Table Of ContentSFB/CPP-13-03
TTP13-02
Four-loop corrections with two closed fermion loops to
fermion self energies and the lepton anomalous
3 magnetic moment
1
0
2
n Roman Leea, Peter Marquardb, Alexander V. Smirnovc,
a
J Vladimir A. Smirnovd,e, Matthias Steinhauserb
8
2
(a) Budker Institute of Nuclear Physics and Novosibirsk State University
]
h 630090 Novosibirsk, Russia
p
- (b) Institut fu¨r Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT)
p
e 76128 Karlsruhe, Germany
h
[ (c) Scientific Research Computing Center, Moscow State University
1 119992 Moscow, Russia
v
1 (d) Skobeltsyn Institute of Nuclear Physics, Moscow State University
8
119992 Moscow, Russia
4
6
(e) Institut fu¨r Mathematik, Humboldt-Universita¨t zu Berlin
.
1
12489 Berlin, Germany
0
3
1
:
v
Abstract
i
X
r We compute the eighth-order fermionic corrections involving two and three
a
closed massless fermion loops to the anomalous magnetic moment of the muon.
Therequired four-loop on-shell integrals are classified and explicit analytical results
forthemasterintegralsarepresented. Asfurtherapplicationswecomputethecorre-
spondingfour-loop QCDcorrections to the mass and wave function renormalization
constants for a massive quark in the on-shell scheme.
PACS numbers: 12.20.Ds 12.38.Bx 14.65.-q
1 Introduction
In the last about ten years several groups have been active in computing four-loop cor-
rections to various physical quantities. Among them are the order α4 corrections to the
s
R ratio and the Higgs decay into bottom quarks [1–3], four-loop corrections to moments
of the photon polarization function [4–8] which lead to precise results for the charm and
bottom quark masses (see, e.g., Ref. [9]), and the free energy density of QCD at high
temperatures [10]. The integrals involved in such calculations are either four-loop mass-
less two-point functions or four-loop vacuum integrals with one non-vanishing mass scale.
In this paper we take the first steps towards the systematic study of a further class of
four-loop single-scale integrals, the so-called on-shell integrals where in the loop massless
and massive propagators may be present and the only external momentum is on the mass
shell.
On-shell integrals enter a variety of physical quantities, where the anomalous magnetic
momentsandon-shellcounterterms areprominent examples. Thefirstsystematic studyof
two-loopon-shellintegralsneededfortheevaluationoftheon-shellmassandwavefunction
renormalization constants (ZOS and ZOS) for a heavy quark in QCD has been performed
m 2
in Refs. [11,12]. Already a few years later, in 1996 the analytical three-loop corrections
to the lepton anomalous magnetic moment a became available [13]. This result has been
l
checked inRefs.[14,15]. InRefs.[14,16]thethree-loopon-shellintegralshavebeenapplied
toQCD,namelytheevaluationofZOS andZOS. ThecalculationofRef.[14]hasconfirmed
m 2
the numerical result of [17,18] which has been available before. Both ZOS and ZOS have
m 2
also been computed in Ref. [15]. Further application of three-loop on-shell integrals are
discussed in Refs. [19,20]. There is no systematic study of four-loop on-shell integrals
availableintheliterature. Nevertheless, somefour-loopresults totheanomalousmagnetic
moment of the muon, a , have been computed analytically, in particular contributions
µ
from closed electron loops. E.g., the contribution where the photon propagator of the
one-loop diagram (see Fig. 1) is dressed by higher order corrections has been considered
in several papers [21–27]. Four-loop corrections where one of the two photon propagators
of the two-loop diagram is dressed by higher orders has been considered in Ref. [28,29].
Contributions where both photon propagators get one-loop electron insertions are still
missing. This gap will be closed in the present work. Let us mention that all four- and
even five-loop results for a are available in the literature in numerical form [27,30–33]
l
(see also the review articles [34,35]).
In this paper we take the first step towards the analytical calculation of four-loop on-shell
integrals by considering the subclass with two or three closed massless fermion loops,
which aremarkedby afactorn . Thus we areconcerned withfour-looptermsproportional
l
to n3 and n2 which we consider for three physical quantities: the anomalous magnetic
l l
moment of the muon, a , the on-shell mass renormalization constant, ZOS, and the on-
µ m
shell wave function renormalization constant, ZOS, for a massive quark. For the latter
2
QCD corrections to the quark two-point functions are computed whereas for the former
muon-photonvertex diagramshave to beconsidered. Some sample Feynmandiagramsare
2
q
p p
1 2
Figure 1: Sample Feyman diagrams for the photon-muon vertex contributing to a . Wavy
µ
and straight lines represent photons and fermions, respectively. In this paper we consider
the contribution where at least two of the closed loops correspond to massless fermions.
The last diagram in the second line is a representative of the so-called “light-by-light”
contribution.
given in Figs. 1 and 2. The precise definition of these quantities is provided in Sections 3
and 4.
Theoutlineofthepaperisasfollows: inthenextsectionweprovidedetailsofthefour-loop
on-shell integrals needed for our calculation. In particular, we identify all master integrals
andprovideanalyticalresultsinAppendix A.TherenormalizationconstantsZOS andZOS
m 2
are discussed in Section 3 and Section 4 is devoted to the anomalous magnetic moment of
the muon. We discuss the relation between the MS and on-shell fine structure constant
and provide analytical results for a . Finally, we conclude in Section 5. Appendix B
µ
contains the analytic results for the relation between the fine structure constant defined
in the MS and on-shell scheme.
3
Figure 2: Sample Feynman diagrams for the QCD corrections to the fermion propagator
contributing to ZOS and ZOS. Curly and straight lines represent gluons and fermions,
m 2
respectively. In this paper we consider the contribution where at least two of the closed
loops correspond to massless fermions.
2 Four-loop on-shell integrals
In this Section we present the setup used for the calculation and discuss the families of
four-loop on-shell integrals needed for the n2 and n3 corrections for ZOS, ZOS and a .
l l 2 m µ
Since all three cases reduce to the calculation of corrections to the fermion propagator we
consider in this Section the corresponding two-point function.
After the generation of the diagrams with QGRAF [36] we use q2e [37,38] to translate the
output into a FORM [39] readable form. In a next step exp [37,38] is applied to map
the momenta to one of five families. During the evaluation of the FORM code we apply
projectors and take traces to end up with integrals which only contain scalar products in
the numerator and quadratic denominators.
In the next step we have to reduce all occurring integrals to a minimal set of master
integrals. This is done using two different programs in order to have a cross check for
the calculation. On the one hand we use crusher [40] and on the other hand the C++
version of FIRE.1 Both programs implements Laporta’s algorithm [42] for the solution of
integration-by-parts identities [43]. We find complete agreement for the expressions where
the physical quantities are expressed in terms of master integrals.
Let us mention that we have performed our calculations for general gauge parameter
which drops out once the four-loop results for ZOS, ZOS and a are expressed in terms of
2 m µ
1The Mathematicaversion of FIRE is publicly available [41].
4
master integrals.2
Altogether we end up with 13 master integrals. Seven of them (shown in Fig. 3) are
products of one- and two-loop integrals whereas the remaining six integrals (cf. Fig. 4)
request a dedicated investigation. We calculate them using the Dimensional Recurrence
and Analyticity (DRA) method introduced in [44]. In order to fix the position and order
of the poles of the integrals, we use FIESTA [45,46]. The remaining constants are fixed
using the Mellin-Barnes technique [47–51]. In order to express the results in terms of
the conventional multiple zeta values we apply the PSLQ algorithm [52] on high-precision
numerical results (with several hundreds of decimal digits).3
The analytic results for the integrals in Fig. 4 are listed in Appendix A. Results in terms
of Gamma functions for the integrals in Fig. 3 are easily obtained recursively using the
formulae from the Appendix of Ref. [49]. For convenience also these results are given in
Appendix A.
All results have been cross-checked numerically with the help of FIESTA [46] where an
accuracy of at least four digits has been achieved.
3 Fermionic n2 and n3 contributions to ZOS and ZOS
l l m 2
Both ZOS and ZOS are obtained from the fermion two-point functions Σ(q) which can be
m 2
cast in the form
Σ(q,m ) = m Σ (q2,m )+(/q −m )Σ (q2,m ). (1)
q q 1 q q 2 q
Here m represents a generic quark mass whereas bare, on-shell and MS quark masses are
q
denoted by m0, M and m¯ .
q q q
Thederivationofready-to-useformulaeforZOS andZOS isdiscussed atlengthinRefs.[14,
m 2
15]. Thus, let us for convenience only repeat the final formulae which are applied in our
calculations. They read
ZOS = 1+Σ (M2,M ), (2)
m 1 q q
ZOS −1 = 1+2M2 ∂ Σ (q2,M ) +Σ (M2,M ). (3)
2 q ∂q2 1 q q2=M2 2 q q
q
(cid:12)
(cid:0) (cid:1)
(cid:12)
The expressions on the right-hand side are comput(cid:12)ed by introducing the momentum Q
2Note that ZOS and a have to be independent of the QCD gauge parameter ξ whereas we expect
m µ
that the n1 and n -independent terms of ZOS do depend on ξ.
l l 2
3Let us mention that the numerical evaluation of the factorizable four-loop master integrals for a
l
which reduce to the evaluation of the correspondingthree-loopmaster integralsin higher ordersof ǫ was
undertaken in Ref. [53] as a warm-up before a future full four-loop calculation. This was done with the
method of [42] based on difference equations. The achieved accuracy of several dozen of decimal digits
was not enough for using PSLQ.
5
L L L
1 2 3
L L L
4 5 6
L
7
Figure 3: Master integrals for the n2 and n3 contribution which are easily obtained by
l l
applying one- and two-loop formulae, see e.g., Ref. [49]. Solid lines carry the mass M
and dashed lines are massless. For L to L we have q2 = M2 where q is the external
1 6
momentum; L is a vacuum integral.
7
M M M
1 2 3
M M M
4 5 6
Figure 4: Non-trivial master integrals contributing to the n2 contribution. The same
l
notation as in Fig. 3 has been used.
6
with Q2 = M2 via q = Q(1+t) which leads to the equation
q
Q/ +M
Tr qΣ(q,M ) = Σ (q2,M )+tΣ (q2,M )
4M2 q 1 q 2 q
(cid:26) q (cid:27)
∂
= Σ (M2,M )+ 2M2 Σ (q2,M ) +Σ (M2,M ) t
1 q q q ∂q2 1 q q2=M2 2 q q
(cid:18) q (cid:19)
(cid:12)
+O(t2). (cid:12) (4)
(cid:12)
Hence, to obtain ZOS one only needs to calculate Σ for q2 = M2. To calculate ZOS, one
m 1 q 2
has to compute the first derivative of the self-energy diagrams. Note that the renormal-
ization of the quark mass is taken into account iteratively by explicitly calculating the
corresponding counterterm diagrams.
We write the perturbative expansion for ZOS in terms of the renormalized strong coupling
m
as (γ is the Euler-Mascheroni number)
E
α (µ) eγE −ǫ α (µ) 2 eγE −2ǫ
ZOS = 1+ s δZ(1) + s δZ(2)
m π 4π m π 4π m
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
α (µ) 3 eγE −3ǫ α (µ) 4 eγE −4ǫ
+ s δZ(3) + s δZ(4) +O α5 . (5)
π 4π m π 4π m s
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
This allows us to take the ratio between the on-shell and MS [54–56] mass renormalization
constant which is given by
m¯ (µ) ZOS
zOS(µ) = q = m
m M ZMS
q m
2 3 4
α (µ) α (µ) α (µ) α (µ)
= 1+ s δz(1) + s δz(2) + s δz(3) + s δz(4)
π m π m π m π m
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
+O α5 (6)
s
(i) (cid:0) (cid:1)
The coefficients δz are by construction finite.
m
In the case of ZOS we choose the bare coupling as expansion parameter which in many
2
applications turns out to be convenient. Furthermore, the dependence on µ/M can be
q
written in factorized form which leads to shorter expressions. Thus we have
α0 eγE −ǫ α0 2 eγE −2ǫ
ZOS = 1+ s δZ(1) + s δZ(2)
2 π 4π 2 π 4π 2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
+ αs0 3 eγE −3ǫδZ(3) + αs0 4 eγE −4ǫδZ(4) +O α0 5 , (7)
π 4π 2 π 4π 2 s
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17)
(cid:0) (cid:1)
where each term δZ(n) contains a factor (µ2/M2)nǫ.
2 q
We refrain from repeating the one-, two- and three-loop results for ZOS and ZOS since
m 2
analytical expressions for general colour coefficients are available in the literature [14–16].
7
We split the four-loopcoefficient according to the number of closed massless fermion loops
and write (i ∈ {m,2})
δZ(4) = δZ(40) +δZ(41)n +δZ(42)n2 +δZ(43)n3. (8)
i i i l i l i l
(4)
with an analog notation for δz .
m
(42) (43) (42) (43)
In the following we present analytical results for δz , δz , δZ and δZ which
m m 2 2
read
l4 13l3 89 π2 ζ 1301 13π2
δz(43) = C T3 M + M + + l2 +l 3 + +
m F 144 216 432 36 M M 3 3888 108
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)
317ζ 71π4 89π2 42979
3
+ + + + , (9)
432 4320 648 186624
(cid:19)
l4 13l3 125l2 2489l 5ζ 19π4 π2 128515
δz(42) = C n T3 M + M + M + M + 3 − + +
m F h 48 72 144 1296 144 480 6 62208
(cid:18) (cid:19)
11l4 91l3 1 ζ π2 6539
+C C T2 − M − M +l2 − π2a − 3 − −
A F 192 144 M 12 1 4 8 2304
(cid:18) (cid:18) (cid:19)
a4 1 11 4a 37ζ π4 29π2 15953
+l 1 + π2a2 − π2a + 4 − 3 − − −
M 18 9 1 18 1 3 16 216 36 2592
(cid:18) (cid:19)
1 11a4 2 11 31π4a 103 44a 8a
− a5 + 1 − π2a3 + π2a2 − 1 − π2a + 4 + 5
45 1 54 27 1 27 1 1080 108 1 9 3
41ζ 13π2ζ 3245ζ 4723π4 527π2 2708353
5 3 3
− − − − − −
24 48 576 51840 384 497664
(cid:19)
11l4 97l3 1 ζ 5π2 157
+C2T2 + M + M +l2 π2a + 3 − −
F 384 576 M 6 1 8 96 2304
(cid:18) (cid:18) (cid:19)
1 2 11 8a 11ζ 11π4 21π2 50131
+l − a4 − π2a2 + π2a − 4 − 3 + − −
M 9 1 9 1 9 1 3 8 216 32 20736
(cid:18) (cid:19)
2a5 11a4 4 22 31 103 88a 16a
+ 1 − 1 + π2a3 − π2a2 + π4a + π2a − 4 − 5
45 27 27 1 27 1 540 1 54 1 9 3
305ζ 3π2ζ 2839ζ 3683π4 5309π2 2396921
5 3 3
+ + − + − − , (10)
48 8 576 51840 3456 497664
(cid:19)
µ2 4ǫ 1 65 89 + 13π2 151ζ3 + 73669 + 845π2
δZ(43) = C T3 + + 192 432 + 216 31104 2592
2 F M2 144ǫ4 864ǫ3 ǫ2 ǫ
(cid:18) q (cid:19)
9815ζ 589π4 1157π2 2106347
3
+ + + + , (11)
1296 4320 576 186624
(cid:19)
µ2 4ǫ 1 187 10957 − 5π2
δZ(42) = C n T3 + + 5184 108
2 M2 F h 36ǫ4 864ǫ3 ǫ2
(cid:18) q (cid:19) "
8
2π2a − 71ζ3 − 1013π2 + 349615 10 20 127
+ 3 1 54 2592 31104 − a4 − π2a2 + π2a
ǫ 9 1 9 1 18 1
80a 21719ζ π4 14027π2 13135057
4 3
− − − − +
3 1296 360 15552 186624
(cid:19)
11 761 −1π2a + ζ3 − 13π2 − 64433
+C C T2 − − + 6 1 16 192 13824
A F 192ǫ4 1152ǫ3 ǫ2
5a41 + 5π2a2 − 163π2a + 20a4 + 37ζ3 − 647π4 − 1627π2 − 18287
+ 18 9 1 72 1 3 288 8640 1152 768
ǫ
5 815a4 50 815 1 281
− a5 + 1 − π2a3 + π2a2 + π4a − π2a
9 1 216 27 1 108 1 18 1 18 1
815a 200a 2079ζ 209π2ζ 27977ζ 7411π4
4 5 5 3 3
+ + − − − −
9 3 32 144 1728 5760
436741π2 60973393
− −
41472 497664
(cid:19)
11 47 1π2a − 5ζ3 − 241π2 + 2363
+C2T2 + + 3 1 16 1152 1536
F 384ǫ4 192ǫ3 ǫ2
−5a4 − 10π2a2 + 163π2a − 40a4 − 773ζ3 + 383π4 − 1181π2 + 2893
+ 9 1 9 1 36 1 3 72 1728 576 2304
ǫ
10a5 815a4 100 815 1 281
+ 1 − 1 + π2a3 − π2a2 − π4a + π2a
9 108 27 1 54 1 9 1 9 1
1630a 400a 7145ζ 187π2ζ 50209ζ 8413π4
4 5 5 3 3
− − + + − +
9 3 48 48 576 6480
75089π2 261181
− − , (12)
4608 55296
(cid:19)(cid:21)
where l = lnµ2/M2, ζ is Riemann’s zeta function, a = ln2 and a = Li (1/2) (n ≥ 1).
M q n 1 n n
In the case of QCD the colour factors take the values C = 3,C = 4/3 and T = 1/2. In
A F
Eqs. (10) and (12) the contributions from closed heavy quark loops are marked by n = 1
h
which has been introduced for illustration.
In order to get an impression of the numerical size of the newly calculated terms we
evaluate zOS for µ = M . After inserting the numerical values for the colour factors we
m q
obtain (A ≡ α (M )/π)
s s q
zOS = 1−A 1.333+A2(−14.229−0.104n +1.041n)
m s s h l
+A3 −197.816−0.827n −0.064n2 +26.946n −0.022n n −0.653n2
s h h l h l l
+A4 −43.465n2 −0.017n n2 +0.678n3 +... +O A5 , (13)
s(cid:0) l h l l s (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1)
where the ellipses indicate n independent contributions and terms proportional to n
l l
which have not been computed. One observes that the n2 contribution at two loops and
l
the n3 contribution at three loops are quite small. This is in contrast to the linear n
l l
9
terms which can become quite sizeable. E.g., setting n = 5, which corresponds to the
l
case of the top quark, we obtain (for n = 1)
h
zOS = 1−A 1.333+A2(−14.332+5.207 )
m s s nl
+A3s −198.707+134.619nl −16.317n2l
(cid:16) (cid:17)
+A4s −1087.060n2 +84.768n3 +... +O A5s . (14)
l l
(cid:16) (cid:17)
(cid:0) (cid:1)
At two-loop order the n contribution is only a factor of three smaller than the n -
l l
independent term, however, with an opposite sign. At three loops the linear-n term
l
has almost the same order of magnitude than the constant contribution but again a dif-
ferent sign. It is remarkable that for n = 5 the coefficient of the four-loop n2 term is
l l
more than a factor of five larger than the n -independent term at order α3.
l s
Let us finally compare our results with the approximate expressions obtained in Ref. [57]
in the large-β approximation. In Ref. [57] one finds for the quantity M /m¯ (m¯ ) the
0 q q q
result (a ≡ α (m¯ )/π)
s s q
M
q = 1+a 1.333+a2(17.186−1.041n)
m¯ (m¯ ) s s l
q q (cid:12)(cid:12)large−β0
(cid:12)
(cid:12) +a3 177.695−21.539n +0.653n2
(cid:12) s l l
+a4 3046.294−553.872n +33.568n2 −0.678n3 , (15)
s(cid:0) l (cid:1) l l
(cid:0) (cid:1)
where for the renormalization scale µ = m¯ has been chosen. The coefficients of Eq. (15)
q
should be compared with our findings which read
M
q = 1+a 1.333+a2(13.443−1.041n)
m¯ (m¯ ) s s l
q q
+a3 190.595−26.655n +0.653n2
s l l
+a4 c +c n +43.396n2 −0.678n3 , (16)
s(cid:0) 0 1 l l (cid:1)l
(cid:0) (cid:1)
where c and c are not yet known. By construction one finds agreement for the coefficient
0 1
of n3 since it has been used as input in Ref. [57]. As far as the n2 term is concerned the
l l
exact coefficient is predicted with an accuracy of about 30%.
4 Fermionic n2 and n3 contributions to a
l l µ
It is convenient to introduce the form factors F and F of the photon-lepton vertex as
1 2
F (q2)
Γµ(q,p) = F (q2)γµ +i 2 σ qν , (17)
1 µν
2M
l
10
