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Virtual photons in chiral perturbation theory
NUCLEAR PHYSICS B ELSEVIER
Nuclear Physics B433 (1995) 234-254
Virtual photons in chiral perturbation theory * Res Urech 1 Institute for Theoretical Physics, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland Received 24 May 1994; accepted 11 August 1994
Abstract
In the framework of chiral perturbation theory virtual photons are included. We calculate the divergences of the generating functional to one loop and determine the structure of the local action that incorporates the counterterms which cancel the divergences. As an application we discuss the corrections to Dashen's theorem at o r d e r eemq.
1. Introduction In the chiral limit where the light up, down and strange quark masses go to zero, the Q C D lagrangian has a SU(3) R × SU(3) L chiral symmetry that is spontaneously broken by the ground state of the theory. T h e r e are eight Goldstone bosons: the 7r's, K's and r/. Their interactions are described by an effective low energy theory, called chiral perturbation theory (CHPT). At low m o m e n t a the chiral lagrangian can be expanded in derivatives of the Goldstone fields and in the masses of the three light quarks. C H P T is a nonrenormalizable theory: Loops produce ultraviolet divergences, which can be absorbed by introducing counterterms. The finite parts of the coupling constants of this counterterm lagrangian are not determined by the lagrangian that generated the loops. The complete information about them is hidden in the Q C D lagrangian, but there is no known way to extract this information from first principle alone. The couplings have to be evaluated from experiments. As shown by Weinberg [1], the loops are suppressed: Every loop and
* Work supported in part by Schweizerischer Nationalfonds. 1 New address since August 1994: Institut fiir Theoretische Teilchenphysik, Universit~it Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Deutschland; e-mail: [email protected]. 0550-3213/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 5 5 0 - 3 2 1 3 0 0 3 4 3 - 2
R. Urech / Nuclear Physics B433 (1995) 234-254
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the associated counterterm correspond to successively higher powers of momenta or quark masses. At low energies, the contributions from higher loops are small. C H P T is an effective theory: It contains all terms allowed by the symmetry of the QCD lagrangian in the chiral limit. The local action is generated by the effective lagrangian, which is in general not determined by the counterterms alone at higher orders in the expansion of momenta and quark masses. Additional terms with finite couplings have to be included. The generating functional to one loop and the corresponding local action in the strong sector has been determined by Gasser and Leutwyler [2]. An extension to the AS = 1 nonleptonic weak interactions to one loop has been given by Kambor, Missimer and Wyler [3]. Here we consider virtual photons and include them in the mesonic sector. We calculate the divergences of the one-loop functional in the presence of external currents and determine the structure of the local action at this order. The article is organized as follows. In Section 2 virtual photons are included in C H P T with three flavours. We evaluate the divergences of the generating functional to one loop and determine the structure of the local action at order p4. In Section 3 we discuss the corrections to Dashen's theorem at order eZmq and compare our estimates with the results in the literature. In Section 4 we summarize our results, and in Appendix A the matrix relations are given that are used to simplify the effective lagrangian at order p4. Finally, in Appendix B we list the renormalized masses of the pseudoscalar mesons at order e2mq.
2. Generating functional We suppose that the reader is familiar with CHPT, otherwise we refer her or him to comprehensive reviews [4]. At leading order, the effective lagrangian of the strong and electromagnetic interactions which respects the chiral symmetry SU(3)R × SU(3)c, P and C invariance is in the mesonic sector [5] =
-
-
½*G
)2
1 2 t~ +~FO(d U + d~U+xU++x+U)+C(QUQU+),
(1)
where F,v is the field strength tensor of the photon field A , , F,~ = OuA~ - O A,. A is the gauge fixing parameter and will from now on be kept at A = 1. U is an unitary 3 × 3-matrix and incorporates the fields of the eight pseudoscalar mesons,
UU += 1,
det U = 1, 8
U=exp
Fo]
q~=
a=lEAa@a'
(2)
R. Urech /Nuclear Physics B433 (1995) 234-254
236
where the Aa'S are the Gell-Mann matrices, and 7/-o
"08
77-+
7r° ___+
Tr
K +
r/s
__
K 0
Ro
(3)
2r/s
K_
d~,U is a covariant derivative, incorporating the couplings to the photon field A~,, the external vector and axial vector currents vu and a~,, respectively, duU = O~U - i(v~, + QA u + a~.)U + iU(v. + Q A . - au),
(4)
where Q is the charge matrix of the three light quarks, e( 2 Q= 5
) - 1
-1
e(
1
)
= "~ A3 + -~-/~'8 "
(5)
We do not consider singlet vector and axial vector currents, and we put therefore tr v~, = tr a~, = 0. X denotes the coupling of the mesons to the scalar and pseudoscalar currents s and p, respectively, X = 2B0(s + ip),
(6)
where s incorporates the mass matrix of the quarks,
/mu )
s = ~ / + ... =
md
+ ....
(7)
ms
F 0 is the pion decay constant in the chiral limit, F~ = F0[1 + O ( m q ) ] . B 0 is related to the quark condensate (0l ~u 10)= -FZB0[1 + O(mq)], and C determines the purely electromagnetic part of the masses of the charged pions and kaons in the chiral limit, C M2± = M 2 ~ = 2e2~--~ + O ( m q ) .
(8)
To ensure the chiral SU(3) R X SU(3) L symmetry of the lagrangian 2 2~Q) we introduce local spurions QR, QL instead of the charge matrix Q. The lagrangian is modified, ~ 2 ( Q ) = __14"F /zv-p/.*v__ 1 ( 0 . A k ~ ) 2
+ 1 F o 2 ( d " U + d . U + x U + + x + U ) + C(QRUQLU+),
(9)
R. Urech/ Nuclear Physics B433 (1995) 234-254
237
and d~U is changed to
d~U = OuU - i(v~, + QRA~, + au)U+ iU(v~ + QLA~, - a~).
(10)
The rules of chiral transformation are [2,5]
giNg{, QI --+glQig~, I = R, L, v. + QRA~, + a~, --+gR(V. + QRA. + au)g ~ + igRO.g~, V~ + QLA~ - a~, ~gL(V~, + QLA. - au)g ~ + igeO~,g{, s +ip--*gR(s +ip)g ~,
(11)
gR,L ~ SU(3)R,L" Since the currents v~ and G, count as O(p), where p means the momentum of the external fields, the term QA~, has the same dimension. We put dim(QR,L) = O ( p ) ,
dim(A~) = O(1),
(12)
the lagrangian ..~2(Q) is therefore of order p2. The convention (12) has the advantage that electromagnetic interactions do not turn upside down the usual chiral counting. The procedure to obtain the generating functional at order O(p 4) is very similar to conventional CHPT. The one-loop graphs generated by .Z~2(Q) are of order O(p4). They contain divergences which are absorbed by adding tree graphs, evaluated with the lagrangian .9'4(Q) of order O(p4), see below. The generating functional becomes, up to and including terms of order O(p4),
exp[i.U(G,a~,,s,p)]=Nf[dU][dA,]exp(ifd4x(..9'(2°'+.~4(°))),
(13)
where the integration over the fields is carried out in the one-loop approximation. (Here and below we disregard contributions from the anomaly altogether.) To evaluate the divergent part of the one-loop functional, we transform the lagrangian 2 2 (°) (9) to euclidean spacetime, £~c~2(0)~.2"(Eo), and expand the fields U and A . around the classical solutions (U, A . ) of the equations of motion,
U=u exp{ Fo i~ }lu =u( =
i
+ =ue.
ro Z,u" =~¢~ ÷ Ep.,
-
1
+i
==ue2.
Z/~o-
Fo +
21;2o+''" u] ....
(14)
238
R. Urech / Nuclear Physics B433 (1995) 2 3 4 - 2 5 4
where we put 0 = u e, and ~ is a traceless hermitian matrix. We insert the expansion (14) in the action 5PE,
,_9"E= fd4XE C.~CEO"+ f d 4 x z ( ¼ ( D . f D ~ f - [ a . ,
f] [Au, sc] +o-~ 2)
1 C 8
F 2 ( [ H R + H L , sC][HR--HL, so]>
+ ½F0(~: [ MR, A/x ] -- H E Dp.~ )~-~ ..[_~Ep.( v l __0~, 0 "~- ~* t ~'2/" 0 l\ , ./42L))E/. ) + ... ,
(15)
where the ellipsis denotes higher order terms in the fluctuations ~ and q,. We used the expressions (see also Ref. [2])
C = ½[U+, ~ U ] __ I.~lU+GARu -1 +dud U + = A~ = ~u
~luG~,l"LU +,
1 --+ u, ~ud~U
oR = Vt* + QR"z~" + au'
G~L_-v~.+ O Y. H R =
(16)
a.,
u+{Q, U}u +=
I ~ + Q R u q-
UQL u+,
= ½(u+xu++ ux+u). Fu and A, are anti-hermitian matrices, whereas HR, L and o- are hermitian ones and fd4XE .~(EQ) represents the classical action of .9'~Q). The normal brackets [., • ] denote the commutator, the curly brackets {., • } stand for the anti-commutator. We use the parametrization sc = E,sc~A~, where the aa's are the Gell-Mann matrices. We define a new covariant derivative Xu =D~, + X,,, with ~('ap~p Xu~ a -_ D ~ a +..~ ~ , 2~ueo = OueP+ X~a~a '
(17)
where D ~ a is understood as
Du~ a = Ou~a + rfb~b
(18)
and F2b
=
X~p ~
:1< [ a a, ab]C>,
-
_
Xu;pa
~
_ ~Fo(HLA 1 a )8~. p
(19)
R. Urech / Nuclear Physics B433 (1995) 234-254
239
The action becomes at the one-loop level ~ E Ioneloop = ~1j fd4x El[ca(l, _ ~ , ~ b ~ a b
+%(-X.Z.
..}_o.ab)~b qt- ~a~12C.ix
+ p)%],
(20)
where now o.ab= _ _ I ( [ A
' /~a] [A/~, /~b]) ..[_ l(o.{/~a ' /~b})
C aF2([HR+HL,
Aa][HR_HL,
Ab])
1 2 ( H L Aa) ( H L A b )' ~F°
(21)
a 1 l~a "Y/.=Fo(([HR, Au] +$D~.HL) ), 3 2 2 p= gF6(H[).
We collect the fluctuations of the mesons and of the photon field and define a new flavour space r/A, where A runs from 1 to 12, r / = (scl . . . . . sc8, e ° . . . . . e3). The action above can now be written as a quadratic form, ~a E [ oneloop = where X~, and
lfd4xE ~A(--XtxXtzsAB -t-AAB)~ B,
A AB are
Z** = Oul +
(22)
12 × 12-matrices,
X2 b
= O~,l + g~,,
~Y A = 5Y 1 o-b p(~o
(23)
This leads to a gaussian integral for the generating functional, exp(--UE I one loop)=
Nfd~trl]
e x p [ - a ( r / , .@r/)]
= U ' ( d e t .@) -1/2, where we defined the differential operator ( f , g) = ~ f d 4 x z A
(24)
~AB = _~,,~6AB + AAB and
TAgA.
(25)
Omitting the constant contribution we arrive at "~E [ one loop = ~1 ln(det .~)
(26)
To renormalize the determinant we use dimensional regularization. In Minkowski spacetime, the one-loop functional in d = 4 dimensions is given by [6] 1
-~oneloop --
167r 2
1 d_4fd'x Tr(-~Y~vY"~+½A2)+finiteparts,
(27)
240
R. Urech /Nuclear Physics B433 (1995) 234-254
Table 1 P and C transformation properties for the fields, the charges and for their derivatives. The definition of cRQR and C~, LQL is given in the text. Spacetime arguments are suppressed P
C
U
U+
UT
d~,U
d~'U +
(d~U)T
X
X+
XT
OR
c~_mRQR
QL cLgQL
QTL (CuLQL)T
QL
OR
cLQL
cRgQR
QT (CgRQR)T
where Tr means the trace in the flavour space ~/A and Ya~ denotes the field strength tensor of Ya,
Ya.=OaY~-O~ra + [Yix, r.].
(28)
In Table 1 we list P and C transformation properties for the fields and quantities used in the one-loop functional. cix RQR and caLQL mean covariant derivatives of QR and QL, respectively,
cixIQI --°aQ, - i[GI, QI] , I = R, L,
(29)
that transform under SU(3)R x SU(3) e in the same way as QR and QL, c' O,
gicixQlgi, ' +
I= R, L,
gR,L ~ SU(3)R,L.
(30)
The divergences in -~one loop can be absorbed by adding the effective lagrangian .~4(Q) that contains all terms of order O(p 4) allowed by the chiral symmetry, P and C invariance. Since in the usual physical picture the charge matrix is a constant and has no chirality, we put again Q R = Q L = Q and OaQ=O. We keep the notation cR'LQ = -i[G R'L, Q] in order to remember the correct chiral transformation of c~Q I. By using the matrix relations that we list in Appendix A, the lagrangian .£~4(Q) can be simplified to
24(Q) = Ll(daU+dixU) 2 + L2(d~U+d"U)(dixU+d~U) + L 3 ( d txU +daUd ~U +d,U)+La(dixU+d~,U)(x+U+x ~+) m
m
"~ L s ( MIXU+MaU(x + U "4- U+X ) ) .-~ L62 + Ls(X + UX+ U + xU+xU +) _iL9(dix U
d v U + G~+dixU+d"U R
L
241
R. Urech/ Nuclear Physics B433 (1995) 234-254
+ L,o + HI(GR""G#. + GL""G#.> + Hz + KIF~ + K2F~
+
+
Q.+>)
+ K4F~
+ KsF2<(d.~+duU + d~.U d.U + )Q2) + K6F~< d"U+d.U QU+QU + d~'U d~,U+QUQU+> + KvFZ(x +U + U+x)(Q2> + KsFZ + K9Fg((x+U + U +X +X ~+ + UX +)02> + KaoFoZ<(x+U+ U+x)QU+QO + (xU++ Ux+)QUQU +) + K11F~( ( x+ U - U+x )QU+ QU + ( xU +- Ux +)QUQU +) + K 1 2 F 2 < d " U + [ c 2 Q , QI ~ + d . U [ c L Q ,
QIu+>
2 R~ QUc~QU -- L --+ )+Ka4F6(c 2 RIz Qc~Q+ R cLla.OcLO) +KI3FO(c ~.. ~ .
+ K15F4(QUQU+) 2 + K16F4(QUQU+)(Q 2) + KaTF~(Q2) 2,
(31)
where G~. and G~; arc the field strength tensors of G~, G#, respectively,
G~v=OuGI-O.G~-i[G~, GI],
/=R,L.
(32)
If we put Q = 0, the terms with the couplings K i vanish and we are left with the contribution to .~4(Q) from the purely strong sector, calculated by Gasser and Leutwyler [2]. In (31) we have only considered contributions made out of the building blocks in Table 1. In particular, terms proportional to polynomials in X . are omitted. The coupling constants L i, H i and Ki are defined as
L i =FiA + L~(p,), H i =Ail~
-I-nir (].£),
(33)
K i = XiA + g [ ( ~ ) . A contains a pole in d = 4 dimensions, ].£d-4 ( A
16~ -2
1 d-
1 [in 47r + F , ( I ) + 1]) 4
2
(34)
242
R. Urech /Nuclear Physics B433 (1995) 234-254
The coefficients ~DAi and the finite parts L~(~) are listed in Ref. [2], whereas H~r and H~ are of no physical significance. The coefficients Si are Z1
3
Z4 : 2 Z ,
~2 = Z,
~3
3
Z5 = -- 4} 9
~ 6 = ~3Z , 4~
1
3
1
--
3
Z14 = 0,
"~13 = 0, "~16 :
1
3
3 - ~Z - 4 Z 2,
~17
--
3
~-
Zl5 = ~ + 3 Z + 20Z 2, 3
~Z + 2 Z 2 ,
(35) with
Z = C / F 4.
(36)
We choose the renormalized coefficients Kr(tz) in such a way that the K i themselves do not depend on the scale Iz,
d tz g K i u/z
/£d-4 d =~iT7"-7 + tz - K r IO'Wd#
'(
Iz) + O ( d - 4) = 0.
(37)
3. Corrections to Dashen's theorem
This chapter is separated in four subsections. In the first subsection we calculate the formal expression for the corrections to Dashen's theorem at order e2mq, i.e. the squared mass difference ( M ~ ± - m ~ 0 ) - ( M 2 ± 2 In the second subsecMe0). tion a numerical estimate is given with an u p p e r limit on the unknown terms. In the third subsection we give an independent approach on the basis of Ref. [7]. Finally, we compare in the fourth subsection the obtained values with the results in the literature.
3.1. Masses at order e2mq In the chiral limit, the masses of the pions and the kaons are of purely electromagnetic nature. A relation between the squared masses at order e 2 is given by Dashen's theorem [8],
( M2± - M :2 ° ) . . . . m- ( M 2 ± - M K 2 ° ) e . m . '
(38)
with C M 2 ± = M K 2 ± = 2 e 2 ~ o 2,
Mffo=MK2o=0,
formq=0.
(39)
At lowest order in the quark mass expansion the squared masses of the pseu-
R. Urech/ NuclearPhysicsB433 (1995)234-254
243
doscalar mesons are denoted by ~ 2 . We neglect the mass difference m a - m u and replace m u, m a by rh = 5(m 1 u + md). If we expand the lagrangian ~ 2 (Q) in the parametrization (2) and use the representation (3) of the physical fields, the squared masses are C M,,_+ ^ 2 = 2e 2 ~ 02 + 2rhB0, ]1,)2o= ~ 2 = 2rhBo, C /~2± = 2e2 _70 2 + (rh + ms)Bo,
(40)
34~,, = M~ = ( rh + ms) B0,
= 2 ( a + 2ms)B0. This decomposition depends on the convention adopted for the quark self-energies. The effect of this ambiguity is small and neglected in this article. The correction to Dashen's theorem at order is denoted by
e2mq 2 AM2-AMZ=(MZ±-MZo)-(M2±-M~o),
for m u = m d = f f / .
(41)
We calculate the Fourier transform of the two-point function of the mesons to one loop, keeping the fields associated with the Gell-Mann matrices ~a instead of the physical ones. For the charged fields it has a cut at p 2 = M d,
if d4x eipx(o[
Tq)a(X)q~a(O) exp(ifd4y(
~(Q)intk Y)+~z~4(Q)(Y)))
[0>1 one loop
Za(M 2 )
_pa/M2),+e2L¢p2)+
= M2( 1
....
a = 1, 2, 4, 5,
(42)
with 1 fa(p2)
= ~8~V
(
M2 ) 1+ 7 '
(43)
whereas for the neutral fields it contains a pole,
if d4x eip'x(01Tga(X)9,(O)exp(if d4y(.2~'z(ie:t (y)+.f'4(O)(y)))10)[ one loop zo(Md) -
M2_p~
+ ....
a = 3, 6, 7, 8.
(44)
Za(M2)
7.a(M2)
~(Q)2 i.t represents the interaction part of the lagrangian G<(Q),2 and are due to the field renormalization and the ellipses denote terms which are not relevant for the mass-shifts at order The contributions to the two-point function are shown in Fig. 1: (b) and (c) are one-photon loops. The first con-
e2mq.
R. Urech/ Nuclear PhysicsB433 (1995) 234-254
244
(a)
(b)
©
© (d)
(c)
m m
(e) Fig. 1. The five diagrams represent the contributions to the two-point function of the pseudoscalar mesons to one loop. The solid lines denote mesons, the external ones carry momentum p. The wavy lines are photons. Graph (b) contributes only to the propagator of the charged fields and (c) vanishes in dimensional regularization. Diagram (e) represents the contribution from the effective lagrangian of order p4. tributes to the two-point function of the charged fields only, the latter vanishes in dimensional regularization. Graph (d) is the tadpole with a four-meson coupling and (e) represents the contributions from the lagrangian .~4~°). The results for the different mesons are explicitly given in Appendix B. Here we consider only the difference A M 2 - A M 2,
AM~-AM~=-e
21(
~
M2 3M2K ln--~-3 M--~ K
eaM~ I n ¢ 8~-z F 4
M2 ln---L M ~- 4) ( M p 2 f- l
-M~
)
2 ln--~- + 1
C - 16e2--~oLrs(~)(M ~ - M 2) + R ~ ( I ~ ) M 2 + RK(P.)MK2
+ O ( e 2 m 2),
(45)
where L~(/x) and R,,,K are the contributions from .~4(o), 2 2 R,~ = 5e ( 6 K 3r - 3K~ + 2K~ - 10K[0 - 12K[1 ) , - r + g ~ _ 6 K [ o _ 6K~I). R K = - - g4e 2/t . ~ k 5
(46)
R. Urech/ Nuclear PhysicsB433 (1995) 234-254
245
At the end we need L} and seven coupling constants K/r from .9~4{°) to determine the correction to Dashen's theorem at order e2mq. 3.2. Numerical results We put F 0 equal to the physical pion decay constant, F~ = 93.3 MeV and the masses of the mesons to a/l= = 135 MeV, MK = 495 MeV. C can be expressed as an integral over the difference of the vector and axial vector spectral functions as established by Das, Guralnik, Mathur, Low and Young [9]. If we consider the low-lying vector and axial vector mesons p and a~, respectively, and use the Weinberg sum rules [10] to eliminate the parameters of the al, we arrive at [11] 3
C-
F2
32rr2M2F2 lnF2_----~2 ,
(47)
where Mp is the mass of the p-meson, Mp = 770 MeV, and Fp denotes the p decay constant, Fp = 154 MeV [5]. Therefore C = 61.1 x 10 -6 ( G e V ) 4.
(48)
At tree level, the difference of the squared pion masses is k 3 1 2 = ( h 4 + + h4~rO)(~t~r±-- M~0)= 2M~ × 4.8 MeV,
(49)
which is very close to the experimental value, (M,~ ±-M~.o)exp. = 4.6 MeV [12]. The spectral functions can also be extracted from data of the r-decay [13]. However, this method contains uncertainties because the spectral functions can only be evaluated for momenta p2 ~< 2 (GeV) 2 and therefore we keep our approach above. L~(/z) is taken from Table 1 of Ref. [5]. Its values at the scale points I* = (0.5 GeV; Mo; 1 GeV) are L~(/,) = (2.4; 1.4; 0.8) _+ 0.5 × 10 -3.
(50)
These values lead to the following result for the squared mass differences, using again the three scales/z = (0.5 GeV; Mp; 1 GeV): A M 2 - A M 2 = ( - 0 . 2 6 ; 0.52; 0.99) _+ 0.13 × 1 0 - 3 ( G e V ) 2 + R , M 2 + R K M 2.
(51)
There is a strong dependence on the scale ~ for all the terms. Of course, this scale-dependence cancels in the full expression. Next we calculate the ratio aM~/aM~.For this purpose we put AM 2 = (M~± and arrive at
-M~0)exp.
AM 2 AM 2
(0.79; 1.42; 1.80) + 0.11 +
R ~ M ~ + R EM2K AM 2
(52)
246
R. Urech /Nuclear Physics B433 (1995) 234-254
In AM 2 we can estimate the coupling constant K~(#) if we neglect the unknown contributions from ~4 ~°) proportional to M 2 (see Appendix B),
AM~= 2e2~02 - e 2 1 6 @ M 2 ( 3 l n - ~ - 4 ) -e 2 - M~ 31n ~" + 1 8,n-2 F 4 ~-
+ M K2 In /./2 ]
- 1 6 e 2 C [(M 2 + 2M2K)Lr4 + M2usl + 8egM2KK~ + O(e2M~).
(53) The values of L](/~) are at the three scalepoints t~ = (0.5 GeV; Mp; 1 GeV), L~(p~) = ( 0 . 1 ; - 0 . 3 ; - 0.5) _+ 0.5 × 10 -3.
(54)
We put AM 2 equal to the observed value and K~(/z) becomes K~(/x) = - ( 1 . 0 ; 4.0; 5.6) + 1.7 × 10 -3 .
(55)
K~ is of the same order as the coefficients L~ [2] and as it was expected by the so-called naive chiral power counting [14]. Note that K~(/x) does not have the usual scale-dependence implied by (37), because we have dropped scale-dependent terms in (53). Now we assume that the K[ which contribute to R~ and R K have the upper limit 1
I gil <.5 167r2 = 6.3 × 10 -3.
(56)
The contributions from R~.M2 + RK M2 to the squared mass difference AM 2 -
AM~ is smaller than I R~M~ + R K M ~ I ~<2.6 × 10 -3 (GeV) 2,
(57)
which is a large number. Similarly the contribution to the mass ratio AM2K/AM~ is
]R,.M 2 + RK M2 ] AM~ ~ 2.1.
(58)
On the basis of (56), large corrections to Dashen's theorem can therefore not be excluded.
3.3. Independent approach We give a further estimate by looking at the QCD mass difference of the kaon. In this calculation we put m a - m , 4= 0, but neglect corrections of order e2(md -mu). The observed squared mass difference of the kaon can be divided into two parts, (M~+ - MK2O)ex,. = (M2+ - MK2,,)OC.D + (M~+ - mK2o)e.m:
(59)
R. Urech / Nuclear Physics B433 (1995) 234-254
247
The same is true for the pions, but the QCD term is of O[(m d --mu)2], which is very small, numerically (M=±-M~0)Qc D = 0.17 MeV [2]. We put therefore 2 2 (M2± - M~o)~xp . _- (M~± - MZ0)e.m..
(60)
The electromagnetic part of the squared mass difference using the notation of (41) becomes
A M 2 - AMZ= ( A M Z - AMZ)exp.-(AMZ)QcD + O(eZm2).
(61)
The experimental part is ( A M ~ - AM2)e×p = - 5 . 2 5 × 10 -3 (GeV)2112!. We refer to Leutwyler [7] for the calculation of (AM~)oc D without using Dashen s theorem, (AMK2)QCD = - ( M 2 - M 2 " ~ m~a-m--u [1 + A M + O ( m 2 ) ] •'1 m s _ r h
(62)
R = (m s - r h ) / ( m d - m u) is the ratio determined from the mass splittings in the baryon octet and from p-w mixing [11] with the value R = 43.7 + 2.7. One way to obtain the correction A M is an estimate of ~--q' mixing. The phenomenological information [2,15] indicates that the mixing angle is somewhere between 20 ° and 25 °. The range [0,,, [ = 22 ° _+ 4 ° corresponds to a value of A M = 0.0 + 0.12. Inserting these uncertainties into the equation above, the QCD squared mass difference of the kaon becomes (AM~)QC D = - (5.19 _+ 1.01) x 10 -3 ( G e V ) 2,
(63)
and the electromagnetic part of the squared mass difference is therefore (we neglect the errors in the experimental values)
A M 2 - A M 2 = - ( 0 . 0 6 _+ 1.01) × 10 -3 ( G e V ) 2
(64)
with a large uncertainty for a small value. Similarly the mass ratio A M Z / A M ~ is
~M~
- 0.95 _+ 0.81.
(65)
3. 4. Comparison with other results Maltman and Kotchan [16] used the effective lagrangian .~2(Q) and calculated the one-loop contributions to the squared mass differences AM~2 K" They considered terms proportional to e 2 M~2 ln(M2//z 2) and e2M 2 ln(M2/)~ 2) at the scale point /z = 1.1 GeV. The scale-independent terms e 2 M~2 and eZMZK as well as the contributions from the effective lagrangian of order p4 were neglected. Their result is
( A M 2 -AM~)log 2 = 0.76 × 10 -3 ( G e V ) 2,
(66)
R. Urech / Nuclear Physics B433 (1995) 234-254
248
which agrees with the contribution from the logarithms in our calculation at /z = 1.1 GeV, ( A M 2 - AM~)log = 0.77 X 10 -3 (GeV) 2.
(67)
The authors expect that the ratio A M K2/ A M ~2 is in the range (at this scale point) AM~ - 1.44 + 0.20.
(68)
Using again AM 2 = (AM2)exp, this corresponds to A M 2 - AMZ~ = (0.55 + 0.25) × 10 -3 ( G e V ) 2.
(69)
Recently two other estimates have been made by Donoghue, Holstein and Wyler [17] and Bijnens [18]. The first work is based on chiral symmetry and vector meson dominance. Their result shows a total ratio which is rather large, AM 2
aM¢
= 1.8.
(70)
The value for AM~z they obtain at this order is about 20% higher than the experimental value, M~Z± - M~0 = 2M,~ × 5.6 MeV.
(71)
Inserting (71) into the ratio above (70), the electromagnetic part of the squared mass difference becomes A M 2 - A M ~ = 1.23 X 10 -3 ( G e V ) 2.
(72)
This result is in good correspondence with the one obtained by Bijnens [18], A M 2 - AM~ = (1.3 + 0.4) × 10 -3 ( G e V ) 2,
(73)
who calculated this value by splitting the contributions to the masses into two parts. A long distance part is estimated using the (1/Nc)-approach and a short distance contribution is determined in terms of the coupling constants L~ of the lagrangian .Z~4~a~. The results we listed in this section show rather large corrections to Dashen's theorem. They coincide only partly with the result we obtained in (64), but they could be in agreement with the estimate on the basis of (56). In order to establish them, we would have to numerically evaluate the relevant coefficients K r of ~4(Q).
4. Summary
1. We have evaluated the divergent part of the generating functional of C H P T including virtual photons to one loop. We have calculated the structure of the local action at order p4 with 29 coupling constants ( L 1. . . . . L10; H1, Ha; K a. . . . . K17), which have in general a single pole in d = 4 dimensions.
R. Urech/ Nuclear Physics B433 (1995) 234-254
249
2. In a next step we have calculated the one-loop contribution to the meson masses in the isospin limit m u = m d = rh and extracted the correction to Dashen's theorem [8],
AM~-AM~=-e
1 (
e-167r2 3M~ In
M_~ 2
-3M~
M2_4(M~_M~))
ln--~-T
1 C[ M_~ ( M ~ ) ] -e:---MK2 In -M 2 21n--~-+l 8~- 2 F 4 C
- 16e2-F-ff~Lrs(tz)(M2 - M 2) + R~(tz)M2~ +RK(tz)M 2 + O(e2m2q),
(74)
where AM 2 = M2± -M~o with P the pseudoscalar meson in question. L~(tx) and R~, K are the contributions from 2 4 (Q), 2
2
r
R,~ = g e ( 6 K 3 - 3 K ~ + 2 K ~ - 1 0 K { 0 -
12K{1),
Rs: = - 4 e e ( K ~ + K~ - 6K{0 - 6K{,).
(75)
At the three scalepoints /x = (0.5; 0.77; 1) GeV, we got numerically
AM 2 - AM 2 = ( - 0 . 2 6 ; 0.49; 0.99) + 0.13 × 10 -3 (GeV) 2 + R~M2~ + RK M2. We assumed the upper limit of R~,M~2+ RKMK2 to be [R~M 2 +RKM 2 [ < 2.6 × 10 -3 (GeV) 2,
(76)
(77)
from which we cannot exclude large corrections to Dashen's theorem. Note that in the numerical part in (76) the correction due to L~(/x) is already included. 3. In addition, we have given an estimate of AM~ - AM 2 on the basis of Ref. [7],
A M ~ - AM~ = - ( 0 . 0 6 _+ 1.01) × 10
3
(GeV)2,
(78)
a small number with a large uncertainty. 4.. We have compared our values to the results in the literature [16-18]. We found that they coincide only partly with the result (78), but they could be well in agreement with the estimate in (76). In order to complete the expression for AMZK AM 2 we need the numerical evaluation of R~, K, i.e. the calculation of the finite parts of the coupling c o n s t a n t s K i. This is beyond the scope of this work. -
-
Acknowledgements
I am grateful to Jiirg Gasser for many discussions and for reading this manuscript carefully. Also I thank H. Neufeld, who pointed out that my original lagrangian
R. Urech /Nuclear Physics B433 (1995) 234-254
250
,.~(Q) contains an overcomplete set of counterterms, J. Bijnens for correspondence concerning effective lagrangians, Urs Biirgi, who helped to draw the Feynman graphs, and Robert Baur for pointing out a numerical error in the coefficients Zi, that he had worked out independently.
Note added in proof One of the work announced in Ref. [4] has been published [19].
Appendix A Matrix relations
For the derivation of the effective lagrangian .Z4(Q) at order p4, we used the equation of motion obeyed by U, C d~d~U+ U - U+d~d~U + ( U+ x - x + U ) + 4-~o2 ( U+ Q U Q - QU+ Q U ) -½(U+x-x+U)
=0.
(79)
The Cayley-Hamilton theorem for a 3 × 3-matrix A, A 3 - ( A ) A 2 + ½ ( ( A ) 2 - ( A Z ) ) A - ~ ( ( A ) 3 - 3 ( A ) ( A 2) + 2(A3))1 = 0,
(80) implies a trace identity for 3 × 3-matrices A, B, C and D, (ABCD) = -(ABDC) - (ACBD) - (ACDB) - (ADBC) - (ADCB) + (AB)(CD) + (AC)(BD) + (AD)(BC) + (A)
(81)
Furthermore we used the observation that Q2 can be written as Q2
2 2 = s1e Q + ~e 1.
(82)
We confirmed the 17 independent terms K i < . . . ) (i = 1,2 . . . . . 17) in _~4(Q) calculating the contributions from tree graphs at order p4 to • the constant term e41, 2 2 • the masses M~_+, M,~0 and M ,2, • the scattering amplitudes at order e2mq, 7/-+,W---> ,W+,TT ,
"n'+'tr'--+ K ° K °, K + K - - o K0~o,
(83)
251
R. Urech/Nuclear Physics B433 (1995) 234-254
• the scattering amplitudes at order e 4, ,/7.+3-/-- ~
,/7"+77"-
(84)
7r+Tr-~ K°g, °, • the matrix elements
(Tr + l~3,sdl~-+rr-), (O[~yjsd[zr-) ,
(~-+ 1~3,5d1~7~7),
(OlTFt(x)y~ysd(x) d(y)y~ysu(y)]0),
(zr + [ T F t ( x ) y ~ d ( x ) d ( y ) y ~ u ( y ) [7r+),
(85)
( O [ T ~ ( x ) y U d ( x ) d ( y ) y u u ( y ) [0).
Appendix B
The formal expressions for the masses at order e2m q
The photon loop (see Fig. 1 in the text) gives a contribution to the masses of the charged particles, 8M~± I,-loop = - e 2
Mc2 3 In--~- - 4
,
(86)
where C stands for ~ and K, respectively. From the tadpole, the masses of the charged particles get a change of the form 8M2± ]tadpole = --e2
A c ± M 2 In
~ +Bc+_M 2 In
7-
K
(87)
'
with (A=+, B=+) = (2, 1) and (AK±, BK±) = (1, 2). The contributions to the masses of ~.0 and rt are 1
C
a v M 2 In 6M 2 I tadpole = e 2 - 48~- 2 F 4
+ 1 + f l e M 2 "~p l n - ~ - + 1
,
(881 with the coefficients (a,~o,/3~o, y~0) = (6, 0, 0) and (a n,/3,, ~/,) = ( - 2 , 1, 4). The masses of the neutral kaons do not get contributions from the loops. At order e2mq, the mass-shift due to .9'4(Q) involves in general only terms with the couplings K s, but for the charged particles, where L 4 and L 5 enter as well, 6M2c + _ [ _~te) = - 1 6 e 2 C [(M 2 + 2 M 2 ) L ] ( / x ) + MZL~(/x)] + . . . .
F0
(89)
R. Urech /Nuclear Physics B433 (1995) 234-254
252
The term proportional to L](/,) drops out in the difference A M ~ - AM~. Explicitly, the masses are, at O(eZmq), =M~_+
16~.2
ln/x"
=
× 2M 2 ln-~5- + M~ ln--~-
4
- 2 F--4 8rr
- 16e2
Fo
,M~ + 2M~)L] + M2Lr5]
4 2 2 - ~e M~(6K 1r + 6K~ + SK i + 5K~ - 6K~ - lSK~ - 5K~ -- 23K~o
-
18K[1)
--1-
8e2M~ K~,
(90)
1 C ( M2 ) 2 =~2o +e 2 M~o 8~r2 F4M~2 l n - - ~ - + l -
2 A2 a a r 2 /
12K[ + a2K~- lSK~
+ 9 K ~ + 1 0 K j + a0Kg - aZK~ - l Z K j - 10K~ - 10K~o),
M~±=~2±-e
2
M 231n-~-4
X M2 l n ~ - + 2 M
)21 -e
(91)
87r2F4
21n
C 2 r 42 2 r r - 16e 2 ~404 [(M2 + 2M2)Lr4 + MKLs] + ~e M~(3K 8 + K9 + K[o )
-
-
4 2 2 ~e MK(6K 1r + 6K~ + 5K[ + 5K~ - 6K-~ - 24K~ - 2K~
- 20K~o - 1 8 K ~ l ) ,
(92)
M2o =M2o = ~ , ~ o -- ~~e2 ~M ~ ( 3 K 1 r + 3 K ~ + g ~ + / ' ; 6 r - 3 K ~ - 3 K j - K 9 - K ~ o ) ,
Mn2--M,~^2-e248~r ~1
[
F4C 2M~ In
+1
(93)
-M 241n¢+1
4 2 2 r 2 2 2 1r + 12K~ - 6K~ + 3 K j + 6 K ~ +-~e Mz~( g9r + Klo) - -~eMn(12K
+6K~ - 12K~ - 12K~ - 4K~ - 4K[o ) . The squared mass differences of the pions and the kaons are
AM2=M2±-M2o = 2e2~d-e21~-~M2(3 ln~--4) e2M2 3 1 n ~ - + l 8,n-2 F 4
+ M ~ In
(94)
R. Urech/ Nuclear Physics B433 (1995) 234-254
- 16e2--~C4[(M2 + 2M2)U4 + M2Lrs] Fo +2e2M2(-2Kj+K~+2K~+aK[o+nK[I) AM~=M2+-M~o
+8e2M~K~,
253
(95)
2C M2 =2e F°2 - e21~-~M~(3 l n ¢ - 4 )
1
_e 2 8~.2F4 __ M~2 In ~/x+2 2 M
21n¢
- 16e2~ [(Mg + 2M~)L: + M~L~] a 2M~(3Ks 2 r +K9r + K ~ o +~e 4 _2aar2/
Tlr
- ~ e ~V, K t ~x 5 + K ~ - 6 K ~
) - 6K~o -
6K~1 ) .
(96)
And finally the difference AM2 -AM 2 is A M ~ - A M 2 = - e 2-167r2 3M2 In
- 3 M 2 ln--~--4(M2-M 2)
1 C[ M2 -e 2-M2 l n ¢ - M 8~ 2 F g
( M~ )] 2 21n--~-+l
C - 16e2~oLrs(tz)(M2K - M 2) + R~(tz)M~ + RK(/Z)M~ , (97) where Rrr,K
are,
2 2 r R~= ~e (6K 33K~ + 2K~- 10Kfo- 12gfl), 4 2 1 r-,-r RK = --3e tas +K~- 6K~0- 6K[~).
(98)
References
[1] [2] [3] [4]
S. Weinberg, Physica A 96 (1979) 327. J. Gasser and H. Leutwyler, Nucl. Phys. B 250 (1985) 465. J. Kambor, J. Missimer and D. Wyler, Nucl. Phys. B 346 (1990) 17. H. Leutwyler, Rapporteur talk given at the XXV1 Int. Conf. on High energy physics, Dallas 1992, ed. J.R. Sanford, AIP Conf. Proc. No. 272 (AIP, New York, 1993); University of Bern preprint BUTP-93/24, Ann. Phys. (NY), to appear; U.-G. Meil3ner, Rep. Prog. Phys. 56 (1993) 903; A. Pich, Lectures given at the V Mexican School of Particles and fields, Guanajuato, M6xico, December 1992, CERN preprint TH 6978/93;
254
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16] [17] [18] [19]
R. Urech /Nuclear Physics B433 (1995) 234-254 G. Ecker, Lectures given at the 6th Indian-Summer School on Intermediate energy physics interaction in hadronic systems, Prague, August 1993, University of Vienna preprint UWThPh1993-31, Proc. (Czech. J. Phys), to appear; G. Ecker, J. Gasser, A. Pich and E. de Rafael, Nucl. Phys. B 321 (1989) 311. J. Gasser and H. Leutwyler, Ann. Phys. (NY) 158 (1984) 142. H. Leutwyler, Nucl. Phys. B 337 (1990) 108. R. Dashen, Phys. Rev. 183 (1969) 1245. T. Das, G.S. Guralnik, V.S. Mathur, F.E. Low and J.E. Young, Phys. Rev. Lett. 18 (1967) 759. S. Weinberg, Phys. Rev. Lett. 18 (1967) 507. J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 77. Particle Data Group, K. Hikasa et al., Review of particle properties, Phys. Rev. D 45 (1992) S1. R.D. Peccei and J. So15, Nucl. Phys. B 281 (1987) 1. A. Manohar and H. Georgi, Nucl. Phys. B 234 (1984) 189. J.F. Donoghue, B.R. Holstein and Y.-C.R. Lin, Phys. Rev. Lett. 55 (1985) 2766; F.J. Gilman and R. Kauffman, Phys. Rev. D 36 (1987) 2761; Riazuddin and Fayyazuddin, Phys. Rev. D 37 (1988) 149. K. Maltman and D. Kotchan, Mod. Phys. Lett. A 5 (1990) 2457. J.F. Donoghue, B.R. Holstein and D. Wyler, Phys. Rev. D 47 (1993) 2089. J. Bijnens, Phys. Lett. B 306 (1993) 343. H. Leutwyler, Ann. Phys. (NY) 235 (1994) 3033.
Nuclear Physics B433 (1995) 234-254
Virtual photons in chiral perturbation theory * Res Urech 1 Institute for Theoretical Physics, University of Bern, Sidlerstr. 5, CH-3012 Bern, Switzerland Received 24 May 1994; accepted 11 August 1994
Abstract
In the framework of chiral perturbation theory virtual photons are included. We calculate the divergences of the generating functional to one loop and determine the structure of the local action that incorporates the counterterms which cancel the divergences. As an application we discuss the corrections to Dashen's theorem at o r d e r eemq.
1. Introduction In the chiral limit where the light up, down and strange quark masses go to zero, the Q C D lagrangian has a SU(3) R × SU(3) L chiral symmetry that is spontaneously broken by the ground state of the theory. T h e r e are eight Goldstone bosons: the 7r's, K's and r/. Their interactions are described by an effective low energy theory, called chiral perturbation theory (CHPT). At low m o m e n t a the chiral lagrangian can be expanded in derivatives of the Goldstone fields and in the masses of the three light quarks. C H P T is a nonrenormalizable theory: Loops produce ultraviolet divergences, which can be absorbed by introducing counterterms. The finite parts of the coupling constants of this counterterm lagrangian are not determined by the lagrangian that generated the loops. The complete information about them is hidden in the Q C D lagrangian, but there is no known way to extract this information from first principle alone. The couplings have to be evaluated from experiments. As shown by Weinberg [1], the loops are suppressed: Every loop and
* Work supported in part by Schweizerischer Nationalfonds. 1 New address since August 1994: Institut fiir Theoretische Teilchenphysik, Universit~it Karlsruhe, Kaiserstr. 12, D-76128 Karlsruhe, Deutschland; e-mail: [email protected]. 0550-3213/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 5 5 0 - 3 2 1 3 0 0 3 4 3 - 2
R. Urech / Nuclear Physics B433 (1995) 234-254
235
the associated counterterm correspond to successively higher powers of momenta or quark masses. At low energies, the contributions from higher loops are small. C H P T is an effective theory: It contains all terms allowed by the symmetry of the QCD lagrangian in the chiral limit. The local action is generated by the effective lagrangian, which is in general not determined by the counterterms alone at higher orders in the expansion of momenta and quark masses. Additional terms with finite couplings have to be included. The generating functional to one loop and the corresponding local action in the strong sector has been determined by Gasser and Leutwyler [2]. An extension to the AS = 1 nonleptonic weak interactions to one loop has been given by Kambor, Missimer and Wyler [3]. Here we consider virtual photons and include them in the mesonic sector. We calculate the divergences of the one-loop functional in the presence of external currents and determine the structure of the local action at this order. The article is organized as follows. In Section 2 virtual photons are included in C H P T with three flavours. We evaluate the divergences of the generating functional to one loop and determine the structure of the local action at order p4. In Section 3 we discuss the corrections to Dashen's theorem at order eZmq and compare our estimates with the results in the literature. In Section 4 we summarize our results, and in Appendix A the matrix relations are given that are used to simplify the effective lagrangian at order p4. Finally, in Appendix B we list the renormalized masses of the pseudoscalar mesons at order e2mq.
2. Generating functional We suppose that the reader is familiar with CHPT, otherwise we refer her or him to comprehensive reviews [4]. At leading order, the effective lagrangian of the strong and electromagnetic interactions which respects the chiral symmetry SU(3)R × SU(3)c, P and C invariance is in the mesonic sector [5] =
-
-
½*G
)2
1 2 t~ +~FO(d U + d~U+xU++x+U)+C(QUQU+),
(1)
where F,v is the field strength tensor of the photon field A , , F,~ = OuA~ - O A,. A is the gauge fixing parameter and will from now on be kept at A = 1. U is an unitary 3 × 3-matrix and incorporates the fields of the eight pseudoscalar mesons,
UU += 1,
det U = 1, 8
U=exp
Fo]
q~=
a=lEAa@a'
(2)
R. Urech /Nuclear Physics B433 (1995) 234-254
236
where the Aa'S are the Gell-Mann matrices, and 7/-o
"08
77-+
7r° ___+
Tr
K +
r/s
__
K 0
Ro
(3)
2r/s
K_
d~,U is a covariant derivative, incorporating the couplings to the photon field A~,, the external vector and axial vector currents vu and a~,, respectively, duU = O~U - i(v~, + QA u + a~.)U + iU(v. + Q A . - au),
(4)
where Q is the charge matrix of the three light quarks, e( 2 Q= 5
) - 1
-1
e(
1
)
= "~ A3 + -~-/~'8 "
(5)
We do not consider singlet vector and axial vector currents, and we put therefore tr v~, = tr a~, = 0. X denotes the coupling of the mesons to the scalar and pseudoscalar currents s and p, respectively, X = 2B0(s + ip),
(6)
where s incorporates the mass matrix of the quarks,
/mu )
s = ~ / + ... =
md
+ ....
(7)
ms
F 0 is the pion decay constant in the chiral limit, F~ = F0[1 + O ( m q ) ] . B 0 is related to the quark condensate (0l ~u 10)= -FZB0[1 + O(mq)], and C determines the purely electromagnetic part of the masses of the charged pions and kaons in the chiral limit, C M2± = M 2 ~ = 2e2~--~ + O ( m q ) .
(8)
To ensure the chiral SU(3) R X SU(3) L symmetry of the lagrangian 2 2~Q) we introduce local spurions QR, QL instead of the charge matrix Q. The lagrangian is modified, ~ 2 ( Q ) = __14"F /zv-p/.*v__ 1 ( 0 . A k ~ ) 2
+ 1 F o 2 ( d " U + d . U + x U + + x + U ) + C(QRUQLU+),
(9)
R. Urech/ Nuclear Physics B433 (1995) 234-254
237
and d~U is changed to
d~U = OuU - i(v~, + QRA~, + au)U+ iU(v~ + QLA~, - a~).
(10)
The rules of chiral transformation are [2,5]
giNg{, QI --+glQig~, I = R, L, v. + QRA~, + a~, --+gR(V. + QRA. + au)g ~ + igRO.g~, V~ + QLA~ - a~, ~gL(V~, + QLA. - au)g ~ + igeO~,g{, s +ip--*gR(s +ip)g ~,
(11)
gR,L ~ SU(3)R,L" Since the currents v~ and G, count as O(p), where p means the momentum of the external fields, the term QA~, has the same dimension. We put dim(QR,L) = O ( p ) ,
dim(A~) = O(1),
(12)
the lagrangian ..~2(Q) is therefore of order p2. The convention (12) has the advantage that electromagnetic interactions do not turn upside down the usual chiral counting. The procedure to obtain the generating functional at order O(p 4) is very similar to conventional CHPT. The one-loop graphs generated by .Z~2(Q) are of order O(p4). They contain divergences which are absorbed by adding tree graphs, evaluated with the lagrangian .9'4(Q) of order O(p4), see below. The generating functional becomes, up to and including terms of order O(p4),
exp[i.U(G,a~,,s,p)]=Nf[dU][dA,]exp(ifd4x(..9'(2°'+.~4(°))),
(13)
where the integration over the fields is carried out in the one-loop approximation. (Here and below we disregard contributions from the anomaly altogether.) To evaluate the divergent part of the one-loop functional, we transform the lagrangian 2 2 (°) (9) to euclidean spacetime, £~c~2(0)~.2"(Eo), and expand the fields U and A . around the classical solutions (U, A . ) of the equations of motion,
U=u exp{ Fo i~ }lu =u( =
i
+ =ue.
ro Z,u" =~¢~ ÷ Ep.,
-
1
+i
==ue2.
Z/~o-
Fo +
21;2o+''" u] ....
(14)
238
R. Urech / Nuclear Physics B433 (1995) 2 3 4 - 2 5 4
where we put 0 = u e, and ~ is a traceless hermitian matrix. We insert the expansion (14) in the action 5PE,
,_9"E= fd4XE C.~CEO"+ f d 4 x z ( ¼ ( D . f D ~ f - [ a . ,
f] [Au, sc] +o-~ 2)
1 C 8
F 2 ( [ H R + H L , sC][HR--HL, so]>
+ ½F0(~: [ MR, A/x ] -- H E Dp.~ )~-~ ..[_~Ep.( v l __0~, 0 "~- ~* t ~'2/" 0 l\ , ./42L))E/. ) + ... ,
(15)
where the ellipsis denotes higher order terms in the fluctuations ~ and q,. We used the expressions (see also Ref. [2])
C = ½[U+, ~ U ] __ I.~lU+GARu -1 +dud U + = A~ = ~u
~luG~,l"LU +,
1 --+ u, ~ud~U
oR = Vt* + QR"z~" + au'
G~L_-v~.+ O Y. H R =
(16)
a.,
u+{Q, U}u +=
I ~ + Q R u q-
UQL u+,
= ½(u+xu++ ux+u). Fu and A, are anti-hermitian matrices, whereas HR, L and o- are hermitian ones and fd4XE .~(EQ) represents the classical action of .9'~Q). The normal brackets [., • ] denote the commutator, the curly brackets {., • } stand for the anti-commutator. We use the parametrization sc = E,sc~A~, where the aa's are the Gell-Mann matrices. We define a new covariant derivative Xu =D~, + X,,, with ~('ap~p Xu~ a -_ D ~ a +..~ ~ , 2~ueo = OueP+ X~a~a '
(17)
where D ~ a is understood as
Du~ a = Ou~a + rfb~b
(18)
and F2b
=
X~p ~
:1< [ a a, ab]C>,
-
_
Xu;pa
~
_ ~Fo(HLA 1 a )8~. p
(19)
R. Urech / Nuclear Physics B433 (1995) 234-254
239
The action becomes at the one-loop level ~ E Ioneloop = ~1j fd4x El[ca(l, _ ~ , ~ b ~ a b
+%(-X.Z.
..}_o.ab)~b qt- ~a~12C.ix
+ p)%],
(20)
where now o.ab= _ _ I ( [ A
' /~a] [A/~, /~b]) ..[_ l(o.{/~a ' /~b})
C aF2([HR+HL,
Aa][HR_HL,
Ab])
1 2 ( H L Aa) ( H L A b )' ~F°
(21)
a 1 l~a "Y/.=Fo(([HR, Au] +$D~.HL) ), 3 2 2 p= gF6(H[).
We collect the fluctuations of the mesons and of the photon field and define a new flavour space r/A, where A runs from 1 to 12, r / = (scl . . . . . sc8, e ° . . . . . e3). The action above can now be written as a quadratic form, ~a E [ oneloop = where X~, and
lfd4xE ~A(--XtxXtzsAB -t-AAB)~ B,
A AB are
Z** = Oul +
(22)
12 × 12-matrices,
X2 b
= O~,l + g~,,
~Y A = 5Y 1 o-b p(~o
(23)
This leads to a gaussian integral for the generating functional, exp(--UE I one loop)=
Nfd~trl]
e x p [ - a ( r / , .@r/)]
= U ' ( d e t .@) -1/2, where we defined the differential operator ( f , g) = ~ f d 4 x z A
(24)
~AB = _~,,~6AB + AAB and
TAgA.
(25)
Omitting the constant contribution we arrive at "~E [ one loop = ~1 ln(det .~)
(26)
To renormalize the determinant we use dimensional regularization. In Minkowski spacetime, the one-loop functional in d = 4 dimensions is given by [6] 1
-~oneloop --
167r 2
1 d_4fd'x Tr(-~Y~vY"~+½A2)+finiteparts,
(27)
240
R. Urech /Nuclear Physics B433 (1995) 234-254
Table 1 P and C transformation properties for the fields, the charges and for their derivatives. The definition of cRQR and C~, LQL is given in the text. Spacetime arguments are suppressed P
C
U
U+
UT
d~,U
d~'U +
(d~U)T
X
X+
XT
OR
c~_mRQR
QL cLgQL
QTL (CuLQL)T
QL
OR
cLQL
cRgQR
QT (CgRQR)T
where Tr means the trace in the flavour space ~/A and Ya~ denotes the field strength tensor of Ya,
Ya.=OaY~-O~ra + [Yix, r.].
(28)
In Table 1 we list P and C transformation properties for the fields and quantities used in the one-loop functional. cix RQR and caLQL mean covariant derivatives of QR and QL, respectively,
cixIQI --°aQ, - i[GI, QI] , I = R, L,
(29)
that transform under SU(3)R x SU(3) e in the same way as QR and QL, c' O,
gicixQlgi, ' +
I= R, L,
gR,L ~ SU(3)R,L.
(30)
The divergences in -~one loop can be absorbed by adding the effective lagrangian .~4(Q) that contains all terms of order O(p 4) allowed by the chiral symmetry, P and C invariance. Since in the usual physical picture the charge matrix is a constant and has no chirality, we put again Q R = Q L = Q and OaQ=O. We keep the notation cR'LQ = -i[G R'L, Q] in order to remember the correct chiral transformation of c~Q I. By using the matrix relations that we list in Appendix A, the lagrangian .£~4(Q) can be simplified to
24(Q) = Ll(daU+dixU) 2 + L2(d~U+d"U)(dixU+d~U) + L 3 ( d txU +daUd ~U +d,U)+La(dixU+d~,U)(x+U+x ~+) m
m
"~ L s ( MIXU+MaU(x + U "4- U+X ) ) .-~ L6
d v U + G~+dixU+d"U R
L
241
R. Urech/ Nuclear Physics B433 (1995) 234-254
+ L,o
+
+
Q.+>)
+ K4F~
+ KsF2<(d.~+duU + d~.U d.U + )Q2) + K6F~< d"U+d.U QU+QU + d~'U d~,U+QUQU+> + KvFZ(x +U + U+x)(Q2> + KsFZ
QIu+>
2 R~ QUc~QU -- L --+ )+Ka4F6(c 2 RIz Qc~Q+ R cLla.OcLO) +KI3FO(c ~.. ~ .
+ K15F4(QUQU+) 2 + K16F4(QUQU+)(Q 2) + KaTF~(Q2) 2,
(31)
where G~. and G~; arc the field strength tensors of G~, G#, respectively,
G~v=OuGI-O.G~-i[G~, GI],
/=R,L.
(32)
If we put Q = 0, the terms with the couplings K i vanish and we are left with the contribution to .~4(Q) from the purely strong sector, calculated by Gasser and Leutwyler [2]. In (31) we have only considered contributions made out of the building blocks in Table 1. In particular, terms proportional to polynomials in X . are omitted. The coupling constants L i, H i and Ki are defined as
L i =FiA + L~(p,), H i =Ail~
-I-nir (].£),
(33)
K i = XiA + g [ ( ~ ) . A contains a pole in d = 4 dimensions, ].£d-4 ( A
16~ -2
1 d-
1 [in 47r + F , ( I ) + 1]) 4
2
(34)
242
R. Urech /Nuclear Physics B433 (1995) 234-254
The coefficients ~DAi and the finite parts L~(~) are listed in Ref. [2], whereas H~r and H~ are of no physical significance. The coefficients Si are Z1
3
Z4 : 2 Z ,
~2 = Z,
~3
3
Z5 = -- 4} 9
~ 6 = ~3Z , 4~
1
3
1
--
3
Z14 = 0,
"~13 = 0, "~16 :
1
3
3 - ~Z - 4 Z 2,
~17
--
3
~-
Zl5 = ~ + 3 Z + 20Z 2, 3
~Z + 2 Z 2 ,
(35) with
Z = C / F 4.
(36)
We choose the renormalized coefficients Kr(tz) in such a way that the K i themselves do not depend on the scale Iz,
d tz g K i u/z
/£d-4 d =~iT7"-7 + tz - K r IO'Wd#
'(
Iz) + O ( d - 4) = 0.
(37)
3. Corrections to Dashen's theorem
This chapter is separated in four subsections. In the first subsection we calculate the formal expression for the corrections to Dashen's theorem at order e2mq, i.e. the squared mass difference ( M ~ ± - m ~ 0 ) - ( M 2 ± 2 In the second subsecMe0). tion a numerical estimate is given with an u p p e r limit on the unknown terms. In the third subsection we give an independent approach on the basis of Ref. [7]. Finally, we compare in the fourth subsection the obtained values with the results in the literature.
3.1. Masses at order e2mq In the chiral limit, the masses of the pions and the kaons are of purely electromagnetic nature. A relation between the squared masses at order e 2 is given by Dashen's theorem [8],
( M2± - M :2 ° ) . . . . m- ( M 2 ± - M K 2 ° ) e . m . '
(38)
with C M 2 ± = M K 2 ± = 2 e 2 ~ o 2,
Mffo=MK2o=0,
formq=0.
(39)
At lowest order in the quark mass expansion the squared masses of the pseu-
R. Urech/ NuclearPhysicsB433 (1995)234-254
243
doscalar mesons are denoted by ~ 2 . We neglect the mass difference m a - m u and replace m u, m a by rh = 5(m 1 u + md). If we expand the lagrangian ~ 2 (Q) in the parametrization (2) and use the representation (3) of the physical fields, the squared masses are C M,,_+ ^ 2 = 2e 2 ~ 02 + 2rhB0, ]1,)2o= ~ 2 = 2rhBo, C /~2± = 2e2 _70 2 + (rh + ms)Bo,
(40)
34~,, = M~ = ( rh + ms) B0,
= 2 ( a + 2ms)B0. This decomposition depends on the convention adopted for the quark self-energies. The effect of this ambiguity is small and neglected in this article. The correction to Dashen's theorem at order is denoted by
e2mq 2 AM2-AMZ=(MZ±-MZo)-(M2±-M~o),
for m u = m d = f f / .
(41)
We calculate the Fourier transform of the two-point function of the mesons to one loop, keeping the fields associated with the Gell-Mann matrices ~a instead of the physical ones. For the charged fields it has a cut at p 2 = M d,
if d4x eipx(o[
Tq)a(X)q~a(O) exp(ifd4y(
~(Q)intk Y)+~z~4(Q)(Y)))
[0>1 one loop
Za(M 2 )
_pa/M2),+e2L¢p2)+
= M2( 1
....
a = 1, 2, 4, 5,
(42)
with 1 fa(p2)
= ~8~V
(
M2 ) 1+ 7 '
(43)
whereas for the neutral fields it contains a pole,
if d4x eip'x(01Tga(X)9,(O)exp(if d4y(.2~'z(ie:t (y)+.f'4(O)(y)))10)[ one loop zo(Md) -
M2_p~
+ ....
a = 3, 6, 7, 8.
(44)
Za(M2)
7.a(M2)
~(Q)2 i.t represents the interaction part of the lagrangian G<(Q),2 and are due to the field renormalization and the ellipses denote terms which are not relevant for the mass-shifts at order The contributions to the two-point function are shown in Fig. 1: (b) and (c) are one-photon loops. The first con-
e2mq.
R. Urech/ Nuclear PhysicsB433 (1995) 234-254
244
(a)
(b)
©
© (d)
(c)
m m
(e) Fig. 1. The five diagrams represent the contributions to the two-point function of the pseudoscalar mesons to one loop. The solid lines denote mesons, the external ones carry momentum p. The wavy lines are photons. Graph (b) contributes only to the propagator of the charged fields and (c) vanishes in dimensional regularization. Diagram (e) represents the contribution from the effective lagrangian of order p4. tributes to the two-point function of the charged fields only, the latter vanishes in dimensional regularization. Graph (d) is the tadpole with a four-meson coupling and (e) represents the contributions from the lagrangian .~4~°). The results for the different mesons are explicitly given in Appendix B. Here we consider only the difference A M 2 - A M 2,
AM~-AM~=-e
21(
~
M2 3M2K ln--~-3 M--~ K
eaM~ I n ¢ 8~-z F 4
M2 ln---L M ~- 4) ( M p 2 f- l
-M~
)
2 ln--~- + 1
C - 16e2--~oLrs(~)(M ~ - M 2) + R ~ ( I ~ ) M 2 + RK(P.)MK2
+ O ( e 2 m 2),
(45)
where L~(/x) and R,,,K are the contributions from .~4(o), 2 2 R,~ = 5e ( 6 K 3r - 3K~ + 2K~ - 10K[0 - 12K[1 ) , - r + g ~ _ 6 K [ o _ 6K~I). R K = - - g4e 2/t . ~ k 5
(46)
R. Urech/ Nuclear PhysicsB433 (1995) 234-254
245
At the end we need L} and seven coupling constants K/r from .9~4{°) to determine the correction to Dashen's theorem at order e2mq. 3.2. Numerical results We put F 0 equal to the physical pion decay constant, F~ = 93.3 MeV and the masses of the mesons to a/l= = 135 MeV, MK = 495 MeV. C can be expressed as an integral over the difference of the vector and axial vector spectral functions as established by Das, Guralnik, Mathur, Low and Young [9]. If we consider the low-lying vector and axial vector mesons p and a~, respectively, and use the Weinberg sum rules [10] to eliminate the parameters of the al, we arrive at [11] 3
C-
F2
32rr2M2F2 lnF2_----~2 ,
(47)
where Mp is the mass of the p-meson, Mp = 770 MeV, and Fp denotes the p decay constant, Fp = 154 MeV [5]. Therefore C = 61.1 x 10 -6 ( G e V ) 4.
(48)
At tree level, the difference of the squared pion masses is k 3 1 2 = ( h 4 + + h4~rO)(~t~r±-- M~0)= 2M~ × 4.8 MeV,
(49)
which is very close to the experimental value, (M,~ ±-M~.o)exp. = 4.6 MeV [12]. The spectral functions can also be extracted from data of the r-decay [13]. However, this method contains uncertainties because the spectral functions can only be evaluated for momenta p2 ~< 2 (GeV) 2 and therefore we keep our approach above. L~(/z) is taken from Table 1 of Ref. [5]. Its values at the scale points I* = (0.5 GeV; Mo; 1 GeV) are L~(/,) = (2.4; 1.4; 0.8) _+ 0.5 × 10 -3.
(50)
These values lead to the following result for the squared mass differences, using again the three scales/z = (0.5 GeV; Mp; 1 GeV): A M 2 - A M 2 = ( - 0 . 2 6 ; 0.52; 0.99) _+ 0.13 × 1 0 - 3 ( G e V ) 2 + R , M 2 + R K M 2.
(51)
There is a strong dependence on the scale ~ for all the terms. Of course, this scale-dependence cancels in the full expression. Next we calculate the ratio aM~/aM~.For this purpose we put AM 2 = (M~± and arrive at
-M~0)exp.
AM 2 AM 2
(0.79; 1.42; 1.80) + 0.11 +
R ~ M ~ + R EM2K AM 2
(52)
246
R. Urech /Nuclear Physics B433 (1995) 234-254
In AM 2 we can estimate the coupling constant K~(#) if we neglect the unknown contributions from ~4 ~°) proportional to M 2 (see Appendix B),
AM~= 2e2~02 - e 2 1 6 @ M 2 ( 3 l n - ~ - 4 ) -e 2 - M~ 31n ~" + 1 8,n-2 F 4 ~-
+ M K2 In /./2 ]
- 1 6 e 2 C [(M 2 + 2M2K)Lr4 + M2usl + 8egM2KK~ + O(e2M~).
(53) The values of L](/~) are at the three scalepoints t~ = (0.5 GeV; Mp; 1 GeV), L~(p~) = ( 0 . 1 ; - 0 . 3 ; - 0.5) _+ 0.5 × 10 -3.
(54)
We put AM 2 equal to the observed value and K~(/z) becomes K~(/x) = - ( 1 . 0 ; 4.0; 5.6) + 1.7 × 10 -3 .
(55)
K~ is of the same order as the coefficients L~ [2] and as it was expected by the so-called naive chiral power counting [14]. Note that K~(/x) does not have the usual scale-dependence implied by (37), because we have dropped scale-dependent terms in (53). Now we assume that the K[ which contribute to R~ and R K have the upper limit 1
I gil <.5 167r2 = 6.3 × 10 -3.
(56)
The contributions from R~.M2 + RK M2 to the squared mass difference AM 2 -
AM~ is smaller than I R~M~ + R K M ~ I ~<2.6 × 10 -3 (GeV) 2,
(57)
which is a large number. Similarly the contribution to the mass ratio AM2K/AM~ is
]R,.M 2 + RK M2 ] AM~ ~ 2.1.
(58)
On the basis of (56), large corrections to Dashen's theorem can therefore not be excluded.
3.3. Independent approach We give a further estimate by looking at the QCD mass difference of the kaon. In this calculation we put m a - m , 4= 0, but neglect corrections of order e2(md -mu). The observed squared mass difference of the kaon can be divided into two parts, (M~+ - MK2O)ex,. = (M2+ - MK2,,)OC.D + (M~+ - mK2o)e.m:
(59)
R. Urech / Nuclear Physics B433 (1995) 234-254
247
The same is true for the pions, but the QCD term is of O[(m d --mu)2], which is very small, numerically (M=±-M~0)Qc D = 0.17 MeV [2]. We put therefore 2 2 (M2± - M~o)~xp . _- (M~± - MZ0)e.m..
(60)
The electromagnetic part of the squared mass difference using the notation of (41) becomes
A M 2 - AMZ= ( A M Z - AMZ)exp.-(AMZ)QcD + O(eZm2).
(61)
The experimental part is ( A M ~ - AM2)e×p = - 5 . 2 5 × 10 -3 (GeV)2112!. We refer to Leutwyler [7] for the calculation of (AM~)oc D without using Dashen s theorem, (AMK2)QCD = - ( M 2 - M 2 " ~ m~a-m--u [1 + A M + O ( m 2 ) ] •'1 m s _ r h
(62)
R = (m s - r h ) / ( m d - m u) is the ratio determined from the mass splittings in the baryon octet and from p-w mixing [11] with the value R = 43.7 + 2.7. One way to obtain the correction A M is an estimate of ~--q' mixing. The phenomenological information [2,15] indicates that the mixing angle is somewhere between 20 ° and 25 °. The range [0,,, [ = 22 ° _+ 4 ° corresponds to a value of A M = 0.0 + 0.12. Inserting these uncertainties into the equation above, the QCD squared mass difference of the kaon becomes (AM~)QC D = - (5.19 _+ 1.01) x 10 -3 ( G e V ) 2,
(63)
and the electromagnetic part of the squared mass difference is therefore (we neglect the errors in the experimental values)
A M 2 - A M 2 = - ( 0 . 0 6 _+ 1.01) × 10 -3 ( G e V ) 2
(64)
with a large uncertainty for a small value. Similarly the mass ratio A M Z / A M ~ is
~M~
- 0.95 _+ 0.81.
(65)
3. 4. Comparison with other results Maltman and Kotchan [16] used the effective lagrangian .~2(Q) and calculated the one-loop contributions to the squared mass differences AM~2 K" They considered terms proportional to e 2 M~2 ln(M2//z 2) and e2M 2 ln(M2/)~ 2) at the scale point /z = 1.1 GeV. The scale-independent terms e 2 M~2 and eZMZK as well as the contributions from the effective lagrangian of order p4 were neglected. Their result is
( A M 2 -AM~)log 2 = 0.76 × 10 -3 ( G e V ) 2,
(66)
R. Urech / Nuclear Physics B433 (1995) 234-254
248
which agrees with the contribution from the logarithms in our calculation at /z = 1.1 GeV, ( A M 2 - AM~)log = 0.77 X 10 -3 (GeV) 2.
(67)
The authors expect that the ratio A M K2/ A M ~2 is in the range (at this scale point) AM~ - 1.44 + 0.20.
(68)
Using again AM 2 = (AM2)exp, this corresponds to A M 2 - AMZ~ = (0.55 + 0.25) × 10 -3 ( G e V ) 2.
(69)
Recently two other estimates have been made by Donoghue, Holstein and Wyler [17] and Bijnens [18]. The first work is based on chiral symmetry and vector meson dominance. Their result shows a total ratio which is rather large, AM 2
aM¢
= 1.8.
(70)
The value for AM~z they obtain at this order is about 20% higher than the experimental value, M~Z± - M~0 = 2M,~ × 5.6 MeV.
(71)
Inserting (71) into the ratio above (70), the electromagnetic part of the squared mass difference becomes A M 2 - A M ~ = 1.23 X 10 -3 ( G e V ) 2.
(72)
This result is in good correspondence with the one obtained by Bijnens [18], A M 2 - AM~ = (1.3 + 0.4) × 10 -3 ( G e V ) 2,
(73)
who calculated this value by splitting the contributions to the masses into two parts. A long distance part is estimated using the (1/Nc)-approach and a short distance contribution is determined in terms of the coupling constants L~ of the lagrangian .Z~4~a~. The results we listed in this section show rather large corrections to Dashen's theorem. They coincide only partly with the result we obtained in (64), but they could be in agreement with the estimate on the basis of (56). In order to establish them, we would have to numerically evaluate the relevant coefficients K r of ~4(Q).
4. Summary
1. We have evaluated the divergent part of the generating functional of C H P T including virtual photons to one loop. We have calculated the structure of the local action at order p4 with 29 coupling constants ( L 1. . . . . L10; H1, Ha; K a. . . . . K17), which have in general a single pole in d = 4 dimensions.
R. Urech/ Nuclear Physics B433 (1995) 234-254
249
2. In a next step we have calculated the one-loop contribution to the meson masses in the isospin limit m u = m d = rh and extracted the correction to Dashen's theorem [8],
AM~-AM~=-e
1 (
e-167r2 3M~ In
M_~ 2
-3M~
M2_4(M~_M~))
ln--~-T
1 C[ M_~ ( M ~ ) ] -e:---MK2 In -M 2 21n--~-+l 8~- 2 F 4 C
- 16e2-F-ff~Lrs(tz)(M2 - M 2) + R~(tz)M2~ +RK(tz)M 2 + O(e2m2q),
(74)
where AM 2 = M2± -M~o with P the pseudoscalar meson in question. L~(tx) and R~, K are the contributions from 2 4 (Q), 2
2
r
R,~ = g e ( 6 K 3 - 3 K ~ + 2 K ~ - 1 0 K { 0 -
12K{1),
Rs: = - 4 e e ( K ~ + K~ - 6K{0 - 6K{,).
(75)
At the three scalepoints /x = (0.5; 0.77; 1) GeV, we got numerically
AM 2 - AM 2 = ( - 0 . 2 6 ; 0.49; 0.99) + 0.13 × 10 -3 (GeV) 2 + R~M2~ + RK M2. We assumed the upper limit of R~,M~2+ RKMK2 to be [R~M 2 +RKM 2 [ < 2.6 × 10 -3 (GeV) 2,
(76)
(77)
from which we cannot exclude large corrections to Dashen's theorem. Note that in the numerical part in (76) the correction due to L~(/x) is already included. 3. In addition, we have given an estimate of AM~ - AM 2 on the basis of Ref. [7],
A M ~ - AM~ = - ( 0 . 0 6 _+ 1.01) × 10
3
(GeV)2,
(78)
a small number with a large uncertainty. 4.. We have compared our values to the results in the literature [16-18]. We found that they coincide only partly with the result (78), but they could be well in agreement with the estimate in (76). In order to complete the expression for AMZK AM 2 we need the numerical evaluation of R~, K, i.e. the calculation of the finite parts of the coupling c o n s t a n t s K i. This is beyond the scope of this work. -
-
Acknowledgements
I am grateful to Jiirg Gasser for many discussions and for reading this manuscript carefully. Also I thank H. Neufeld, who pointed out that my original lagrangian
R. Urech /Nuclear Physics B433 (1995) 234-254
250
,.~(Q) contains an overcomplete set of counterterms, J. Bijnens for correspondence concerning effective lagrangians, Urs Biirgi, who helped to draw the Feynman graphs, and Robert Baur for pointing out a numerical error in the coefficients Zi, that he had worked out independently.
Note added in proof One of the work announced in Ref. [4] has been published [19].
Appendix A Matrix relations
For the derivation of the effective lagrangian .Z4(Q) at order p4, we used the equation of motion obeyed by U, C d~d~U+ U - U+d~d~U + ( U+ x - x + U ) + 4-~o2 ( U+ Q U Q - QU+ Q U ) -½(U+x-x+U)
=0.
(79)
The Cayley-Hamilton theorem for a 3 × 3-matrix A, A 3 - ( A ) A 2 + ½ ( ( A ) 2 - ( A Z ) ) A - ~ ( ( A ) 3 - 3 ( A ) ( A 2) + 2(A3))1 = 0,
(80) implies a trace identity for 3 × 3-matrices A, B, C and D, (ABCD) = -(ABDC) - (ACBD) - (ACDB) - (ADBC) - (ADCB) + (AB)(CD) + (AC)(BD) + (AD)(BC) + (A)
(81)
Furthermore we used the observation that Q2 can be written as Q2
2 2 = s1e Q + ~e 1.
(82)
We confirmed the 17 independent terms K i < . . . ) (i = 1,2 . . . . . 17) in _~4(Q) calculating the contributions from tree graphs at order p4 to • the constant term e41, 2 2 • the masses M~_+, M,~0 and M ,2, • the scattering amplitudes at order e2mq, 7/-+,W---> ,W+,TT ,
"n'+'tr'--+ K ° K °, K + K - - o K0~o,
(83)
251
R. Urech/Nuclear Physics B433 (1995) 234-254
• the scattering amplitudes at order e 4, ,/7.+3-/-- ~
,/7"+77"-
(84)
7r+Tr-~ K°g, °, • the matrix elements
(Tr + l~3,sdl~-+rr-), (O[~yjsd[zr-) ,
(~-+ 1~3,5d1~7~7),
(OlTFt(x)y~ysd(x) d(y)y~ysu(y)]0),
(zr + [ T F t ( x ) y ~ d ( x ) d ( y ) y ~ u ( y ) [7r+),
(85)
( O [ T ~ ( x ) y U d ( x ) d ( y ) y u u ( y ) [0).
Appendix B
The formal expressions for the masses at order e2m q
The photon loop (see Fig. 1 in the text) gives a contribution to the masses of the charged particles, 8M~± I,-loop = - e 2
Mc2 3 In--~- - 4
,
(86)
where C stands for ~ and K, respectively. From the tadpole, the masses of the charged particles get a change of the form 8M2± ]tadpole = --e2
A c ± M 2 In
~ +Bc+_M 2 In
7-
K
(87)
'
with (A=+, B=+) = (2, 1) and (AK±, BK±) = (1, 2). The contributions to the masses of ~.0 and rt are 1
C
a v M 2 In 6M 2 I tadpole = e 2 - 48~- 2 F 4
+ 1 + f l e M 2 "~p l n - ~ - + 1
,
(881 with the coefficients (a,~o,/3~o, y~0) = (6, 0, 0) and (a n,/3,, ~/,) = ( - 2 , 1, 4). The masses of the neutral kaons do not get contributions from the loops. At order e2mq, the mass-shift due to .9'4(Q) involves in general only terms with the couplings K s, but for the charged particles, where L 4 and L 5 enter as well, 6M2c + _ [ _~te) = - 1 6 e 2 C [(M 2 + 2 M 2 ) L ] ( / x ) + MZL~(/x)] + . . . .
F0
(89)
R. Urech /Nuclear Physics B433 (1995) 234-254
252
The term proportional to L](/,) drops out in the difference A M ~ - AM~. Explicitly, the masses are, at O(eZmq), =M~_+
16~.2
ln/x"
=
× 2M 2 ln-~5- + M~ ln--~-
4
- 2 F--4 8rr
- 16e2
Fo
,M~ + 2M~)L] + M2Lr5]
4 2 2 - ~e M~(6K 1r + 6K~ + SK i + 5K~ - 6K~ - lSK~ - 5K~ -- 23K~o
-
18K[1)
--1-
8e2M~ K~,
(90)
1 C ( M2 ) 2 =~2o +e 2 M~o 8~r2 F4M~2 l n - - ~ - + l -
2 A2 a a r 2 /
12K[ + a2K~- lSK~
+ 9 K ~ + 1 0 K j + a0Kg - aZK~ - l Z K j - 10K~ - 10K~o),
M~±=~2±-e
2
M 231n-~-4
X M2 l n ~ - + 2 M
)21 -e
(91)
87r2F4
21n
C 2 r 42 2 r r - 16e 2 ~404 [(M2 + 2M2)Lr4 + MKLs] + ~e M~(3K 8 + K9 + K[o )
-
-
4 2 2 ~e MK(6K 1r + 6K~ + 5K[ + 5K~ - 6K-~ - 24K~ - 2K~
- 20K~o - 1 8 K ~ l ) ,
(92)
M2o =M2o = ~ , ~ o -- ~~e2 ~M ~ ( 3 K 1 r + 3 K ~ + g ~ + / ' ; 6 r - 3 K ~ - 3 K j - K 9 - K ~ o ) ,
Mn2--M,~^2-e248~r ~1
[
F4C 2M~ In
+1
(93)
-M 241n¢+1
4 2 2 r 2 2 2 1r + 12K~ - 6K~ + 3 K j + 6 K ~ +-~e Mz~( g9r + Klo) - -~eMn(12K
+6K~ - 12K~ - 12K~ - 4K~ - 4K[o ) . The squared mass differences of the pions and the kaons are
AM2=M2±-M2o = 2e2~d-e21~-~M2(3 ln~--4) e2M2 3 1 n ~ - + l 8,n-2 F 4
+ M ~ In
(94)
R. Urech/ Nuclear Physics B433 (1995) 234-254
- 16e2--~C4[(M2 + 2M2)U4 + M2Lrs] Fo +2e2M2(-2Kj+K~+2K~+aK[o+nK[I) AM~=M2+-M~o
+8e2M~K~,
253
(95)
2C M2 =2e F°2 - e21~-~M~(3 l n ¢ - 4 )
1
_e 2 8~.2F4 __ M~2 In ~/x+2 2 M
21n¢
- 16e2~ [(Mg + 2M~)L: + M~L~] a 2M~(3Ks 2 r +K9r + K ~ o +~e 4 _2aar2/
Tlr
- ~ e ~V, K t ~x 5 + K ~ - 6 K ~
) - 6K~o -
6K~1 ) .
(96)
And finally the difference AM2 -AM 2 is A M ~ - A M 2 = - e 2-167r2 3M2 In
- 3 M 2 ln--~--4(M2-M 2)
1 C[ M2 -e 2-M2 l n ¢ - M 8~ 2 F g
( M~ )] 2 21n--~-+l
C - 16e2~oLrs(tz)(M2K - M 2) + R~(tz)M~ + RK(/Z)M~ , (97) where Rrr,K
are,
2 2 r R~= ~e (6K 33K~ + 2K~- 10Kfo- 12gfl), 4 2 1 r-,-r RK = --3e tas +K~- 6K~0- 6K[~).
(98)
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