Leading two-loop Yukawa corrections to the pole masses of SUSY fermions in the MSSM

0.4 MB·English

by  R. Schofbeck

HEPHY-PUB 860/07
Leading two-loop Yukawa corrections to the pole
masses of SUSY fermions in the MSSM
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R. Sch¨ofbeck and H. Eberl
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p Institut fu¨r Hochenergiephysik der O¨sterreichischen Akademie der Wissenschaften,
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p A–1050 Vienna, Austria
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Abstract
2
.
1
We have calculated the leading Yukawa corrections to the chargino, neutralino
1
′
7 andgluinopolemassesintheDR schemeintheMinimalSupersymmetricStandard
0 Model (MSSM) with the full set of complex parameters. We have performed a
:
v numerical analysis for a particular point in the parameter space and found typical
Xi corrections of a few tenths of a percent thus exceeding the experimental resolution
as expected at the ILC. We provide a computer program which calculates two-loop
r
a pole masses for SUSY fermions with complex parameters up to (αα ,Y4,α Y2).
S S
O
1 Introduction
In the Minimal Supersymmetric Standard Model (MSSM), each Standard Model particle
is part of a SUSY multiplet containing its superpartners of opposite statistics and a spin
differing by 1/2 unit. Moreover, due to anomaly cancellation the SUSY Higgs-sector
consists of two chiral multiplets whose fermionic degrees of freedom are called higgsinos.
On the other hand, the vector supermultiplets contain the standard model gauge bosons
and their spin-1/2 superpartners called gauginos.
Organizing these particles in mass eigenstates one finds four mixing neutral Majorana
fermions composed of the superpartners of the photon, the Z0-boson, and the neutral
Higgs bosons H0 . Then there are two charginos χ˜± and χ˜±, which are the fermion mass
1,2 1 2
eigenstates of the partners of the W± and the charged Higgs bosons H± . Finally, the
1,2
gluino is the superpartner of the SU(3) vector-boson, the gluon.
C
The next generation of future high-energy physics experiments at Tevatron, LHC
and a future e+e− linear collider (ILC) will hopefully discover some of these particles
if supersymmetry (SUSY) is realized at low energies. The physics interplay of these
experiments has been studied in [1] and it turned out that particularly a linear collider
[1, 2, 3] will allow to test the underlying SUSY model with great accuracy. In fact, the
accuracies of the masses of the lighter SUSY fermions are expected to be in the permille
region which makes the inclusion of higher order loop-corrections indispensible.
Within the real MSSM important results on quark self-energies were obtained in [4]-
[6]. In [7]-[10] the gluino pole mass was calculated to two-loop order (α2). Moreover,
O S
the scalar MSSM Higgs-sector has been studied in detail, also in the full complex model
[11]-[15] and even some three-loop results have been included recently [16].
In previous works [17, 18] we studied the SUSYQCD two-loop corrections to chargino
and neutralino pole masses and found that these effects have to be taken into account
when matching DR -input with experimental data. However, the reference scenario used
for the numerical analysis was the benchmark point SPS1a’ [19], a scenario where the
lighter mixing partners have a dominating gaugino component. But once the higgsino
component becomes more important one can expect that also leading two-loop Yukawa
corrections are of the same size.
In this paper we therefore study these leading two-loop Yukawa corrections to the
pole masses of charginos, neutralinos and the gluino. Since the one-loop correction to the
gluino pole mass is of order (α ) the leading two-loop Yukawa correction is of order
S
O
(α Y2). On the contrary, leading Yukawa corrections for charginos and neutralinos are
S
O
of order (Y4). We neglect the Yukawa coupling to leptons. The reference point for the
O
numerical study is chosen such that the lighter mixing partners have a dominant higgsino
component and their masses are roughly of the size of their SPS1a’-values so that we can
compare the magnitude of the corrections to the expected experimental accuracy at this
point.
Weconcludethattheleadingtwo-loopYukawa correctionstypically exceed experimen-
tal resolution for the case of neutralinos. For charginos we find smaller corrections com-
parable to their two-loop SUSYQCD corrections [18] which should be taken into account
when extracting DR parameters from experiment. The leading two-loop Yukawa correc-
tions to the gluino are comparably small but extending the known two-loop SUSYQCD
2
corrections [9, 10] to include the gluino phase for thenumerical evaluation gives important
results.
For the calculation of the self-energy amplitudes we employ semi-automatic Math-
ematica tools [20, 21] and cross-check the results with the generic analytic formulae
for two-loop corrections to fermion pole masses in a general renormalizable field-theory
derived in [9].
In principle, some of the complex phases of the full complex MSSM can be rotated
awaybyrigidfieldtransformations, e.g. thephaseofthegluinomassparameter. However,
we do not pose restrictions on these phases for convenience. Instead we generalize the
explicit results on the SUSYQCD corrections to SUSY fermions derived in [9] to the case
of a complex gluino mass parameter. For the above reason this is not a generalization of
the physics.
All of these new results are implemented in the C-program Polxino 1.1 [22]. It
currently is the only publicly available program for the calculation of even the one-loop
chargino and neutralino pole masses with complex parameters and described in [18].
We list all the analytic results in the appendix.
2 Diagrammatics and calculation
The masses of a set of mixing fermions are defined through the complex poles of the full
renormalized propagator of the mixing system as gauge-invariant and renormalization
scale invariant quantities [23]-[31]. The tree-level values are the eigenvalues of the tree-
level DR mass-matrix m.
This complex pole masses can be inferred from the zeros of the determinant of the
renormalized effective action Γ(2)(p2) in terms of the renormalized self-energies Σˆ
iΓˆ(2)(p) = Gˆ(2)−1(p)
−
m+ΣˆLf(p2) σ p(1+ΣˆRf(p2))
= − m · p .
σ¯ p(1+ΣˆLf(p2)) m† +ΣˆRf(p2) !
· p − m
(1)
Here, we use the usual definition for the Lorentz-generators σµ = (1,σi) and σ¯µ =
(1, σi). The diagonalized tree-level mass-matrix m could be chosen real and semi-
−
positive definite by a rigid reparametrization of the fields. However, in the case of a
complex gluino mass parameter we find it more natural to keep its complex phase in the
bilinear Lagrangian and therefore we keep the complex conjugates in Eq. (1).
The pole mass condition in the Weyl representation (1) becomes particularly simple
in the rest frame where kinematics is trivial
0 = det(s (m ΣˆL(s)) (1+ΣˆL(s))−1 (m† ΣˆR(s)) (1+ΣˆR(s))−1). (2)
− − m · p · − m · p
Note that the objects ΣˆL,R are finite matrix valued functions of the external momentum
m,p
invariant s = p2. The iterative solution to Eq. (2) is [9]
3
s M2 iΓ M
i ≡ i − i i
∂Π(1) Π(1)Π(1)
= Π(1) +Π(1) ii + ij ji Π(2)
− ii ii ∂s m2 m2 − ii
Xi6=j i − j
Π(1) = m m†Σˆ(1)L + m 2Σˆ(1)R +m†Σˆ(1)L +m Σˆ(1)R
ij i j k ij | j| k ij j m ij i m ij
Π(2) = m†Σˆ(2)L +m Σˆ(2)R + m 2(Σˆ(2)L +Σˆ(2)R )
ii ii m ii ii m ii | ii| k ii k ii
m 2 Σ(1)R Σ(1)R +Σ(1)L Σ(1)L +m†(Σ(1)R Σ(1)L +Σ(1)L Σ(1)L )
− | i| k ij k ji k ij k ji i k ij m ji m ij k ji
(cid:16) (cid:17)
+m Σ(1)L Σ(1)R +m Σ(1)R Σ(1)R +m†m Σ(1)R Σ(1)L +Σ(1)L Σ(1)R
i k ij m ji j k ij m ji i j k ij k ji m ij m ji
!
(3)
All self-energies have to be evaluated at an external momentum invariant s = m 2 with
i
| |
an infinitesimal positive imaginary part [32].
′
According to the DR -scheme the UV-divergencies of the Feynman-integrals are reg-
ulated dimensionally by d = 4 ǫ, and the unphysical scalar mass parameter m for the
ǫ
−
evanescent fields is absorbed according to [33].
The resulting two-loop integrals have a rather complex tensor structure which has
already been worked out in detail in [9], implemented in a C-program [10] and tested
thoroughly[17,18]. In thiswork we therefore use thenotationof [9]forthe basisintegrals.
2.1 One-loop level
InFig.1-3weshowtheone-loopdiagramsforthegluino,theneutralinosandthecharginos,
respectively. Note, that diagrams with charge conjugated inner particles are not shown
explicitly. We checked our analytic one-loop calculation against previous work [34, 35] in
the on-shell scheme and found agreement.
In the case of the gluino theone-loopcorrection is proportionalto g2, thus any squared
s
one-loop self-energies cannot contribute to leading two-loop Yukawa corrections (g2Y2).
O s
The result is
Πgg(1) = 8π2g2 C m 2 3ln m 2 5 B (q,q )+2B (q,q )m Re(m Lq∗Rq)
s A| g| | g| − − FS s FS s q g s s
(cid:16) (cid:16) (cid:17) (cid:17)(4)
ee
e e
e e e
which slightly generalizes Eq.(5.2) of [9] for a complex gluino mass parameter.
The (α)one-loopmassshifts forcharginos andneutralinos aregiven inEq.(5.15) and
O
Eq.(5.18) of [9] and are not repeated here. However, for a fixed order calculation (α,Y4)
O
we need the (Y2) one-loop self-energies because their squares appear in Eq. (3).
O
The results are
n m
ΣˆL− = (ΣˆR−)† = c qY Y B (q,Q )
m ij m ij −16π2 qχ−j Q∗s χ−i ∗q¯Qs FS s
n
ΣˆL− = c Y∗ Y B (q,Q )/s e
k ij −16π2 qχ−i Q∗s qχ−j Q∗s eFSe e s e
e
e e e e 4
n
ΣˆR− = c Y∗ Y B (q,Q )/s
k ij −16π2 χ−j∗q¯Qs χ−i ∗q¯Qs FS s
n m
ΣˆL0 = (ΣˆR0)† = c q Y Y e +Y Y B (q,Q )
m ij m ij e −e16πe2 e qχ0iqs∗ χ0jq¯qs qχ0jqs∗ χ0iq¯qs FS s
(cid:18) (cid:19)
n
ΣˆL0 = (ΣˆR0)T = c Y∗ Y +Y∗ Y B (q,Qe )/s (5)
k ij k ij −16π2 χe0iq¯eqs χe0jq¯qes qχe0iqes∗ qeχ0jqes∗ FS s
(cid:18) (cid:19)
e
where summation over the squark index s and q = (u,d,c,s,t,b) is understood implicitly.
e e e e e e e e
Q denotes the iso-spin partner of q and is not summed over independently. The Yukawa
couplings are given by
Yχ0iu¯us = −Ni∗4LusYu Yχ−i u¯ds = Vi∗2LdsYu
Yeχ0id¯eds = −Ni∗3LedsYd Yχe−i ∗d¯ues = Ui∗2LeusYd (6)
Yueχ0iue∗s = −Ni∗4Resu∗Yu Yeuχ−i ed∗s = Ui∗2Resd∗Yd
Ydeχ0ied∗s = −Ni∗3Resd∗Yd Ydχe−i ∗ue∗s = Vi∗2Rsue∗Yu
e e
e e e e
where we employ the conventions of [36] for the mixing parameters. Furthermore,
g m g m
2 u 2 d
Y = and Y = . (7)
u d
√2M sinβ √2M cosβ
W W
In Eq. (6) and Eq. (7) the symbols u and d denote the up-type and down-type quark,
respectively. The generation index is suppressed.
2.2 Two-loop results
InFig.4andFig.5we show theleading two-loopself-energy diagramscontributing (Y4)
O
to the pole mass. For simplicity we put the external neutralinos and the gluino in Fig. 4
into a generic Majorana field η. Without loss of generality neutralino and chargino masses
are taken to be real, the gluino mass however is allowed to have a complex phase thus
avoiding the introduction of a one-dimensional rotation matrix for this particle.
The renormalization scheme adopted is the familiar DR -scheme which regulates UV-
divergencies dimensionally but introduces an unphysical scalar field for any gauge field
in the theory in order to restore the counting of degrees of freedom in supersymmetry.
These unphysical mass parameters can be absorbed in the sfermion mass parameters [33],
the resulting scheme is called DR′.
Thediagramsandtheamplitudes weregeneratedinFeynArts 3.2[20]andsimplified
analytically using FeynCalc 4.0.2 [21]. The resulting tensor integrals were translated
into the conventions of [9]. We then independently derived the same results from the
generic formulae given in Eq. (4.4, 4.5) of [9].
The arguments of these self-energy tensor-functions are the squared masses of the
inner particles and we abbreviate them by their respective symbol. The external mo-
mentum invariant is the squared tree-level DR -input mass-parameter s = m 2 with an
ii
| |
infinitesimal positive imaginary part.
5
For the numerical evaluation of the tensor integrals in our program Polxino [22] we
use the implementation of [10] and the two-loop self-energy library [32].
We split the rather lengthy analytic expression into five parts according to which
particles occur in the loops, neutral (scalar and pseudoscalar) Higgs bosons, charged
Higgs bosons, neutralinos, charginos or only quarks and squarks
1
Πgg(2) = Π +Π +Π +Π +Π (8)
(16π2)2 g|φ0 g|φ− g|q g|χ0 g|χ−
(cid:16) (cid:17)
ee 1
χχ(2)
Πii = (16π2)2 Πeχie|φ0 +Πeχei|φ− +Πeeχi|q +eΠeχi|χ0 +eeΠχi|χ− (9)
(cid:16) (cid:17)
ee
where χ in Eq. (9) can be either theeeneutraelineo or teheecharegieno. Theeeexplicit form of
the self-energy functions on the r.h.s. are given in the App. (A.1)-(A.3). Due to charge
conserveation there are quark isospin-partners in the chargino diagrams in the loop (Fig.5)
denoted by (q,Q). In order to get the leading Yukawa correction (Y4) at the two-loop
O
level we have to use the Yukawa part of the couplings given in Eq. (6) and Eq. (29-32).
g q~ q~
g~ g~ g~ g~ g~ g~
g~ q q
Figure 1: Gluino one-loop self-energy diagrams
h0;H0;A0;G0 H+;G+ (cid:23)e ee
0 0 0 0 0 0 0 0
(cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~
(cid:31)~0 (cid:31)~+ (cid:23) e
ue de Z0 W+
0 0 0 0 0 0 0 0
(cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~
u d (cid:31)~0 (cid:31)~+
Figure 2: Neutralino one-loop self-energy diagrams
6
h0;H0;A0;G0 H+;G+ (cid:23)e ee
+ + + + + + + +
(cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~
(cid:31)~+ (cid:31)~0 e (cid:23)
qe (cid:13) Z W+
+ + + + + + + +
(cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~
Q (cid:31)~+ (cid:31)~+ (cid:31)~0
Figure 3: Chargino one-loop self-energy diagrams
(cid:30)0i;(cid:30)(cid:0)j;qe;Qg
qe qe qe q qe qe Qg qe
(cid:17) (cid:17) (cid:17) (cid:31)f0 (cid:17) (cid:17) (cid:30)0i (cid:17) (cid:17) (cid:30)(cid:0)i (cid:17)
q q qe q q Q q
(cid:31)f0 Q q Q
qe qe qe qe q q q q
(cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17)
q (cid:31)f(cid:0) (cid:30)0i (cid:30)(cid:0)i
q q qe qe
(cid:31)f0 (cid:31)(cid:0) (cid:30)0i (cid:30)(cid:0)i
q q q q qe qe qe qe
(cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17) (cid:17)
qe Qg qe Qg
qe qe q q
Figure 4: Leading Yukawa gluino and neutralino two-loop self-energy diagrams, η =
(g,χ0). Q denotes the isospin partner of q = (u,d). φ0 labels the neutral Higgs scalars
(h0,H0,G0,A0) and φ− the charged ones (G−,H−). The direction of the charge flow is
i
coenesistent with q = u. The arrows on quark and squark lines have to be reversed for q = d.
3 Numerical results
Our reference scenario for the numerical analysis is a higgsino scenario defined through
M = 200GeV, M = 400GeV and µ = 130GeV at Q = 1 TeV. All other parameters
1 2
are taken from SPS1a’, in fact tanβ = 10, A = 532.38GeV, A = 938.91GeV,
t b
− −
M = 470.91GeV,M = 385.32GeVandM = 501.37GeV,forfurtherdetailssee[19].
Q˜3 U˜3 D˜3
EvaluatingtheDR-spectrum, theone-loopcorrectionsandourtwo-loopresultsEq.(8,
9) as well as the respective two-loopSUSYQCD corrections [9] we find the following result
7
(cid:30)0i;(cid:30)(cid:0)j;qe;Qg
(cid:31)f0
qe qe qe q qe qe
(cid:31)~(cid:0) (cid:31)~(cid:0) (cid:31)~(cid:0) (cid:31)f0 (cid:31)~(cid:0) (cid:31)~(cid:0) (cid:30)0i (cid:31)~(cid:0) (cid:31)~(cid:0) qe q qe (cid:31)~(cid:0)
g
Q Q Q Q Q
Q
(cid:0)
Q q Q (cid:31)~
(cid:0) qe qe (cid:0) (cid:0) q q (cid:0) (cid:0) q q (cid:0) (cid:0) q q (cid:0)
(cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~
(cid:31)f(cid:0) (cid:30)0i (cid:30)(cid:0)i Qg
g g g
Q Q Q Q
(cid:31)f0 (cid:30)0i (cid:30)(cid:0)i
(cid:0) q q (cid:0) (cid:0) qe qe (cid:0) (cid:0) qe qe (cid:0)
(cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~ (cid:31)~
qe qe Qg
g
Q Q Q
Figure 5: Leading Yukawa chargino two-loop self-energy diagrams. Q denotes the isospin
partner of q = (u,d). φ0 labels the neutral Higgs scalars (h0,H0,G0,A0) and φ− the
i
charged ones (G−,H−). The direction of the charge flow is consistent with q = u. The
arrows on quark and squark lines have to be reversed for q = d.
for the real parts of the one- and two-loop masses :
Mχ˜01 = 106.54 −0.86 −0.08GeV Mχ˜−1 = 121.54 +0.86 −0.05GeV
Mχ˜02 = 137.22 +1.92 +0.15GeV Mχ˜−2 = 417.75 −0.09 +0.02GeV (10)
Mχ˜03 = 212.81 −5.66 +0.03GeV Mg = 572.33 +39.1 +10.9GeV
Mχ˜04 = 417.87 −0.40 +0.04GeV
e
In this scenario the lighter mixing partners of charginos and neutralinos (χ˜0,χ˜0,χ˜−)
1 2 1
haveadominatinghiggsinocomponentbuttheirtree-levelmassesareroughlyofthesizeof
the SPS1a’ values. Therefore, we can compare the magnitude of the two-loop corrections
to the expected experimental accuracy at SPS1a’.
InFig.6we showtheneutralino polemasses mχ˜01 andmχ˜02 andthetwo-loopmass shifts
on the right as a function of tanβ. The dotted line contains the two-loop SUSYQCD
corrections only whereas the solid line is the full (αg2,Y4) result. Despite the obvious
O s
cancellation between SUSYQCD- and Yukawa corrections for χ˜0 the corrections exceed
1
the expected experimental accuracy of about 0.05GeV [1].
±
For χ˜− in Fig. 7 we find corrections of a few hundred MeV for rather high values
1
of tanβ which should be included when matching DR-parameters to on-shell masses.
However, corrections to χ˜− tend to stay below expected experimental accuracy.
1
From the dependence of δ(2)mχ˜01 and δ(2)mχ˜02 on At in Fig. 8 we learn that At severely
8
effects these two-loop corrections and typical results exceed experimental resolution by an
order of magnitude. On the contrary, δ(2)mχ˜−1 in Fig. 9 as a function of At is again smaller
than the expected experimental accuracy. In Fig. 10 we show the pole masses and the
corrections as a function of φ for the two light neutralinos. The two-loop corrections
At
substantially depend on this phase.
Note that for χ˜0 and χ˜− the Yukawa corrections tend to balance the SUSYQCD
1 1
corrections.
In Fig. 11 and Fig. 12 we investigate the dependence on φ of the lightest mass-
M3
eigenstates and the gluino. For the gluino it turns out that only the variation of the one-
loopcorrection is numerically significant. Notealso, thatthe two-loopYukawa corrections
to the gluino pole mass are very small. On the other hand, δ(2)mχ˜01 and δ(2)mχ˜−1 have a
comparably strong dependence on φ coming from the SUSYQCD corrections alone.
M3
4 Conclusions
We have calculated leading two-loop Yukawa corrections to the pole masses of charginos,
neutralinos and the gluino. The magnitude of these corrections typically exceeds exper-
imental resolution for neutralinos. For charginos we find smaller corrections which still
need to be taken into account when analyzing precision experiments. The leading two-
loopYukawa corrections to the gluino pole mass are rather small but we extended existing
results on the SUSYQCD corrections to the case of a complex gluino phase giving rise to
numerically relevant one- and two-loop effects.
Finally, we included all results in the publicly available program Polxino 1.1 [22]
which has an convenient interface to the commonly used SUSY Les Houches accord [37]
for further numerical studies.
A Analytic results
In the following q is summed over (u,d,c,s,t,b) and Q denotes the iso-spin partner of
q. The chargino index c, the neutralino indicex n as well as the squark-indices s,t,u are
summed over independently. The generation index g appearing in Eq. (14) is summed
over (1,2,3). Finally, φ0 is summed over (h0,H0,A0,G0) and φ− over (H−,G−). φ0 is 0
| |
for the scalar (h0,H0) and +1 for the pseudo-scalar Higgs (A0,G0). The tensor-integral
functions are listed in [9], their arguments denote the squared DR -masses of the particles.
A.1 Gluino
For the gluino we get
1
Πgg(2) = Π +Π +Π +Π +Π (11)
(16π2)2 g|φ0 g|φ− g|q g|χ0 g|χ−
(cid:16) (cid:17)
ee
with ee ee ee ee ee
Πg|φ0 = gs2 4m2qYqq¯φ0Re(mgλφ0qsqt∗Rsq∗Lqt)MSSFFS(qs,qt,q,q,φ0)
(cid:16)
e e ee 9 e e
+4Yqq¯φ0Re(m∗gλφ0qsqt∗RtqLqs∗)MSSFFS(qs,qt,q,q,φ0)
−4mqYqq¯φ0Ree(λφ0qesqet∗(Lqs∗Lqt +(−)|φ0|Resq∗eRtq))MSSFFS(qs,qt,q,q,φ0)
+4m Re(m λ λ∗ Rq∗Lq)V (q,q ,q ,q ,φ0)
q g φ0qsqeu∗eφ0qtqu∗ s t FSSSS s t u e e
2 λ 2V (q,q ,q ,q ,φ0)
− | φ0qsqt∗| e FSeSSeS ese s t e e e
2m Re(m λ Rq∗Lq)Y (q,q ,q ,φ0)
− q ee g φ0φ0∗qsqt∗ es et e FSSS s t
+Re(λ )Y (q,q ,q ,φ0)
φ0φ0∗qesqs∗ FSeSeS s s e e
+4( )|φ0|m3Y2 Re(RqLq∗m∗)V (q ,q,q,q,φ0)
− eqeqq¯φ0 s es eg SFFFS s
2m2Y2 (V +2( )|φ0|V )(q ,q,q,q,φ0)
− q qq¯φ0 SFFFS − e SFFFS es
+4m Y2 Re(m∗RqLq∗)(2V +( )|φ0|V )(q ,q,q,q,φ0)
q qq¯φ0 g s s SFFFS −e SFFFS s
2Y2 V (q ,q,q,q,φ0) (12)
− qq¯φ0 SFFFS es e
(cid:17)
e
Π = g2 4m m Re(m∗Y λ RqLQ∗)M (q ,Q ,q,Q,φ−)
g|φ− s q Q g qQ¯φ− φ−∗Qtqs∗ s t SSFFS s t
+(cid:16)4Re(m Y λ RQ∗Lq)M (q ,Q ,q,Q,φ−)
e g qQ¯φ−eφ−∗Qtqs∗ t e es SSFFS s t e e
4m Re(λ (LQ∗Lq Y +RQ∗Rq Y∗ ))M (q ,Q ,q,Q,φ−)
− Q e φ−∗Qtqs∗ ete s qQ¯φ− t es Qeq¯φ−∗ SSFFS s t
+4m Re(m∗λ∗ λ RqLq∗)V (q,q ,q ,Q ,φ−)
q g φe−Qeuqt∗ φ−Quqs∗ s t FSSSS s t u e e
−2|λφ−Qtqs∗|e2VFSeSSeS(q,qes,eqs,Qt,φ−) e e e
4m Re(m λ Rq∗Lq)Y (q,q ,q ,φ−)
− q e e g φ−φ−∗qsqt∗ es et eFSSS s t
+2Re(λ )Y (q,q ,q ,φ−)
φ−φ−∗qsqs∗ FSSS s s e e
e ee
+4m2qmQRe(m∗gYqQ¯φ−YQq¯φ−e∗ResqLqs∗)VSFFFS(qs,q,q,Q,φ−)
ee
−2m2q(|YQq¯φ−∗Lqs|2 +|YqQ¯φ−Rsq|2)VSFFFS(qs,eq,q,Q,φ−)
−4mqmQRe(YQeq¯φ−∗YqQ¯φ−)VSFFFS(qs,q,q,eQ,φ−)
+4mq(|YqQ¯φ−|2 +|YQq¯φ−∗|2)Re(RsqLeqs∗m∗g)VSFFFS(qs,q,q,Q,φ−)
+4mQRe(mgYQq¯φ−∗YqQ¯φ−Rsq∗Lqs)VSFFFS(qs,q,q,Qe,φ−)
e
−2(|YQq¯φ−∗Resq|2 +|YqQ¯φ−Lqs|2)VSFFFS(qs,eq,q,Q,φ−) (13)
(cid:17)
e
π = 2g2 m Re((m Rq∗Lq +m∗Lq∗Rq)λ )Y (q,q ,q ,X)
g|X − s q g s t g s t qsqt∗XX∗ FSSS s t
(cid:16)
Re(λ Y (q,q ,q ,X)
e − qsqs∗XX∗ eFSSS s es ee e e
Π = n π +π +π (cid:17) (14)
g|q c g|qg,u g|qu g|Qu e e
ee
ee ee ee ee
Π = g2 4Re(RqLqY Y∗ )M (q ,q,q,q ,χ0)
g|χ0 s s t qχ0nqs∗ χ0nq¯qt SFFSF s t n
(cid:16)4m Re(m Y Y∗ LqLq +m∗Y Y∗ RqRq)M (q ,q,q,q ,χ0)
ee − q g qχe0nqes∗ χe0nq¯qet s t geqχ0nqt∗ eχ0nq¯eqs s t SFFSF s t n
e e e e e 10e e e e e e e e