Table Of ContentDESY-13-253, LPN14-003, SFB/CPP-14-03, SI-HEP-2014-01, TTP14-002, TTK-14-01, TUM-HEP-922/14
Three-loop matching of the vector current
Peter Marquard,1,2 Jan H. Piclum,3,4 Dirk Seidel,5 and Matthias Steinhauser1
1Institut fu¨r Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany
2Deutsches Elektronen Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen, Germany
3Institut fu¨r Theoretische Teilchenphysik und Kosmologie, RWTH Aachen, D-52056 Aachen, Germany
4Physik-Department T31, Technische Universita¨t Mu¨nchen, D-85748 Garching, Germany
5Theoretische Physik 1, Universita¨t Siegen, D-57068 Siegen, Germany
(Dated: January 15, 2014)
We evaluate the three-loop corrections to the matching coefficient of the vector current between
Quantum Chromodynamics (QCD) and non-relativistic QCD. The result is presented in the MS
scheme where large perturbative corrections are observed. The implications on the threshold pro-
duction of top quark pairs are briefly discussed.
4
1
PACSnumbers: 12.38.Bx,14.65.Ha,14.65.Fy
0
2
n I. INTRODUCTION abletoobtainthecompletenext-to-next-to-next-tolead-
a ing orderQCDprediction ofthe crosssectione+e− →tt¯
J
close to threshold or the decay width of the Υ(1S) me-
3 In the recent years effective field theories constructed
son to leptons. The results for the latter are presented
1 fromQuantumChromodynamics(QCD)havebeenenor-
in an accompanying paper [20] where all building blocks
mously successful to describe phenomena where masses
are combined to a phenomenological analysis.
] and momenta follow certain limits. Among them is non-
h
relativistic QCD (NRQCD) [1, 2] which is applicable to
p
- a systemoftwoheavyquarksmovingwithsmallrelative II. VECTOR CURRENTS IN QCD AND NRQCD
p velocity. Next to properties of the ψ and Υ families also
e
thethresholdproductionoftopquarkpairsisamongthe
h The vector current in the full theory (QCD) is given
prominent examples (see, e.g., Ref. [3] for a review).
[
by
The common method to construct an effective theory
1
is based on the so-called matching procedure: appropri- jµ = Q¯γµQ, (1)
v v
ately chosen Green’s functions are computed in the full
4
where Q denotes a generic heavy quark with mass m .
0 andeffectivetheoryandequalityisrequireduptopower- Q
On the other hand in the effective theory (NRQCD) the
0 suppressed terms. In this way the couplings of the effec-
3 tive operators (i.e. the matching coefficients) are deter- current is represented by an expansion in 1/mQ where
1. mined which completely specifies the effective theory. at each order effective operators have to be considered
whicharemultipliedbycoefficientfunctions. Theleading
0 A crucial operator both in QCD and NRQCD is the
contribution involves one operator given by
4
vector current of a heavy quark-antiquark pair. The
1
corresponding matching coefficient enters as a building ˜jk =φ†σkχ, (2)
:
v block in a variety of physical observables, for exam-
i ple the bottom-quark mass from Υ sum rules (see, e.g., whereφandχaretwo-componentPaulispinorsforquark
X and anti-quark, respectively, and σk (k =1,2,3) are the
Refs. [4, 5] for recent analyses) and top-quark threshold
r Pauli matrices. Hence, the matching coefficient of the
a productionatafutureelectronpositronlinearcollider[6].
vector current is defined through
The latter process allows for an extraction of the top-
quark mass with an accuracy below 100 MeV [7–9] —
1
animprovementofabout anorderof magnitude as com- jk = c (µ)˜jk+O . (3)
v v m2
pared to the current results from the Fermilab Tevatron Q!
or the the CERN Large Hadron Collider [10].
Note that the 0-component of jµ is only relevant for the
v
Several quantities are needed in order to perform a
power-suppressedcontributions.
third-order analysis of a heavy quark-antiquark system
The purpose of this paper is the evaluation of the
at threshold. Ultrasoft effects have been considered in
purelygluonicthree-loopcorrectionstoc . Thefermionic
v
Refs. [11, 12], the three-loop static potential has been
contributions have already been considered in Refs. [21,
computed in Refs. [13–15] and in Ref. [16, 17] a prelimi-
22].
naryanalysisofthetop-quarkthresholdproductioncross
In orderto compute c it is convenientto consider on-
v
sectionhas beenpresentedincluding alsothird-orderpo- shell vertex corrections involvingthe currents jk and˜jk.
v
tentialeffects. Detailsonthe potentialcontributionscan
Aftertakingintoaccountthewavefunctionrenormaliza-
be found in Refs. [18, 19]. In this paper we compute the
tion one obtains (see also Ref. [18])
three-loop matching coefficient between the vector cur-
rentinQCDandNRQCD.Thusallingredientsareavail- Z Γ = c Z˜ Z˜−1Γ˜ +... , (4)
2 v v 2 v v
2
ǫ=(4−D)/2,whereDisthespace-timedimension. The
lasttwotasksbecomemorecomplicatedbytheadditional
condition q2 =4m2 on the external momentum. An au-
Q
tomated setup for the calculation has been described in
Ref. [22] and applied to the fermionic contributions. Its
(a) (b) core parts are a powerful implementation of Laporta’s
algorithminthe programCRUSHER[28], andFIESTA[29–
31] which is based on sector decomposition and is used
forthenumericalintegrationofthemasterintegrals. The
maindifferencestothegluonicpartconsideredinthispa-
per are the larger number of diagramsand the increased
(c) (d) complexity of the integrals which have to be reduced to
master integrals. Furthermore, the master integrals are
FIG. 1: Feynman diagrams contributing to Γv. Straight and more numerous and more involved.
curly lines denote heavy quarks with mass mQ and gluons,
respectively.
III. MATCHING COEFFICIENT TO ORDER α3s
where the quantities witha tilde are definedin the effec-
Beforediscussingthematchingcoefficientitisinstruc-
tive theory and the ellipsis represents terms suppressed
by the heavy-quark mass. Z˜−1 is the renormalization tive to consider the renormalization constant Z˜v. The
constant of the current ˜jk whivch is used to subtract the analyticalresultscanbeextractedfromRefs.[11,21,32,
33].
remaining poles after renormalization. These poles are
due to the separation of long and short distance contri-
butionsintheeffectivetheory. Inordertoevaluatephys- Z˜v =
ical quantities it is important that the same subtraction α(nl)(µ) 2 C π2 1 1
s F
scheme is also adopted in the contributions originating 1+ CF + CA
π ǫ 12 8
from the effective theory [11, 12]. It is well-known that ! (cid:18) (cid:19)
inthefulltheorytherenormalizationconstantofthevec- α(nl)(µ) 3
tor current is equal to one. + s C π2
F
π
InEq.(4)Z istheon-shellwavefunctionrenormaliza- !
2
tion constantwhichhas been computed up to three-loop
5 43 1 5 1
accuracy in Refs. [23–25]. Γv denotes the one-particle × CF2 144ǫ2 + 144 − 2ln2+ 48Lµ ǫ
irreducible vertex diagrams with on-shell quarks carry- ( (cid:20) (cid:18) (cid:19) (cid:21)
ing momenta q and q . It incorporates all one-particle 1 113 1 5 1
1 2 +C C + + ln2+ L
irreducible vertex graphs and the corresponding coun- F A 864ǫ2 324 4 32 µ ǫ
(cid:20) (cid:18) (cid:19) (cid:21)
terterms for m and α . Sample Feynman diagrams are
Q s 1 2 1 1 1
shown in Fig. 1. +C2 − + + ln2+ L
A 16ǫ2 27 4 24 µ ǫ
The counterparts of Γ and Z in the effective theory (cid:20) (cid:18) (cid:19) (cid:21)
v 2
1 25 1 37
canbefoundontheright-handsideofEq.(4). Itisconve-
+Tn C − +C −
nienttoapplythethresholdexpansion[26,27]toEq.(4). l F 54ǫ2 324ǫ A 36ǫ2 432ǫ
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
This requires the identification of the hard, soft, poten-
1
tialandultrasoftmomentumregionsintheintegralscon- +C Tn +O(α4), (5)
tributing to Γ and Γ˜ . Since NRQCD is obtained from F h60ǫ) s
v v
QCD by integratingout the hardmodes one has by con-
struction that the soft, potential and ultrasoft modes where CA = Nc, CF = (Nc2 −1)/(2Nc) and T = 1/2
agreeinΓv andΓ˜v andthus dropoutfromEq.(4). As a for a SU(Nc) gauge group and Lµ = ln(µ2/m2Q). Note
consequence Γ is evaluated for q2 = (q +q )2 = 4m2, thatthe strongcouplingisdefinedinthe effectivetheory
v 1 2 Q
which corresponds to the leading term of the hard in- with nl active quarks where nl+nh is the total number
tegration region, and Γ˜v = 1. Furthermore, we have of quark flavors. In our case we have nh = 1, however,
Z˜2 =1. we keep nh in the formulae for convenience.
There are several technical difficulties which one has From our calculation we can extract the renormaliza-
to overcome in order to compute the vertex corrections. tion constant Z˜v and compare with Eq. (5). The central
Among them are the large number of diagrams which values of our numerical coefficients agree at the per cent
leads to several thousand Feynman integrals to be eval- level with the analytical result of Eq. (5) which consti-
uated in the first place, their reduction to a small set tutes a first non-trivial check and provides quite some
of about 100 basis integrals, so-called master integrals, confidence in the overallset-up of our calculation.
and the evaluation of the latter in an expansion in We write the perturbative expansion of the matching
3
coefficient in the form c = −97.81(0.38)
FAA
α(nl)(µ) α(nl)(µ) 2 + −2 − 3ln2 π2L + 1π2L2,
c = 1+ s c(1)+ s c(2) 9 4 µ 6 µ
v π v π v (cid:18) (cid:19)
! 25 1
c = 46.691(0.006)+ π2L − π2L2,
α(nl)(µ) 3 FFL 108 µ 18 µ
+ s c(3)+O(α4), (6) 37 1
π ! v s cFAL = 39.624(0.005)+ 144π2Lµ− 12π2L2µ,
557 26
and decompose c(v3) according to the color structures as cFHL = − + π2,
162 81
c(3) = 163 4
v c = − − π2,
FLL
C C2c +C C c +C2c 162 27
F F FFF F A FFA A FAA
1
+Tn (C c +C c +Tn c +Tn c ) c = −0.846(0.006)− π2L ,
(cid:2) l F FFL A FAL h FHL l FLL FFH 20 µ
+Tn (C c +C c +Tn c )
h F FFH A FAH h FHH c = −0.098(0.051),
FAH
+singlet terms. (cid:3) (7) c = −427 + 158 π2+ 16ζ(3). (8)
FHH
Note that all color factors of the non-singlet part can be 162 2835 9
expressedintermsofCF,CA,andT. Inthispaperwedo All uncertainties originating from the individual master
not consider the singlet contribution where the external integrals are added quadratically. In order to obtain a
current does not couple to the fermion line in the final conservative error estimate we interpret the uncertainty
state. At two-loop order such contributions have been of the numerical integration as one standard deviation
computed [43] for axial-vector, scalar and pseudo-scalar from a Gaussian distribution and multiply it by a factor
currentsinRef.[34]. Theirnumericaleffectinthosecases of five which is accounted for in Eq. (8) [45]. The coeffi-
isbelow3%ascomparedtothenon-singletcontributions cients of L could be obtained in analytic form since all
µ
and thus quite small. renormalizationconstants are known analytically.
Whereas the one- [35] and two-loop [32, 34, 36] terms In most applications it is sufficient to know the result
havebeenknownsincelongthefermionicthree-loopcor- for the matching coefficient with numerically evaluated
rections became available only a few years ago [21, 22]. color factors. Setting C = 4/3, C = 3, T = 1/2
F A
The so-called renormalon contribution, consisting of the and n = 1 before inserting the master integrals and
h
one-loop diagram with arbitrary many massless quark combining the numerical uncertainties we get
loop insertions in the gluon propagator, has been com-
puted in Ref. [37]. Supersymmetric one-loop corrections α(nl) α(nl) 2
s s
to cv have been computed in Ref. [38]. cv ≈ 1−2.667 π + π [−44.551+0.407nl]
!
In the following we present the results for the indi-
vidual coefficients in Eq. (7) parameterized in terms of α(nl) 3
s
α(snl)(mQ). The reconstruction of the full dependence + π ! [−2091(2)+120.66(0.01)nl
on the renormalization scale is straightforward; the cor-
responding expressions can be obtained from [39]. Our −0.823n2l +singlet terms, (9)
results read [44]
where µ = m h(cid:3)as been chosen. Note that the n -
Q l
c(1) = −2C , independent three-loop term contains the contribution
v F
with a closed massive quark loop which amounts to
151 89 5 13
c(v2) = − 72 + 144π2− 6π2ln2− 4 ζ(3) CACF c(v3)|nnlh=0 ≈−0.93(8)[22]. Oneobservesthatfornl =3,4
(cid:18) (cid:19) and 5 all coefficients in Eq. (9) have the same sign and
+ 23 − 79π2+π2ln2− 1ζ(3) C2 that they grow quite rapidly when going from NLO to
8 36 2 F NNNLO. At NNLO and NNNLO the fermionic correc-
(cid:18) (cid:19)
22 2 11 tions screen the non-fermionic ones, but even for nl = 5
+ − π2 CFTnh+ CFTnl only a reduction of at most 30% is obtained. A first
9 9 18
(cid:18) (cid:19) glance at Eq. (9) would suggest that for the quantity c
v
1 1 1
− π2 C + C C L , perturbation theory breaks down even though the mo-
A F F µ
2 2 3 mentumscaleinvolvedintheproblem,m ,isquitelarge.
(cid:18) (cid:19) Q
cFFF = 36.55(0.53) However, as already mentioned above, cv itself does not
representaphysicalquantity. Ithastobecombinedwith
9 3 5
+ −16 + 2ln2 π2Lµ− 32π2L2µ, contributions originating from soft, potential and ultra-
(cid:18) (cid:19) soft momentum regions which can compensate the large
c = −188.10(0.83)
FFA coefficients in Eq. (9). Further discussions on this topic
59 3 47 can be found in Ref. [20]. It might very well be that the
+ − − ln2 π2L − π2L2,
108 4 µ 576 µ MS scheme adopted in our calculation is not well suited
(cid:18) (cid:19)
4
default basis (cf. Eq.(8)) alternative basis
q
1 cFFF 36.55(0.11) 36.61(2.93)
cFFA −188.10(0.17) −188.04(2.91)
cFAA −97.81(0.08) −97.76(2.05)
c(v3) (nl =4) −1621.7(0.4) −1621(23)
c(v3) (nl =5) −1508.4(0.4) −1507(23)
TABLE I: Comparison of the purely gluonic coefficients of
Eq. (8) and c(v3) with nl = 4 and nl = 5 for two different
q choicesofthemasterintegralbasis. Forconvenienceµ=mQ
2 has been adopted. The given uncertainties are obtained by
combiningthenumericaluncertaintiesofeachmasterintegral
contribution in quadrature. In contrast to Eqs. (8) and (9)
FIG.2: Typicalmasterintegralappearinginourcalculation. nofactor fivehas been introduced for this comparison.
Thesolid lines and dashed lines represent massive and mass-
lesslines,respectively. Fortheexternalmomentawehavethe
conditions q2 =q2 =m2 and (q +q )2=4m2.
1 2 Q 1 2 Q
1.2
NNNLO (ferm.)
for separating the divergences occuring in the different
1.1
regimes. In fact, also the ultrasoft contribution studied
NNLO
in Ref. [11, 12] shows large numerical effects. )S 1
µ
Wehaveperformedseveralchecksonthecorrectnessof (
O
our resultwhichwe wantto mention inthe following. In ZLt 0.9 NNNLO
ourcalculationweallowedforageneralgaugeparameter µ)/
ξ which manifests as a polynomial dependence of the in- Z(t 0.8
dividualdiagrams. After summing the three-loopresults NLO
for Z2 and Γv (taking into account the corresponding 0.7 LO
quark mass counterterm contribution) we concentrated
0.6
on the coefficient of the linear ξ-dependence and have 20 40 60 80 100 120 140 160
verified that it vanishes. As a further check we recom- µ (GeV)
putedthen contribution[21]usingourautomatedsetup.
l
In this context we want to mention that in Ref. [21] all
FIG. 3: Residue for the top quark system normalized to
occuringmasterintegralshavebeencomputedeitheran- ZtLO(µS)asafunctionoftherenormalizationscaleµ. Dotted,
alyticallyorusinganumericalmethoddifferentfromthe dash-dotted, short-dashed and solid lines correspond to LO,
one used in the present paper. As already mentioned NLO, NNLO and NNNLO prediction. In the long-dashed
above, with our calculation we could also reproduce the curve only the fermion contributions to c(v3) are taken into
renormalization constant in Eq. (5) with high accuracy account (NNNLO(ferm.)).
whichchecksallbutthehighestǫcoefficientsofthemas-
ter integrals. We note in passing that we have a similar
accuracy for the cancellation of the spurious poles up to
different, in general more complicated ones. This trans-
seventh order occuring due to our reduction procedure.
formation is done analytically for general space-time di-
At this point it is instructive to show a result for a
mension D. In a next step the new master integrals are
typical master integralcontributing to cv. For the Feyn- againevaluatedwith FIESTAandinsertedinthe new ex-
man diagram in Fig. 2, which we need up to order ǫ, we
pression for c . In Tab. I we compare the results for
obtain with the help of FIESTA [29–31] v
the purely gluonic coefficients and the complete result
for c(3) obtained in the two bases. We observe an ex-
3ǫ v
e3ǫγE µ2 0.411236(3) 3.4860(1) cellent agreement within the uncertainties. In the case
M = + +
m4 m2 ǫ2 ǫ of the “alternative basis” one has to keep in mind that
Q Q!
theintegralstobeevaluatednumericallyaresignificantly
morecomplicatedwhichexplainsthelargeruncertainties
+34.520(2)+339.68(4)ǫ+O(ǫ2) . (10)
for the coefficients in Tab. I.
!
As a further check on the numerical evaluation of the
A very powerful check on the correctness of our result master integrals we have used a different momentum as-
is provided by the change of basis for the master inte- signment in the input for FIESTA.As a consequence dif-
grals. We employ the integral tables generated during ferent expressions are generated in intermediate steps
the reduction procedure in order to re-express the mas- leading to different numerical integrations. The final re-
ter integrals, which are not known analytically, through sults are in complete agreement with Eq. (9).
5
We are now in the position to have a first look to the ofthe 1/(m r2)potential (b ǫ), andfromthe three-loop
Q 2
phenomenological consequences of our result for cv. We fermion (cf) and purely gluonic (cg) contribution to c(v3)
considerthe residue ofthe two-pointfunction ofthe vec- separately. For this choice of µ one observes quite big
tor currents NNNLOcontributionswhicharedominatedbyc . Thus,
g
it is instructive to investigate the µ-dependence of Z
t
−q2gµν +qµqν Π(q2) = i dxeiqxh0|Tjµ(x)jν†(0)|0i, which is shown in Fig. 3. Around the soft scale no con-
Z vergenceisobserved. Allowing,however,forhigherscales
(cid:0) (cid:1) (11)
onefindsaquiteflatbehavioroftheNNNLOcurve. Fur-
thermore, the NNNLO corrections become quite small.
which is obtained by considering Π(q2) close to the QQ¯
E.g., considering the top quark system for µ ≈ 80 GeV,
threshold. In this limit Π(q2) is dominated by pole con-
the NLO terms amount to about +15% and the NNLO
tributions originating from bound-state effects
toroughly+20%. Thethird-ordercontributionispracti-
N Z cally zero. Similar observations also hold for the bottom
Π(q2) E→=En c n +... , (12) quark case, see Ref. [20].
2m2 E −(E+i0)
Q n
where the ellipsis denotes contributions from the conti-
nuum. Z and E are the residue and energy of the nth
n n
resonancewhichdeterminetheheightandpositionofthe
IV. CONCLUSIONS
threshold cross section, respectively. In the following we
consider the residue of the 1S (pseudo) bound state of
The third-order contribution to the matching coeffi-
top quarksandextend the considerationsof Ref. [16][46]
by including the non-fermionic contribution of c(3) and cient of the vector current between QCD and NRQCD
v hasbeencomputed. Anautomatedsetuphasbeendevel-
the O(ǫ) term of the 1/m potential [20]. Choosing the
Q oped where even the occuring master integrals are iden-
potential subtracted scheme [40] with µ = 20 GeV to
f tified automatically, processed with the help of the com-
define the top quark mass we obtain mPS = 171.4 GeV
t puter program FIESTA, and prepared for the insertion
which leads to
into the analytic reduction of c(3).
v
Z = (CFmPtSαs)3 [1+(−2.131+3.661L)α +(8.38 Inthe MSschemethenumericalimpactofc(v3) isquite
t s
8π big as can be seen from Eq. (9) which constitutes the
+1.27x −7.26lnα −13.40L+8.93L2 α2 main result of this paper. In a dedicated analysis one
f s s
+(5.46+(−2.23+0.78L )x +2.21 hastoinvestigatetheconsequencesforthebottom-quark
f f a3 (cid:1)
mass extracted from Υ sum rules and the top-quark
+21.48 +37.53 −134.8(0.1)
b2ǫ cf cg threshold production cross section at a future linear col-
+(−9.79−44.27L)lnαs−16.35ln2αs lider.
+(53.17+4.66x )L−48.18L2+18.17L3 α3 Ananalysisoftheresidueofthe1S stateindicatesthat
f s
at energy scales around two to three times the soft scale
+O(α4s) (cid:1) good convergence of the perturbative series is observed.
= (CFmPtSα(cid:3)s)3 1−2.13α +23.66α2
8π s s
−113.0(0.1)α(cid:2)3+O(α4) , (13)
s s
Acknowledgments
where x = µ /(mPSα ), L =(cid:3)ln(µ/(mPSC α )), and
f f t s t F s
Lf = ln(µ2/µ2f). We have used αs(MZ) = 0.1184 to We would like to thank Martin Beneke and Alexan-
compute α = α (µ ) ≈ 0.141 where the soft scale der Penin for many useful discussions and communi-
s s S
µ = m C α (µ ) ≈ 32.16 GeV has been adopted af- cations and Alexander Smirnov for continuous support
S Q F s S
ter the second equality sign. In order to get an impres- on FIESTA. This work is supported by DFG through
sionabouttheimportanceoftheindividualcontributions SFB/TR9“ComputationalParticlePhysics”. TheFeyn-
we mark the µ-independent coefficients from the three- man diagrams were drawnwith the help of Axodraw[41]
loop static potential (a ), from the two-loop O(ǫ) term and JaxoDraw [42].
3
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