν (p) = −e
2
d4 l 1 1 Tr γµ γν 4 /l + p/ + m /l + m (2π) "
Z
#
(23)
Following Ramond [6] this simplifies to
Πµν (p) = 8e2 (pµ pν − δµν p2 )
Z 1
dx
Z
0
d4 l x(1 − x) 4 2 2 (2π) [l + m + p2 x(1 − x)]2
(24)
which is taken as the common starting point for both dimensional and operator regularization. 5.2.1
Operator regularization (one loop photon correction)
Proceeding with operator regularization, and again following the same general route that would be taken with dimensional regularization
Πµν (p) = 8e2 (pµ pν − δµν p2 )
Z 1 0
d Z d4 l ε(1 + αε) x(1 − x) ε→0 dε (2π)4 [l2 + m2 + p2 x(1 − x)]ε+2
dx lim
Performing the momentum integrals using equation 35 Z 1 d e2 2 (pµ pν − δµν p ) dx lim =8 2 ε→0 (4π) dε 0
ε(1 + αε) Γ(ε) 2 Γ(ε + 2) [m + p2 x(1 − x)]ε
again using d lim ε→0 dε
ε(1 + αε) Γ(ε) Γ(ε + 2) uε
yielding the finite part: 10
!
= α − 1 − ln u
!
Z 1 e2 m2 + p2 x(1 − x) 2 −8 (p p − δ p ) dx ln µ ν µν (4π)2 µ2 0
!
(25)
and the divergent part 4 e2 (pµ pν − δµν p2 )(α − 1) (26) 3 (4π)2 Compare this against the result of dimensional regularization; the finite part Z 1 m2 + p2 x(1 − x) e2 2 dx ln (pµ pν − δµν p ) −8 (4π)2 2πµ2 0
!
(27)
and the divergent part 1 4 e2 (pµ pν − δµν p2 ) −γ 2 3 (4π) ε which are seen to agree in form.
5.3
(28)
One loop correction to the vertex
Lastly the Feynman diagram for the one loop correction to the vertex (Γρ (p, q)) l
p
Γρ (p, q) = −ie3
p+l
Z
q
q+l
d4 l 1 1 δλσ γ γ γ λ ρ σ (2π)4 p/ + /l + m /q + /l + m l2
(29)
Following Ramond [6] and retaining only the divergent part for this investigation, this simplifies to 3 Γ(1) ρ (p, q) = −2ie
Z 1 0
dx
Z 1−x 0
11
dy
Z
d4 l γσ /lγρ /lγσ (2π)4 (l2 + M 2 )3
(30)
where M 2 ≡ m2 (x + y) + p2 x(1 − x) + q 2 y(1 − y) − 2p.q xy which is taken as the common starting point for both dimensional and operator regularization. 5.3.1
Operator regularization (one loop vertex correction)
Proceeding with operator regularization, and again following the same general route that would be taken with dimensional regularization
3 Γ(1) ρ (p, q) = −2ie
Z 1 0
dx
Z 1−x 0
d Z d4 l γσ /lγρ /lγσ ε(1 + αε) ε→0 dε (2π)4 (l2 + M 2 )ε+3
dy lim
Performing the momentum integrals (equation 36) Z 1−x e2 Z 1 d = −ie dy lim dx 2 ε→0 (4π) 0 dε 0
!
ε(1 + αε) Γ(ε) γσ γτ γρ γτ γσ Γ(ε + 3) (M 2 )ε
then using d lim ε→0 dε
ε(1 + αε) Γ(ε) Γ(ε + 3) uε
!
1 = (−3 + 2α − 2 ln u) 4
with γσ γµ γρ γν γσ = 2γν γρ γµ and γµ γρ γµ = 2γρ leads to the finite part ! Z 1−x M2 e2 Z 1 dy ln 2ieγρ dx (4π)2 0 µ2 0
(31)
where M 2 ≡ m2 (x + y) + p2 x(1 − x) + q 2 y(1 − y) − 2p.q xy, and the divergent part e2 3 − ieγρ − +α 2 (4π) 2
(32)
Compare this to the result from dimensional regularization; the finite part ! Z 1−x e2 Z 1 M2 2ieγρ dx dy ln (4π)2 0 4πµ2 0
12
(33)
where again M 2 ≡ m2 (x + y) + p2 x(1 − x) + q 2 y(1 − y) − 2p.q xy, and the divergent part − ieγρ
e2 (4π)2
1 −γ−1 ε
(34)
which agree in form, recalling that operator regularization goes further than dimensional regularization in so much as that it actually removes the divergences.
6
Conclusion
The above is suggestive that operator regularization can in fact be used in conjunction with Feynman diagrams. The calculation using operator regularization is actually somewhat simpler than that using dimensional regularization as the gamma matrices are not dimensionally continued when using operator regularization.
Integrals 1 d2w l 1 Γ(A − w) = 2w 2 2 A w (2π) (l + M ) (4π) Γ(A) (M 2 )A−w
(35)
d2w l lµ lν 1 δµν Γ(A − 1 − w) = 2w 2 2 A w (2π) (l + M ) (4π) Γ(A) 2 (M 2 )A−1−w
(36)
Z
Z
References [1] D. G. C. McKeon and T. N. Sherry, Operator regularization of Green’s functions, Phys. Rev. Lett. 59, p.532 (1987) [2] D. G. C. McKeon and T. N. Sherry, Operator regularization and oneloop Green’s functions, Phys. Rev. D 35, p.3854 (1987) 13
[3] R. B. Mann, L. Tarasov, D. G. C. McKeon and T. Steele, Operator regularization and quantum gravity, Nuclear Physics B, Volume 311, Issue 3, p.630 (1989) [4] A.Y. Shiekh, Quantizing Orthodox Gravity, Can. J. Phys., 74, p.172 (1996) [5] A.Y. Shiekh, Zeta-function regularization of quantum field theory, Can. J. Phys., 68, p.620 (1990) [6] P. Ramond, Field Theory: A Modern Primer, 2nd ed. (Westview Press, 2001) [7] D. Binisi and L. Theussl, JaxoDraw: A graphical user interface for drawing Feynman diagrams, Comp. Phys. Comm. 161, p.76 (2004) [8] Wolfram Research, Inc., Mathematica, Version 7.0 (2008)
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