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a Masses and Decay Constants of Pseudoscalar Mesons to Two Loops
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2 in Two-Flavor Partially Quenched Chiral Perturbation Theory
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6 Johan Bijnens and Timo A. L¨ahde
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Abstract
This paper presents a first study of the masses and decay constants of the
charged,orflavor-off-diagonal,pseudoscalar mesonstotwoloopsfortwoflavors
of sea-quarks, in Partially Quenched Chiral Perturbation Theory (PQχPT).
Explicit analytical expressions up to (p6) in the momentum expansion are
O
given. The calculations have been performed within the supersymmetric for-
mulation of PQχPT. A numerical analysis is done to indicate the size of the
corrections.
PACS: 12.38.Gc, 12.39.Fe, 11.30.Rd
Masses and Decay Constants of Pseudoscalar Mesons to Two Loops
in Two-Flavor Partially Quenched Chiral Perturbation Theory
Johan Bijnens1 and Timo A. La¨hde1
1Department of Theoretical Physics, Lund University,
S¨olvegatan 14A, S 223 62 Lund, Sweden
This paper presents a first study of the masses and decay constants of the charged, or flavor-
off-diagonal, pseudoscalar mesons to two loops for two flavors of sea-quarks, in Partially Quenched
ChiralPerturbationTheory(PQχPT).ExplicitanalyticalexpressionsuptoO(p6)inthemomentum
expansionaregiven. Thecalculations havebeenperformed withinthesupersymmetricformulation
of PQχPT. A numerical analysis is doneto indicate thesize of thecorrections.
PACSnumbers: 12.38.Gc, 12.39.Fe,11.30.Rd
I. INTRODUCTION resultsforthemassofapseudoscalarmesonattwoloops
(NNLO)inPQχPThavealreadyappeared,forthreefla-
vorsofdegeneratesea-quarks[8]. Thedecayconstantsof
At the presenttime, the mostpromisingwayto derive
the pseudoscalarmesonsarealsoknowntoNNLO [9]for
information on low-energy hadronic observables directly
three sea-quarks. A full PQχPT calculationof the pseu-
from QCD is by means of numerical Lattice QCD simu-
doscalar meson masses for three flavors of sea-quarks is
lations. However, it is well known that such simulations
in progress [10]. However, as many Lattice QCD sim-
are computationally very demanding if dynamical sea-
ulations are performed with two rather than three sea-
quarkeffects (unquenched Lattice QCD) areto be taken
quark flavors, such expressions for the pseudoscalar me-
intoaccount. Althoughmuchprogresshasbeenmadere-
sonmasses anddecayconstantsto NNLO arealso called
cently, simulations with sea-quark masses that are close
for.
to the physical u,d quark masses of a few MeV are still
This paper presents the full calculation of the masses
not available, whereas sea-quark masses down to several
anddecayconstantsofthecharged,orflavoroff-diagonal,
tens of MeV are becoming possible. The physical values
pseudoscalarmesonsinNNLOPQχPT,fortwoflavorsof
of the quark masses then have to be reached by extrap-
sea-quarks (n = 2). The results are characterized by
olation from these simulations. sea
thenumberofnondegeneratevalenceandsea-quarks,de-
A generalization of QCD where the valence quark
notedd andd ,respectively. Forthedecayconstants
masses are different from the sea-quark masses is pos- val sea
ofthe chargedpseudoscalarmesons,the maximumnum-
sible, and is called Partially Quenched QCD (PQQCD).
ber of nondegenerate valence quark masses is d = 2.
QCD may then be obtained as a continuous limit from val
The degree of quark mass degeneracy in each result is
this theory, in contrast to the quenched calculations
sometimes also referred to with the notation d +d .
where no such smooth connection exists. The reason for val sea
The decay constant of the charged pion in the SU(2)
doing PQQCD Lattice simulations is that the valence
symmetriclimitforbothvalenceandsea-quarksthuscor-
quark masses can be varied much more efficiently than
responds to the 1+1 case.
the sea-quark masses, which allows for more extensive
The analytical expressions for the NNLO shifts of the
studies of the quark mass dependence of the various ob-
meson masses and decay constants are in general very
servables.
long,but the expressionssimplify considerablywhen the
The generalization of χPT to sea-quark masses differ-
sea or valence quark masses become degenerate. In view
ent from the valence ones, i.e. to the partially quenched
of this, the NNLO loop results are given separately for
case [1, 2], provides a practical method for the extrap-
each case of d +d considered. In the next sections,
olation to physical quark masses from the lattice simu- val sea
a short technical overview of the NNLO calculations is
lations of PQQCD. The quark mass dependence of par-
given, with an emphasis on the parts not covered in the
tially quenched chiral perturbation theory (PQχPT) is
previouspublications[8,9],alongwiththefullresultsfor
explicit, and thus the limit where the sea-quark masses
the NNLO contributions to the meson masses and decay
become equal to the valence quark masses can be taken.
constants. Finally, a numerical analysis of the results is
As a consequence, unquenched χPT [3] is included in
given, along with a concluding discussion.
PQχPT, and the free parameters, or low-energy con-
stants (LEC:s), of χPT can be directly obtained from
those of PQχPT [2, 4]. The need for PQχPT calcula-
tions to next-to-next-to-leading order (NNLO) has also II. TECHNICAL OVERVIEW
been demonstrated by Lattice QCD simulations [5, 6].
The calculation of the charged pseudoscalar meson The technical aspects of the PQχPT calculations to
masses and decay constants to one loop (NLO) in NLO have been thoroughly covered in Refs. [4, 7]. The
PQχPT has been carried out in Refs. [2, 4, 7], and first new parts needed for NNLO are treated in Refs. [8, 9].
2
Forthesakeofcompleteness,someoftheissueswhichare they are Lr(npq) through Lr(npq). The two possible ex-
0 10
directly relevant for the present two-flavor calculations tra terms Lr(npq) and Lr(npq) can be removedusing field
11 12
are repeated here. redefinitions [11, 12]. For the case of three sea-quarks,
Most significantly, the Lagrangians of PQχPT at the Lr are simple linear combinations of the Lr(3pq),
(p4) and (p6) can be directly obtained from the cor- i i
O O and these relations can be found in Refs. [9, 11]. Sim-
responding Lagrangians of unquenched n flavor χPT
F ilarly at NNLO, the Cr are linear combinations of the
by replacement of traces with supertraces. This follows i
Kr(3pq), and the corresponding relations can be found
because the derivation of these Lagrangians in Ref. [11] i
in Ref. [11]. For three flavors of sea-quarks, one simply
only used relations which remain valid when all traces
are replaced by the corresponding supertraces. For the takes Lr(3pq) =Lr(3pq) =0.
11 12
definitionofthe supertracesinPQχPTseeRefs [1,2, 4]. The unquenched two-flavor Lagrangian as defined in
We work here in the supersymmetric version of PQχPT the second paper of Ref. [3] differs by an Lr type term.
11
withoutthesingletΦ asdiscussedinRef.[7]. Thediver- At NLO, this makes no difference since that term does
0
gencesofthisversionofPQχPTalsofollowdirectlyfrom not contribute, but in order to get the correct values at
those of nf flavor χPT calculated in Ref.[12], provided NNLO for the lr,cr and the Lr(2pq),Kr(2pq) one should
i i i i
that nf is set equal to the number of sea-quark flavors. take Lr(2pq) = lr/4 and Lr(2pq) = 0. The relations
This is so since all the manipulations in Ref. [12] for nf between11the NLO−L4EC:s are g1i2ven in Table II, and simi-
flavorsremainvalidwhentracesarereplacedwithsuper-
lar linear relations between the NNLO LEC:s Kr(2pq) of
traces. Alternatively, this derivation of the Lagrangians i
PQχPT and the cr of unquenched two-flavor χPT may
ofPQχPTcanbe arguedforwiththe Replicamethodof i
befoundinRef.[11]. Theequationsgivenintheremain-
Ref. [13].
ingsectionsofthispaperonly dealwithn =2 PQχPT,
f
andthereforethesuperscript(2pq)oftheLr(2pq) andthe
i
A. Quark Masses and Low-Energy Constants Kr(2pq) have been suppressed in most of the formulas in
i
the following sections.
The number of independent low-energy constants
(LEC:s) in unquenched and partially quenched χPT is
slightly different, but the former are always linear com- TABLEII:ThecorrespondencebetweentheNLOlow-energy
binations of the latter. The number of LEC:s is also constantsLri oftwo-flavorPQχPTandthelirofstandardtwo-
flavor χPT. The relations relevant for the NNLO low-energy
dependent on the number of flavors. Table I gives an
constants Kr are given in Ref. [11].
overview of the various sets of LEC:s and the number of i
parametersuptoNNLOforthedifferentversionsofχPT
n =2 χPT n =2 PQχPT
and PQχPT. f f
lr −2Lr(2pq)+4Lr(2pq)+2Lr(2pq)
TABLEI:TherelevantsetsofLEC:s,wherethei+jnotation 1 0 1 3
indicatesthenumberofphysicallyrelevant(i)andcontact(j) l2r 4Lr0(2pq)+4Lr2(2pq)
termsintherespectiveLagrangians. AtNLO,thelatterones lr −8Lr(2pq)−4Lr(2pq)+16Lr(2pq)+8Lr(2pq)
3 4 5 6 8
are usually denoted hri for nf = 2 and Hir for nf = 3. The l4r 8Lr4(2pq)+4Lr5(2pq)
relations between the various sets of LEC:s is discussed in lr Lr(2pq)
thetext. Forsimplicity,thesuperscript(npq)ofthepartially 5 10
lr −2Lr(2pq)
quenchedLEC:s has been suppressed in most of this paper. 6 9
lr −16Lr(2pq)−8Lr(2pq)
7 7 8
χPT χPT χPT PQχPT PQχPT
n 2 3 n 2 3
f
The input quark masses m enter into the calculation
i
LO F,B F0,B0 F0(n),B0(n) F,B F0,B0 in terms of the lowest order squared meson masses χi,
whicharedefinedasusualinχPT,byχ =2Bm . Inthe
i i
NLO lr Lr Lr(n) Lr(2pq) Lr(3pq) presentcalculations,there are two valence inputs χ1,χ2,
i i i i i two sea inputs χ ,χ , and two ghost inputs χ ,χ . In
i+j 7+3 10+2 11+2 11+2 11+2 3 4 5 6
orderforthedisconnectedvalencequarkloopstobecan-
celed, the masses of the ghost quarks are always equal
NNLO cr Cr Kr(n) Kr(2pq) Kr(3pq) to those of the corresponding valence quarks, such that
i i i i i
i+j 53+4 90+4 112+3 112+3 112+3 χ5 = χ1 and χ6 = χ2. Explicitly, for dval = 1, we have
χ =χ andford =2,wehaveχ =χ . Similarly,for
1 2 val 1 2
6
the sea-quark sector d = 1 implies χ = χ , while for
sea 3 4
At lowest order, the LEC:s of PQχPT with n fla- d = 2 we have χ = χ . The most general results (in
sea 3 4
6
vors of sea-quarks are the same as those of n flavor un- the quark masses) for n = 2 are therefore those with
sea
quenched χPT. At NLO with n flavors of sea-quarks, d =2 and d =2.
val sea
3
B. Chiral Logarithms indicates that the first propagator appears squared and
n=3 that the second propagator appears squared. The
Thefinite partsoftheone-loopintegralswhichareen- cases with two double propagators that can appear in
countered in the NNLO calculation of the masses and the calculationsaren=5, forwhichthe firstandsecond
decay constants of the pseudoscalar mesons may be ex- propagatorsappearsquared,andn=7forwhichthesec-
pressedinterms ofthe chirallogarithmsA,B andC. Of ond and third propagators are squared. Explicit expres-
these, the logarithms A are defined as sions for n=1 can be found in Ref. [14], and the other
cases may be obtained by differentiation with respect to
A¯(χ ) = π χ log(χ /µ2), (1) the masses of those expressions.
i 16 i i
−
A¯(χ ,ε) = A¯(χ )2/(2π χ )
i i 16 i
+ π χ (π2/12+1/2), (2)
16 i D. Notation for Meson Masses and Decay
Constants
where π = 1/(16π2). Note that the chiral logarithms
16
whichdependonεarenaturallygeneratedbytheprocess
As explained in great detail in Refs. [4, 7, 9], the neu-
of dimensional regularization and can be reexpressed in
tral meson propagators in PQχPT have a complicated
terms of the standard logarithms A and B. However,
structure. In order to obtain a result which can be ex-
they have been retained in the final results for the sake
pressed in terms of the chiral logarithms and two-loop
ofnotationalefficiency. Thefinitepartsofthelogarithms
integrals outlined in the previous subsection, that prop-
B are
agator has to be treated by partial fractioning, which
B¯(χ ,χ ,0) = π 1+log(χ /µ2) , (3) produces expressions that consist of sums of single and
i i 16 i
− double poles. The residues, which are rational functions
χ log(χ /µ2) χ log(χ /µ2)
B¯(χ ,χ ,0) = π (cid:0)i i −(cid:1) j j , of the sea and valence quark masses, pile up and pro-
i j 16
− χi χj duce very long and awkward end results. With a major
−
B¯(χ ,χ ,0,ε) = A¯(χ )B¯(χ ,χ ,0)/(π χ ) (4) effort, these can be simplified and in some cases the ex-
i i i i i 16 i
A¯(χ )2/(2π χ2) pressions have been compressed by more than an order
− i 16 i of magnitude.
+π (π2/12+1/2),
16 The residues R which are naturally generated in the
B¯(χi,χj,0,k) = χiB¯(χi,χj,0)+A¯(χj), (5) calculations are, for dsea = 2, the single-pole residues
Ri and the double-pole residue Rd. The residues sim-
where the B¯(χi,χj,0,k) integrals, which are symmetric pljifky when pairs of sea-quark masises become degener-
undertheinterchangeofχi andχj,havebeenintroduced ate, and hence the number of indices in the single-pole
in order to make the symmetries in the end results more residue depends on the degree of degeneracy in the sea-
explicit. Finally the remaining one-loop integral C is quarkmasses. The residues relevantfor the presenttwo-
given by flavor PQχPT calculations may be written in terms of
the quantities Rz , which are defined as
C¯(χ ,χ ,χ ,0) = π /(2χ ). (6) a...b
i i i 16 i
−
Rz = χ χ ,
In all of these cases, the dependence on the subtraction ab a− b
χ χ
scale µ has been moved into the loop integrals. Rz = a− b,
abc χ χ
a c
−
(χ χ )(χ χ )
Rz = a− b a− c , (8)
C. Two-loop Integrals abcd χ χ
a d
−
and so on. Thus Rz has the same dimension as χ for
Thefinitetwo-loopintegralsHF,HF,HF thatappear a...b i
1 21 an even number of indices and is dimensionless for an
in the NNLO calculation of the meson masses and de-
odd number of indices. It should be noted that the Rz
cay constants may be evaluated using the methods of
notation may also appear independently in the final re-
Ref. [14]. Note that the corresponding primed integrals
sults. In such cases, the Rz have been generatedby sim-
HF′,HF′,HF′ indicatedifferentiationwithrespecttop2
1 21 plification procedures. For dsea = 2, the residues which
and appear only for the decay constants. The notation
originate in the partial fractioning of the neutral meson
usedfortheH integralsinthispaperisotherwisesimilar
propagator are
to that of Ref. [14], except that an extra integer argu-
ment now indicates the propagator structure, such that Ri = Rz ,
jk i34jk
e.g.
Rd = Rz ,
i i34π
HF(χ ,χ ,χ ,p2) HF(n,χ ,χ ,χ ,p2). (7) Rc = Ri +Ri Ri , (9)
i j k → i j k i 3π 4π − ππ
The case of n = 1 indicates that the integral consists of where the neutral pion mass in the sea-quark sector is
single propagators only, as in Ref. [14], whereas n=2 simply given by χ = χ = (χ +χ )/2. Because of
π 34 3 4
4
this trivial relation, the above residue notation is highly III. TWO-FLAVOR PSEUDOSCALAR MESON
redundant, a fact which has been exploited in the sim- DECAY CONSTANTS
plification of the end results. The quantity Rv, which is
defined as ThedecayconstantsF ofthepseudoscalarmesonsare
a
obtained from the definition
Rv = Ri +Ri 2Ri (10)
ijk jj kk− jk
0Aµ(0)φ (p) =i√2pµF , (14)
appears throughout the final results and has therefore h | a | a i a
been given a separate name. For the case of dsea = 1, in terms of the axialcurrentoperator Aµ. The diagrams
a
all residues associated with the sea-quark sector reduce which contribute to that operator at (p6), or NNLO,
to numbers. Some nontrivial residues may still appear if are shown in Fig. 1. Diagrams at O(p2) and (p4)
the valence quarks are nondegenerate, according to O O
also contribute to Eq. (14) via the renormalization of
the pseudoscalar meson wave function φ (p). The decay
Ri = Rz , a
j i3j constants are functions of the valence inputs χ ,χ and
1 2
Rd = Rz . (11) the sea inputs χ ,χ , which are defined in terms of the
i i3 3 4
quark masses through χ =2Bm , and of the quantities
i i
For convenience, Rid is also used for dval = 1 in order to χij =(χi+χj)/2, which correspond to the lowest order
maintain a notation similar to the more general cases. chargedmesonmasses. Otherparametersincludethede-
cayconstantinthechirallimit(F),thequarkcondensate
in the chiral limit, via q¯q = BF2, and the LEC:s of
E. Summation Conventions (p4) and (p6), i.e. thhe Lir(2p−q) and the Kr(2pq), which
Ofor simpliciOty are denoted biy Lr and Kr. i
i i
The expressions for the meson masses and decay con-
stants are symmetric, within the sea and valence quark
sectors,under the interchangeofquark masses,andmay
thus be compactified by the introduction of summation
conventions which exploit these symmetries. The sum-
mationsinthevalencequarksectoraresimilartothosein
thethree-flavorworkofRef.[9],whereasthoseofthesea-
quark sector are considerably simpler in the two-flavor
case.
The sea-quark sector of the results involves the sum-
mation index s, which may appear for the quark masses
χ , and among the indices of the residue functions R. A
i
simple example is χ = χ +χ . On the other hand,
s 3 4
if the term in question consists of products of integrals,
residues and quark masses, then the summation sign al-
FIG. 1: Feynman diagrams at O(p6) or two-loop for the
ways applies to the term as a whole, such that e.g.
matrix element of the axial current operator. Filled circles
denote vertices of the L Lagrangian, whereas open squares
A¯(χs)Rη1sχs = A¯(χi)Rη1iχi. (12) and shaded diamonds de2note vertices of the L4 and L6 La-
i=3,4 grangians, respectively.
X
Any contribution which is written in terms of the sum-
mation index s explicitly fulfills the required symmetry
The decay constants of the pseudoscalar mesons are
properties in the sea-quark sector.
given in the form
The summation indices p and q, which refer to the
valence quark sector, are only needed for the case of δ(4)vs δ(6)vs+δ(6)vs
dqvuaalr=k m2aasnsedstχhenanadlwχay.sIofctchuerifnodrexthqeissqpuraerseedntv,atlehnecree Fphys =F "1+ F2 + ct F4 loops +O(p8)#, (15)
1 2
will always be an index p and the resulting sum is over
the pairs (p,q) = (1,2) and (p,q) = (2,1). If only p is where the (p4) and (p6)contributions havebeen sep-
present,the sumis overthe indices 1and2. Anexample arated. ThOe contribuOtion from the (p6) counterterms
O
is δ(6) has also been separated from the remaining part
ct
δ(6) ,whichconsistsmostlyofthecontributionsfromthe
A¯(χ )Rp χ = A¯(χ )R1 χ + [1 2]. (13) loops
p qπ p 1 2π 1 ↔ chiral loops of (p6). It should be noted that δ(6) also
O loops
Any contribution written in terms of the (p,q) notation receives a contribution from the chiral loops and coun-
is thus symmetric under the interchange of the valence terterms of (p4), which originate in the expansion of
quark masses χ and χ . Eq. (14) to O(p6). The superscripts (v) and (s) indicate
1 2
O
5
thevaluesofd andd ,respectively. Asexpected,the (loops and counterterms) for d =2, is
val sea sea
infinities in all expressions for the decay constant have
canceled. The appearance of ’unphysical’ B¯ logarithms δ(4)12 = 8Lrχ +4Lrχ + 1/2A¯(χ ), (16)
4 34 5 1 1s
is, in part, due to the partial quenching, and to the fact
that the results havebeen expressedin terms ofthe low- andthecontributionfromthe (p6)countertermstothe
O
estordermassesratherthanthefullphysicalones. Thus NNLO expression for the decay constant is
not all of them correspond to the quenched chiral loga-
rithms which are ill-behaved in the chiral limit. δ(6)12 = 8Kr χ2 + 16Kr χ χ + 8Kr χ2
ct 19 1 20 1 34 21 s
+ 32Kr χ2 + 8Kr χ2. (17)
22 34 23 1
A. Results for dval =1 At NNLO, the chiral loops form, by far, the largest con-
tribution to the decay constant. As a straightforward
The NLO result for d =1 is short, and will only be derivation of the results for d = 1 from the d = 2
val sea sea
given for d =2. The result for d =1 can readily be caseistediousandcomplicated,theexpressionsforthese
sea sea
derivedfrom that expression. The combined NLO result two cases are given separately below.
δ(6)11 = π Lr 4χ χ +2χ2 χ2 2π Lrχ2 π Lr χ2+3χ2
loops 16 0 − 1 3 1− 3 − 16 1 1 − 16 2 1 3
+ π16L(cid:2)r3 −3χ1χ3+5/2χ21(cid:3)−1/2χ23 + π126 −23/96(cid:2)χ1χ3+1/(cid:3)64χ21−47/192χ23 − 32Lr4Lr5χ1χ3
− 32Lr42χ(cid:2)23 − 8Lr52χ21 + 8Lr112χ21 +(cid:3)6A¯(χ1)(cid:2)Lr0 χ1+R1d − 4A¯(χ1)Lr1χ1 − 10(cid:3)A¯(χ1)Lr2χ1
+ 6A¯(χ1)Lr3 χ1+R1d − 2A¯(χ1)Lr5χ1 − 1/2A¯(cid:2)(χ1)B¯(χ1(cid:3)3,χ13,0)χ13 + 1/8A¯(χ1,ε)π16χ3
A¯(χ )π [1/6χ +2/3χ ] 8A¯(χ )Lrχ 20A¯(χ )Lrχ 8A¯(χ )Lrχ + 4A¯(χ )Lrχ
− 13 16 (cid:2) 1 (cid:3) 3 − 13 0 13 − 13 3 13 − 13 4 3 13 5 1
+ A¯(χ ,ε)π χ 24A¯(χ )Lrχ 6A¯(χ )Lrχ + 12A¯(χ )Lrχ + 3/8A¯(χ ,ε)π χ
13 16 3 − 3 1 3 − 3 2 3 3 4 3 3 16 3
+ 6B¯(χ ,χ ,0)LrRdχ + 6B¯(χ ,χ ,0)LrRdχ 2B¯(χ ,χ ,0)LrRdχ
1 1 0 1 1 1 1 3 1 1 − 1 1 5 1 1
1/8B¯(χ ,χ ,0,ε)π Rdχ 16B¯(χ ,χ ,0)Lrχ χ 8B¯(χ ,χ ,0)Lrχ2
− 1 1 16 1 1 − 13 13 4 13 3 − 13 13 5 13
+ 32B¯(χ ,χ ,0)Lrχ χ + 16B¯(χ ,χ ,0)Lrχ2 1/8HF(1,χ ,χ ,χ ,χ )χ
13 13 6 13 3 13 13 8 13 − 1 13 13 1 3
3/8HF(1,χ ,χ ,χ ,χ )χ + 1/8HF(2,χ ,χ ,χ ,χ )Rdχ + 5/12HF′(1,χ ,χ ,χ ,χ )χ2
− 13 13 3 1 3 1 13 13 1 1 1 1 1 1 1 1
+ HF′(1,χ ,χ ,χ ,χ ) 1/8χ χ 1/2χ2 + 3/8HF′(1,χ ,χ ,χ ,χ )χ χ
1 13 13 1 1 3− 1 13 13 3 1 1 3
+ 1/2HF′(2,χ1,χ1,χ1,χ1(cid:2))R1dχ21 + 3/8HF′(cid:3)(2,χ1,χ13,χ13,χ1)R1dχ21
+ 1/4HF′(5,χ ,χ ,χ ,χ )(Rd)2χ2 2HF′(3,χ ,χ ,χ ,χ )Rdχ2
1 1 1 1 1 1 − 1 13 1 13 1 1 1
+ 3/8HF′(1,χ ,χ ,χ ,χ )χ2 + 9/8HF′(1,χ ,χ ,χ ,χ )χ2
21 1 13 13 1 1 21 3 13 13 1 1
3/8HF′(2,χ ,χ ,χ ,χ )Rdχ2, (18)
− 21 1 13 13 1 1 1
δ(6)12 = π Lr 4χ χ 3χ2 +2χ2+2χ χ 2π Lrχ2 π Lr 3χ2 +χ2
loops 16 0 − π 1− π 1 3 4 − 16 1 1 − 16 2 π 1
+ π16L(cid:2)r3 −3χπχ1−2χ2π+5/2χ21+3/(cid:3)2χ3χ4 + π126 −23/96χ(cid:2)πχ1−1/3(cid:3)χ2π+1/64χ21+17/192χ3χ4
− 32Lr4Lr5(cid:2)χ1χ34 − 32Lr42χ234 − 8Lr52χ21 + 8(cid:3)Lr112χ21 +(cid:2) 6A¯(χπ)Lr0R1π1χπ − 8A¯(χπ)Lr1χπ (cid:3)
2A¯(χ )Lrχ + 6A¯(χ )LrRπ χ + 4A¯(χ )Lrχ 2A¯(χ )LrRπ χ
− π 2 π π 3 11 π π 4 π − π 5 11 1
1/4A¯(χ )B¯(χ ,χ ,0)Rπ χ + 1/8A¯(χ ,ε)π Rcχ + 6A¯(χ )Lr Rcχ +Rd 4A¯(χ )Lrχ
− π 1s 1s 1s 1s π 16 1 π 1 0 1 1 1 − 1 1 1
− 10A¯(χ1)Lr2χ1 + 6A¯(χ1)Lr3 R1cχ1+R1d − 2A¯(χ1)Lr5R1cχ1 − 1/4A¯(cid:2)(χ1)B¯(χ1s,χ(cid:3)1s,0)Rs1πχ1s
+ 1/8A¯(χ1,ε)π16Rπ1πχ34 − 16(cid:2)A¯(χ34)Lr1χ(cid:3)34 − 4A¯(χ34)Lr2χ34 + 8A¯(χ34)Lr4χ34
+ 1/4A¯(χ ,ε)π χ A¯(χ )π [1/4χ +1/6χ ] 4A¯(χ )Lrχ 10A¯(χ )Lrχ
34 16 34 − 1s 16 34 1s − 1s 0 1s − 1s 3 1s
4A¯(χ )Lrχ + 2A¯(χ )Lrχ + 1/4A¯(χ ,ε)π [χ +χ ] + 6B¯(χ ,χ ,0)LrRdχ
− 1s 4 34 1s 5 1 1s 16 34 s 1 1 0 1 1
+ 6B¯(χ ,χ ,0)LrRdχ 2B¯(χ ,χ ,0)LrRdχ 1/8B¯(χ ,χ ,0,ε)π Rdχ
1 1 3 1 1 − 1 1 5 1 1 − 1 1 16 1 1
8B¯(χ ,χ ,0)Lrχ χ 4B¯(χ ,χ ,0)Lrχ2 + 16B¯(χ ,χ ,0)Lrχ χ
− 1s 1s 4 34 1s − 1s 1s 5 1s 1s 1s 6 34 1s
+ 8B¯(χ ,χ ,0)Lrχ2 + 1/16HF(1,χ ,χ ,χ ,χ )Rv χ HF(1,χ ,χ ,χ ,χ ) 1/8R1 χ
1s 1s 8 1s π 1s 1s 1 π1s π − 1 1s 1s 1 sπ 1s
− 1/16R1cχ1] − 1/4HF(1,χ13,χ14,χ34,χ1)χ34 + 1/16HF(2,χ1,χ1s,χ1s,χ1)R1dχ1 (cid:2)
6
+ 1/4HF′(1,χ ,χ ,χ ,χ )(Rπ )2χ2 + 1/2HF′(1,χ ,χ ,χ ,χ )Rπ Rcχ2
π π 1 1 11 1 π 1 1 1 11 1 1
HF′(1,χ ,χ ,χ ,χ ) 1/4Rπ χ2+1/16Rv χ χ + HF′(1,χ ,χ ,χ ,χ ) 1/6χ2
− π 1s 1s 1 11 1 π1s π 1 1 1 1 1 1
+ 1/4(R1c)2χ21 + HF′(1,(cid:2)χ1,χ1s,χ1s,χ1) 1/8Rs1πχ1χ(cid:3)1s−1/2Rs1πχ21+3/16R1cχ(cid:2)21
+ 1/4HF′(1,χ(cid:3)13,χ14,χ34,χ1)χ1χ34 + 1/2(cid:2)HF′(2,χ1,χπ,χ1,χ1)R1π1R1dχ21 (cid:3)
+ 1/2HF′(2,χ ,χ ,χ ,χ )RcRdχ2 + 3/16HF′(2,χ ,χ ,χ ,χ )Rdχ2
1 1 1 1 1 1 1 1 1s 1s 1 1 1
+ 1/4HF′(5,χ ,χ ,χ ,χ )(Rd)2χ2 1/2HF′(1,χ ,χ ,χ ,χ )Rπ Rz χ2
1 1 1 1 1 1 − 1 π 1s 1s 1 1s 1sπ 1
HF′(1,χ ,χ ,χ ,χ )Rπ Rz χ2 HF′(3,χ ,χ ,χ ,χ )Rdχ2
− 1 1s 1s 1 1 1s 1sπ 1 − 1 1s 1 1s 1 1 1
3/16HF′(1,χ ,χ ,χ ,χ )Rv χ2 + HF′(1,χ ,χ ,χ ,χ ) 3/8R1 χ2 3/16Rcχ2
− 21 π 1s 1s 1 π1s 1 21 1 1s 1s 1 sπ 1− 1 1
+ 3/4H2F1′(1,χ34,χ13,χ14,χ1)χ21 − 3/16H2F1′(2,χ1,χ1s,χ1s,χ1)R(cid:2)1dχ21. (cid:3) (19)
B. Results for dval =2 The NLO contribution for dval = 2 and dsea = 1 may
be obtained from Eq. (20) by carefully taking the limit
cthoomInsbeginfeoanrteirdoavnla,sltt=hhea1te,xcpaarsnestashpieopnnesaufrmoirbnedrvtaholef=iinn2tdeaegrpreaenllsodniesgnetsrigmtnhaiafisns- χtree4rmm→aianniχnd3g.rreepEsliffadceuicnetsgivrAee¯ldy(χutpchse)iss→umch2eAat¯nh(χsatpd3Rr)o.pcpF→puinr1tghaetnrhmdeoRAr¯qpe(πχt→hπe)
cantlylarger. Themostgeneraltwo-flavorexpressionfor Rp. The contribution from the (p6) counterterms to
q O
the pseudoscalarmesondecayconstantis the resultwith the NNLO expression is
d = 2 and d = 2. However, in contrast with the
val sea
resultsford =1,the limitwherethe sea-quarkmasses
val
become degenerate is nontrivialeven at NLO. In the no- δ(6)22 = 4Kr χ2 + 16Kr χ χ + 8Kr χ2
ct 19 p 20 12 34 21 s
tationofEq.(15),theNLOcontributionford =2and
val + 32Kr χ2 + 8Kr χ χ , (21)
d =2 is 22 34 23 1 2
sea
δ(4)22 = 8Lrχ + 4Lrχ + A¯(χ ) 1/4Rp
4 34 5 12 p qπ
fromwhichthecorrespondingexpressionford =2and
− 1/8Rpc + 1/4A¯(χps) − 1/(cid:2)8A¯(χπ)Rπv12 dsea = 1 may readily be inferred. The remainvianlg NNLO
− 1/8B¯(χ(cid:3)p,χp,0)Rpd. (20) contributionswithdval =2are,fordsea =1anddsea =2,
δ(6)21 = π Lr 2χ χ 4χ χ +4χ2 χ2 2π Lrχ2 π Lr χ2 +3χ2
loops 16 0 − 1 2− 12 3 12− 3 − 16 1 12 − 16 2 12 3
+ π16L(cid:2)r3 −3/2χ1χ2−3χ12χ3+4χ212(cid:3)−1/2χ23 + π126 −17/192(cid:2)χ1χ2−23/(cid:3)96χ12χ3+5/48χ212
− 47/192(cid:2)χ23 − 32Lr4Lr5χ12χ3 − 32Lr42χ23 − 8(cid:3)Lr52χ212(cid:2)+ 8Lr112χ212 + A¯(χp)π16 [1/48χq+1/48χ12
+ 1/16χ3−(cid:3)1/12Rqpχ12−1/8Rqpχ3 + A¯(χp)Lr0 χp+4Rqpχp+Rpd + A¯(χp)Lr3 5/2χp+Rqpχp
+ 5/2Rpd + A¯(χp)Lr4 χ3−2Rqpχ3(cid:3) − A¯(χp)Lr5(cid:2)1/2χ12+Rqpχ12 (cid:3)+ A¯(χp)2 9/(cid:2)128−1/32RqpRpq
− A¯(χp)A¯(cid:3)(χp3) 1/16+(cid:2)1/12Rqp − A(cid:3)¯(χp)A¯(χq3) 3(cid:2)/16+1/12Rqp +(cid:3) 1/8A¯(χp)A(cid:2)¯(χ12) (cid:3)
+ A¯(χp)B¯(χp,χ(cid:2)p,0) 3/8χp−1/(cid:3)8Rqpχp−1/32Rqp(cid:2)Rpd+1/64Rpd −(cid:3) A¯(χp)B¯(χq,χq,0) 1/32RqpRqd
− 1/64Rqd − 1/4A¯((cid:2)χp)B¯(χp3,χp3,0)χp3 − 1/8A¯(χp)B¯(χ1,χ2,(cid:3)0)Rpqχp (cid:2)
+ 1/8A¯(χp(cid:3))C¯(χp,χp,χp,0)Rpdχp − 1/8A¯(χp,ε)π16 χp−Rqpχ3 + A¯(χp3)π16 [1/12χp3−1/4χq3
− 1/4χ3] − 4A¯(χp3)Lr0χp3 − 10A¯(χp3)Lr3χp3 − 4(cid:2)A¯(χp3)Lr4χ3(cid:3) + 2A¯(χp3)Lr5χ12 − 1/8A¯(χp3)2
+ A¯(χ )B¯(χ ,χ ,0) [1/8χ 5/8χ ] 1/16A¯(χ )B¯(χ ,χ ,0)Rd + 1/6A¯(χ )B¯(χ ,χ ,0)χ
p3 p p p− p3 − p3 q q q p3 1 2 3
+ 1/3A¯(χ )B¯(χ ,χ ,0,k) + 1/2A¯(χ ,ε)π χ A¯(χ )A¯(χ ) 1/64 1/16R1R2 4A¯(χ )Lrχ
p3 1 2 p3 16 3 − 1 2 − 2 1 − 12 1 12
− 10A¯(χ12)Lr2χ12 + 1/8A¯(χ12)2 − 1/2A¯(χ12)B¯(χ1,χ2,0,k) + 1(cid:2)/4A¯(χ12,ε)π16χ12(cid:3)
+ 1/4A¯(χ )A¯(χ ) 24A¯(χ )Lrχ 6A¯(χ )Lrχ + 12A¯(χ )Lrχ + 3/16A¯(χ )B¯(χ ,χ ,0)χ
13 23 − 3 1 3 − 3 2 3 3 4 3 3 p p 3
3/8A¯(χ )B¯(χ ,χ ,0)χ + 3/8A¯(χ ,ε)π χ + B¯(χ ,χ ,0)π 1/96Rdχ +1/32Rdχ
− 3 1 2 3 3 16 3 p p 16 p p p q
+ 1/16Rpdχ3 + B¯(χp,χp,0)Lr0Rpdχp + 5/2B¯(χp,χp,0)Lr3Rpdχp +(cid:2) B¯(χp,χp,0)Lr4 [2χpχ3
(cid:3)
7
4Rpχ χ +3Rdχ + B¯(χ ,χ ,0)Lr χ2 2Rpχ χ 1/2Rdχ + B¯(χ ,χ ,0)Lr 4χ2
− q p 3 p 3 p p 5 p− q 1 2− p 12 p p 6 3
− 8Rpqχpχ3 + 4B¯(χ(cid:3)p,χp,0)Lr7(Rpd)2 +(cid:2)B¯(χp,χp,0)Lr8 2χ23−4Rpqχ(cid:3)2p + B¯(χp,χp,0)2 (cid:2)1/8RpqRpdχp
+ 1/128(Rpd(cid:3))2 − 1/8B¯(χp,χp,0)B¯(χ1,χ2,0)RpqRpdχp +(cid:2)1/8B¯(χp,χp,0(cid:3))C¯(χp,χp,χp,0)(R(cid:2)pd)2χp
− 1/8B¯(χp,χp(cid:3),0,ε)π16Rpdχp3 − 8B¯(χp3,χp3,0)Lr4χp3χ3 − 4B¯(χp3,χp3,0)Lr5χ2p3
+ 16B¯(χ ,χ ,0)Lrχ χ + 8B¯(χ ,χ ,0)Lrχ2 + 1/64B¯(χ ,χ ,0)B¯(χ ,χ ,0)RdRd
p3 p3 6 p3 3 p3 p3 8 p3 1 1 2 2 1 2
8B¯(χ ,χ ,0)LrRdRd 4B¯(χ ,χ ,0)LrRdRd + 4C¯(χ ,χ ,χ ,0)LrRdχ χ
− 1 2 7 1 2 − 1 2 8 1 2 p p p 4 p p 3
+ 2C¯(χ ,χ ,χ ,0)LrRdχ2 8C¯(χ ,χ ,χ ,0)LrRdχ χ 4C¯(χ ,χ ,χ ,0)LrRdχ2
p p p 5 p p − p p p 6 p p 3 − p p p 8 p p
+ HF(1,χ ,χ ,χ ,χ ) 1/8χ 5/64χ +1/16RpRqχ + HF(1,χ ,χ ,χ ,χ ) [ 1/16χ
p p 12 12 p− 12 q p 12 p 13 23 12 − p
+ 1/16χq−1/8Rqpχ3 +(cid:2)HF(1,χ1,χ12,χ2,χ12) 1/32χ12−(cid:3)1/8R21R12χ12
1/8HF(1,χ12,χ12,χ(cid:3)12,χ12)χ12 3/8HF(1,χ1(cid:2)3,χ23,χ3,χ12)χ3 (cid:3)
− −
+ HF(2,χ ,χ ,χ ,χ ) 1/16RpRdχ 1/32Rdχ + HF(2,χ ,χ ,χ ,χ ) 1/16RqRdχ
p p 12 12 q p 12− p 12 p 12 q 12 p p 12
− 1/32Rpdχ12 + 1/8HF(cid:2)(2,χp,χ13,χ23,χ12)Rpdχp3 −(cid:3)1/64HF(5,χp,χp,χ12,χ12(cid:2))(Rpd)2χ12
− 1/32HF(5,χ(cid:3)1,χ2,χ12,χ12)R1dR2dχ12 − HF′(1,χp,χp,χ12,χ12) 1/8χpχ12+15/64χ212−3/16Rqpχ212
+ 1/16(Rqp)2χ212 + HF′(1,χp,χ13,χ23,χ12) 1/8χ212+3/8Rpqχpχ(cid:2)12+1/8Rpqχqχ12−1/8Rpqχ12χ3
− HF′(1,χ1,χ12,(cid:3)χ2,χ12) 1/32χ212−3/8R21R(cid:2)12χ212 + 1/8HF′(1,χ12,χ12,χ12,χ12)χ212 (cid:3)
+ 3/8HF′(1,χ13,χ23,χ3,χ(cid:2)12)χ12χ3 − HF′(2,χp,χ(cid:3)p,χ12,χ12) 1/16RqpRpdχ212−5/32Rpdχ212
− HF′(2,χp,χ12,χq,χ12) 1/16RpqRpdχ212−1/32Rpdχ212 − 1/8(cid:2)HF′(2,χp,χ13,χ23,χ12)Rpdχp(cid:3)3χ12
+ 5/64HF′(5,χp,χp,χ12,(cid:2)χ12)(Rpd)2χ212 + 1/32HF′(5,χ(cid:3)1,χ2,χ12,χ12)R1dR2dχ212
+ HF′(1,χ ,χ ,χ ,χ ) 5/4χ2 1/2RpRqχ2 HF′(1,χ ,χ ,χ ,χ )Rqχ2
1 p p 12 12 12− q p 12 − 1 p 13 23 12 p 12
− H1F′(1,χp3,χq3,χp,χ12)(cid:2)χ212 + H1F′(1,χ12,χ1,χ(cid:3)2,χ12) 1/4χ212−1/2R21R12χ212
− 1/4H1F′(3,χ12,χp,χp,χ12)RpqRpdχ212 + 1/4H1F′(3,χ12,(cid:2)χp,χq,χ12)RpqRpdχ212 (cid:3)
1/8HF′(7,χ ,χ ,χ ,χ )(Rd)2χ2 3/8HF′(1,χ ,χ ,χ ,χ )χ2
− 1 12 p p 12 p 12 − 21 p p 12 12 12
+ 3/8HF′(1,χ ,χ ,χ ,χ )Rqχ2 + 3/8HF′(1,χ ,χ ,χ ,χ )Rpχ2
21 p 13 23 12 p 12 21 p3 q3 p 12 q 12
3/8HF′(1,χ ,χ ,χ ,χ )Rpχ2 + HF′(1,χ ,χ ,χ ,χ ) 15/64χ2 3/16RpRqχ2
− 21 p3 q3 q 12 q 12 21 12 p p 12 12− q p 12
− H2F1′(1,χ12,χ1,χ2,χ12) 3/32χ212−3/8R21R12χ212 + 3/8H2F1′((cid:2)1,χ12,χ12,χ12,χ12)χ212 (cid:3)
+ 9/8H2F1′(1,χ3,χ13,χ23,χ(cid:2)12)χ212 − 3/8H2F1′(3,χp3(cid:3),χp,χq3,χ12)Rpdχ212
HF′(3,χ ,χ ,χ ,χ ) 3/16RpRdχ2 3/32Rdχ2 HF′(3,χ ,χ ,χ ,χ ) 3/16RqRdχ2
− 21 12 p p 12 q p 12− p 12 − 21 12 p q 12 p p 12
− 3/32Rpdχ212 + 3/64H2F(cid:2)1′(7,χ12,χp,χp,χ12)(Rpd)2χ212(cid:3)+ 3/32H2F1′(7,χ12,χ1,χ2,χ(cid:2)12)R1dR2dχ212, (22)
(cid:3)
δ(6)22 = π Lr 4χ χ 3χ2 2χ χ +4χ2 +2χ χ 2π Lrχ2 π Lr 3χ2 +χ2
loops 16 0 − π 12− π− 1 2 12 3 4 − 16 1 12 − 16 2 π 12
+ π16L(cid:2)r3 −3χπχ12−2χ2π−3/2χ1χ2+4χ212+3/(cid:3)2χ3χ4 + π126 −23/96χπ(cid:2)χ12−1/3χ(cid:3)2π−17/192χ1χ2
+ 5/48χ21(cid:2)2+17/192χ3χ4 − 32Lr4Lr5χ12χ34 − 32Lr42χ23(cid:3)4 − 8Lr5(cid:2)2χ212 + 8Lr112χ212
+ A¯(χp)π16 −1/12Rqpπχ1(cid:3)2−1/8Rqpπχ34+1/48Rpcχq+1/48Rpcχ12+1/16Rpcχ34
+ A¯(χp)Lr0 (cid:2)4Rqpπχp+Rpcχp+Rpd + A¯(χp)Lr3 Rqpπχp+5/2Rpcχp+5/2Rpd (cid:3)
− A¯(χp)Lr4 (cid:2)2Rqpπχ34−Rpcχ34 −(cid:3)A¯(χp)Lr5 Rqpπ(cid:2)χ12+1/2Rpcχ12 + A¯(χp)2(cid:3) 1/16+1/32(Rqpπ)2
− 1/32RqpπR(cid:2)pc +1/128(Rpc)2 −(cid:3) A¯(χp)A¯(χps)(cid:2) 1/24Rqpπ+5/48Rspπ(cid:3)−7/96Rpc (cid:2)
− A¯(χp)A¯(χqs) 1/24Rqpπ+1(cid:3)/16Rspπ+1/32Rpc(cid:2) − A¯(χp)A¯(χπ) 1/32RqpπRπv1(cid:3)2−1/64RpcRπv12
+ 1/8A¯(χp)A¯(χ(cid:2)12) + 1/12A¯(χp)A¯(χ34)Rpπp +(cid:3)A¯(χp)B¯(χp,χp,0)(cid:2) 1/4χp−1/8RqpπRpcχp−1/(cid:3)32RqpπRpd
+ 1/8(Rpc)2χp+1/64RpcRpd + 1/8A¯(χp)B¯(χp,χπ,0)RqzπpRpcRπv12(cid:2)χp − A¯(χp)B¯(χq,χq,0) 1/32RqpπRqd
(cid:3) (cid:2)
8
1/64RcRd 1/8A¯(χ )B¯(χ ,χ ,0)Rp χ 1/8A¯(χ )B¯(χ ,χ ,0)Rq Rcχ
− p q − p ps ps sπ ps − p 1 2 pπ p p
+ 1/8A¯(χp)C¯(cid:3)(χp,χp,χp,0)RpcRpdχp + A¯(χp,ε)π16 −1/8χp+1/8Rqpπχ34+1/8Rpπpχ34
+ A¯(χps)π16 [1/24χps−1/8χqs−1/8χ34] − 2A¯(χp(cid:2)s)Lr0χps − 5A¯(χps)Lr3χps − 2A¯(χ(cid:3)ps)Lr4χ34
+ A¯(χ )Lrχ 1/32A¯(χ )2 + A¯(χ )A¯(χ ) 7/96Rπ 5/48Rπ 1/32Rπ +5/48Rπ 1/24Rπ
ps 5 12 − ps ps π pp− ps− qq qs− 12
+ A¯(χps)B¯(χp,χp,0) [1/16Rspπχp−5/16Rspπχps](cid:2)− A¯(χps)B¯(χp,χπ,0) 1/12RpπsRqzpπχp (cid:3)
+ 1/6RpπsRqzpπχps − 1/32A¯(χps)B¯(χq,χq,0)Rqd + 1/12A¯(χps)B¯(χ1,χ2(cid:2),0)Rsqπχs
+ 1/6A¯(χps)B¯(χ1,(cid:3)χ2,0,k)Rsqπ + 1/8A¯(χps,ε)π16 [χ34+χs] − 1/16A¯(χp3)A¯(χp4)
+ 1/16A¯(χ )A¯(χ ) + A¯(χ )π [1/24Rv χ +1/16Rv χ ] + A¯(χ )Lr [6Rπ χ +Rv χ ]
p3 q4 π 16 π12 π π12 12 π 0 12 π π12 π
8A¯(χ )Lrχ 2A¯(χ )Lrχ + A¯(χ )Lr [6Rπ χ +5/2Rv χ ] + A¯(χ )Lr [4χ +Rv χ ]
− π 1 π − π 2 π π 3 12 π π12 π π 4 π π12 π
A¯(χ )Lr [2Rπ χ +1/2Rv χ ] + 1/128A¯(χ )2(Rv )2 + 1/12A¯(χ )A¯(χ )Rv
− π 5 12 12 π12 12 π π12 π 34 π12
+ A¯(χ )B¯(χ ,χ ,0) 1/8Rp Rπ χ +3/16Rcχ 1/8(Rc)2χ 1/16Rd+1/64RdRv
π p p − qπ pp p p p− p p− p p π12
− 1/8A¯(χπ)B¯(χp,χπ(cid:2),0)RπpπR1π2Rpzqπχp − 1/8A¯(χπ)B¯(χps,χps,0)Rpπsχps (cid:3)
1/16A¯(χ )B¯(χ ,χ ,0)Rv χ 1/8A¯(χ )B¯(χ ,χ ,0)R1 R2 χ
− π π π π12 34 − π 1 2 ππ ππ 34
+ 1/8A¯(χ )C¯(χ ,χ ,χ ,0)Rπ Rdχ + 1/8A¯(χ ,ε)π [χ Rπ χ Rv χ ]
π p p p pp p p π 16 π − 12 π − π12 π
+ A¯(χ )A¯(χ ) 1/32Rp Rc+1/16R1 R2 +1/64RcRc 4A¯(χ )Lrχ 10A¯(χ )Lrχ
1 2 − qπ q 2π 1π 1 2 − 12 1 12 − 12 2 12
+ 1/8A¯(χ12)2(cid:2)− 1/2A¯(χ12)B¯(χ1,χ2,0,k) + 1/4A¯(χ12,ε(cid:3))π16χ12 − 16A¯(χ34)Lr1χ34
4A¯(χ )Lrχ + 8A¯(χ )Lrχ + 1/8A¯(χ )B¯(χ ,χ ,0)Rp χ
− 34 2 34 34 4 34 34 p p ππ 34
+ A¯(χ )B¯(χ ,χ ,0) 1/12Rz Rv χ 1/6Rz Rv χ 1/6A¯(χ )B¯(χ ,χ ,0,k)Rz Rv
34 p π qπp π12 p− qπp π12 34 − 34 p π qπp π12
+ 1/8A¯(χ34)B¯(χπ,χπ(cid:2),0)Rπv12χ34 + A¯(χ34)B¯(χ1,χ2,0)[−1(cid:3)/4χ34+1/6R1π2χ12−1/4R1π2χ34]
1/6A¯(χ )B¯(χ ,χ ,0,k)Rπ + 1/4A¯(χ ,ε)π χ + 1/16A¯(χ )A¯(χ )
− 34 1 2 12 34 16 34 1s 2s
+ B¯(χ ,χ ,0)π 1/96Rdχ +1/32Rdχ +1/16Rdχ + B¯(χ ,χ ,0)LrRdχ
p p 16 p p p q p 34 p p 0 p p
+ 5/2B¯(χp,χp,0)(cid:2)Lr3Rpdχp + B¯(χp,χp,0)Lr4 −4Rqpπχpχ(cid:3)34+2Rpcχpχ34+3Rpdχ34
+ B¯(χp,χp,0)Lr5 −2Rqpπχ1χ2+Rpcχ2p−1/2(cid:2)Rpdχ12 + B¯(χp,χp,0)Lr6 8Rqpπχpχ34(cid:3)−4Rpcχpχ34
− 4Rpd χ34] + 4B¯(cid:2)(χp,χp,0)Lr7(Rpd)2 + B¯(χp,χp,0)(cid:3)Lr8 4Rqpπχ1χ2−2R(cid:2)pcχ2p+2(Rpd)2
− B¯(χp,χp,0)2 1/8RqpπRpdχp−1/8RpcRpdχp−1/128(R(cid:2)pd)2 (cid:3)
+ 1/8B¯(χp,χp,0(cid:2))B¯(χp,χπ,0)RqzπpRpdRπv12χp − 1/8B¯(χp,χ(cid:3)p,0)B¯(χ1,χ2,0)RpqπRpdχp
+ 1/8B¯(χ ,χ ,0)C¯(χ ,χ ,χ ,0)(Rd)2χ 1/16B¯(χ ,χ ,0,ε)π Rdχ +Rdχ
p p p p p p p − p p 16 p p p 34
+ 8B¯(χp,χπ,0)Lr7RqzpπRpdRπz34p + 4B¯(χp,χπ,0)Lr8RqzpπRpdRπz34p −(cid:2) 4B¯(χps,χps,0(cid:3))Lr4χpsχ34
2B¯(χ ,χ ,0)Lrχ2 + 8B¯(χ ,χ ,0)Lrχ χ + 4B¯(χ ,χ ,0)Lrχ2
− ps ps 5 ps ps ps 6 ps 34 ps ps 8 ps
+ 2B¯(χ ,χ ,0)LrRv χ2 + B¯(χ ,χ ,0)LrRv χ2 4B¯(χ ,χ ,0)LrRv χ2
π π 4 π12 π π π 5 π12 π − π π 6 π12 π
4B¯(χ ,χ ,0)Lr(Rz )2Rv B¯(χ ,χ ,0)Lr Rv χ2+Rv χ2
− π π 7 π4 π12 − π π 8 π12 3 π12 4
+ 1/64B¯(χ1,χ1,0)B¯(χ2,χ2,0)R1dR2d − 8B¯(χ1,χ2(cid:2),0)Lr7R1dR2d − 4B¯(cid:3)(χ1,χ2,0)Lr8R1dR2d
+ 4C¯(χ ,χ ,χ ,0)LrRdχ χ + 2C¯(χ ,χ ,χ ,0)LrRdχ2 8C¯(χ ,χ ,χ ,0)LrRdχ χ
p p p 4 p p 34 p p p 5 p p − p p p 6 p p 34
4C¯(χ ,χ ,χ ,0)LrRdχ2 + HF(1,χ ,χ ,χ ,χ ) 1/8χ 1/16χ 1/16(Rp )2χ
− p p p 8 p p p p 12 12 p− 12− qπ 12
+ 1/16RqpπRpcχ12−1/64(Rpc)2χ12 + HF(1,χp,χ1s,χ(cid:2)2s,χ12) −1/8Rqpπχqs+1/16Rqpπχ12
− 3/32Rspπχp+1/32Rspπχq +1/1(cid:3)6Rpcχps − 1/8HF(1,χp3,χ(cid:2)q4,χ34,χ12)χ34
+ HF(1,χπ,χp,χ12,χ12) 1/16RqpπRπv12χ12(cid:3)−1/32RpcRπv12χ12 − 1/64HF(1,χπ,χπ,χ12,χ12)(Rπv12)2χ12
+ 1/32HF(1,χπ,χ1s,χ2s(cid:2),χ12) −Rpπsχp+Rpπsχq+Rπvpsχπ+(cid:3)Rπv12χs
+ HF(1,χ1,χ12,χ2,χ12) 1/16R(cid:2) qpπRqcχ12−1/8R21πR12πχ12−1/32R1cR(cid:3)2cχ12
− 1/8HF(1,χ12,χ12,χ12,(cid:2)χ12)χ12 + HF(2,χp,χp,χ12,χ12) 1/16RqpπRpdχ12(cid:3)−1/32RpcRpdχ12
− 1/32HF(2,χp,χπ,χ12,χ12)RpdRπv12χ12 + HF(2,χp,χ12,χ(cid:2)q,χ12) 1/16RpqπRpdχ12−1/32R(cid:3)qcRpdχ12
+ 1/16HF(2,χp,χ1s,χ2s,χ12)Rpdχps − 1/64HF(5,χp,χp,χ12,χ12)(cid:2)(Rpd)2χ12 (cid:3)
9
1/32HF(5,χ ,χ ,χ ,χ )RdRdχ + HF′(1,χ ,χ ,χ ,χ ) 1/8χ χ 3/16χ2
− 1 2 12 12 1 2 12 p p 12 12 − p 12− 12
− 1/16(Rqpπ)2χ212+3/16RqpπRpcχ212−3/64(Rpc)2χ212 + HF′(1,χp(cid:2),χ1s,χ2s,χ12) −5/16Rqpπχqχ12
+ 1/8Rp χ χ +3/32Rp χ χ +7/32Rp χ χ 1/16Rc χ χ ]
qπ ps 12 sπ q 12 sπ 12 s(cid:3)− p ps 12 (cid:2)
+ 1/8HF′(1,χ ,χ ,χ ,χ )χ χ + HF′(1,χ ,χ ,χ ,χ ) 1/4Rp Rπ χ2 1/16Rp Rv χ2
p3 q4 34 12 12 34 π p 12 12 qπ 12 12− qπ π12 12
+ 1/8RpπpRpcχ212+1/32RpcRπv12 χ212 + HF′(1,χπ,χπ,χ12,χ12) 1(cid:2)/4(R1π2)2χ212+1/4R1π2Rπv12χ212
+ 5/64(Rπv12)2χ212 + HF′(1,χπ,χ1(cid:3)s,χ2s,χ12) −3/32Rpπsχpχ12(cid:2)−5/32Rpπsχqχ12+1/4R1π2χ212
− 1/32Rπvpsχπχ12(cid:3)−1/32Rπv12 χ12χs] + HF′(1(cid:2),χ1,χ12,χ2,χ12) −1/16RqpπRqcχ212+3/8R21πR12πχ212
+ 1/32R1cR2c χ212 + 1/8HF′(1,χ12,χ12,χ12,χ12)χ212 + HF′(2,χ(cid:2)p,χp,χ12,χ12) −1/16RqpπRpdχ212
+ 5/32RpcRpd χ212(cid:3) + HF′(2,χp,χπ,χ12,χ12) 1/8RpπpRpdχ212+1/32RpdRπv12χ212 (cid:2)
− HF′(2,χp,χ12,(cid:3)χq,χ12) 1/16RpqπRpdχ212−1(cid:2)/32RqcRpdχ212 − 1/16HF′(2,χp,(cid:3)χ1s,χ2s,χ12)Rpdχpsχ12
+ 5/64HF′(5,χp,χp,χ12,χ(cid:2)12)(Rpd)2χ212 + 1/32HF′(5,χ1,χ(cid:3)2,χ12,χ12)R1dR2dχ212
+ HF′(1,χ ,χ ,χ ,χ ) χ2 +1/2(Rp )2χ2 1/2Rp Rcχ2 +1/4(Rc)2χ2
1 p p 12 12 12 qπ 12− qπ p 12 p 12
+ 1/2H1F′(1,χp,χ1s,χ2s,χ(cid:2)12)RqpπRszqpχ212 − 1/2H1F′(1,χps,χqs,χp,χ12)Rspπχ21(cid:3)2
1/2HF′(1,χ ,χ ,χ ,χ )Rπ Rz χ2 + 1/4HF′(1,χ ,χ ,χ ,χ ) Rp Rv χ2
− 1 ps qs π 12 12 spπ 12 1 12 p π 12 qπ π12 12
− RpπpRqzpπRpc χ212 − H1F′(1,χ12,χπ,χπ,χ12) 1/4R1π2Rπv12χ212+1/8(Rπv1(cid:2)2)2χ212
+ H1F′(1,χ12,χ1,χ(cid:3)2,χ12) 1/4RqpπRqcχ212−1/2(cid:2)R21πR12πχ212 + 1/4H1F′(3,χ12,χp(cid:3),χp,χ12) RqpπRpdχ212
− RpcRpd χ212 + 1/4H1F′(3(cid:2),χ12,χp,χq,χ12)RpqπRpdχ212 + 1/(cid:3)4H1F′(3,χ12,χp,χπ,χ12)R1π2Rpz(cid:2)qπRpdχ212
− 1/8H1F′(7(cid:3),χ12,χp,χp,χ12)(Rpd)2χ212 − 3/8H2F1′(1,χp,χp,χ12,χ12)χ212
3/16HF′(1,χ ,χ ,χ ,χ )Rp Rz χ2 + 3/16HF′(1,χ ,χ ,χ ,χ ) Rp χ2 +Rp χ2 Rcχ2
− 21 p 1s 2s 12 qπ sqp 12 21 ps qs p 12 qπ 12 sπ 12− p 12
+ 3/16H2F1′(1,χps,χqs,χq,χ12)RpqπRszpqχ212 + 3/16H2F1′(1,χps,χqs,χπ,χ12)R(cid:2)1π2RpzqπRszpπχ212 (cid:3)
3/16HF′(1,χ ,χ ,χ ,χ )Rπ Rz Rz χ2 + HF′(1,χ ,χ ,χ ,χ ) 3/16χ2 +3/16(Rp )2χ2
− 21 π 1s 2s 12 12 s1π s2π 12 21 12 p p 12 12 qπ 12
− 3/16RqpπRpcχ212+3/64(Rpc)2χ212 + H2F1′(1,χ12,χp,χπ,χ12) −3/16RqpπR(cid:2)πv12χ212+3/32RpcRπv12χ212
+ 3/64H2F1′(1,χ12,χπ,χπ,χ12)(Rπv(cid:3)12)2χ212 + H2F1′(1,χ12,χ1,χ2(cid:2),χ12) −3/16RqpπRqcχ212+3/8R21πR12πχ(cid:3)212
+ 3/32R1cR2c χ212 + 3/8H2F1′(1,χ12,χ12,χ12,χ12)χ212 + 3/8H2F1′(1,χ(cid:2)34,χp3,χq4,χ12)χ212
− 3/16H2F1′(3,χp(cid:3)s,χp,χqs,χ12)Rpdχ212 − H2F1′(3,χ12,χp,χp,χ12) 3/16RqpπRpdχ212−3/32RpcRpdχ212
− H2F1′(3,χ12,χp,χq,χ12) 3/16RpqπRpdχ212−3/32RqcRpdχ212 + 3/(cid:2)32H2F1′(3,χ12,χp,χπ,χ12)RpdRπv1(cid:3)2χ212
+ 3/64H2F1′(7,χ12,χp,χp,χ(cid:2)12)(Rpd)2χ212 + 3/32H2F1′(7,χ12,(cid:3)χ1,χ2,χ12)R1dR2dχ212. (23)
IV. TWO-FLAVOR PSEUDOSCALAR MESON of (p6) LEC:s that contribute to the meson mass is
O
MASSES slightlydifferentfromthatofthe decayconstant,inpar-
ticularthenumberof (p6)LEC:sislargerforthemeson
O
The pseudoscalar meson squared masses M2 are cal- mass. Theresultsforthe pseudoscalarmesonmassesare
PS
culated from the resummation of the irreducible self- given in the form
energies, according to
δ(4)vs δ(6)vs+δ(6)vs
MP2S =M02+Σ(MP2S), (24) MP2S =M02+ F2 + ct F4 loops +O(p8), (25)
where M is the lowest order meson mass, and Σ is the where the (p4) and (p6)contributions havebeen sep-
0
O O
sum of the irreducible self-energy diagrams. The self- arated. It should be noted that the lowest order mass
energycontributions iΣat (p6),orNNLO,areshown has not been factored out of the results, as this is not
− O
in Fig. 2. The pseudoscalar meson mass is a function meaningfulforthe results withd =2. Similarlyto the
val
of the same mass inputs and LEC:s as outlined for the treatment of the decay constant, the contribution from
decayconstants inthe previoussection, althoughthe set the (p6)countertermshasalsobeenseparatedfromthe
O
