Neutron EDM Formulas

In this subsection, we will write down the effective Lagrangian for nEDM and then calculate the coefficients of these effective operators in the mirror fermion model.

Model-Independent Expressions of Neutron EDM

The effective Lagrangian for nEDM at the hadronic scale up to dimension six was given by (for reviews see [101, 102])

L_{e f f} = ¯q O_q

Â

q

d_q

2 O_q

Â

q

d_q^c

2 O_q^c+ CWO_W +

Â

f , f⁰

C_{f f}0O_{f f}0. (4.97)

Here, ¯q = q_QCD+q_Weak isq term coefficient and the coefficients dq, d_q^c, CW and Cf f⁰ are the EDM, chromo-EDM for light quarks (q = u,d), chromo-EDM of the gluon [104, 114]

and four fermion interaction for light fermions ( f , f⁰=e,u,d) respectively. The operators in (4.97) are defined by

O_q = g²_s

64pG^a_µne^µnrsG^a_rs, Oq=i ¯qs^µng5qF_µn, O_q^c = ig_sT^a¯qs^µng₅qG^a_µn, O_W = 1

6 f^abcG^a_µne^nbrsG^b_rsG^µ,c_b ,

O_{f f}0 = ( ¯f f )( ¯f⁰ig₅f⁰) , (4.98)

where O_q is theq term operator, Oqand O_q^care EDM and chromo-EDM operators for light quarks (q = u,d); F_µn and G^a_µn are the electromagnetic and the gluonic field strength tensors

4.4 Neutron EDM 83

respectively; OW is the Weinberg’s gluonic operator [104, 114]; T^a= ^l₂^a and f^abc are the generators and structure constants of the Lie algebra of the SU(3)Ccolor group respectively;

O_{f f}0is the four fermion interaction operator for light fermions ( f , f⁰=e,u,d).

For the q term, one can employ the Peccei-Quinn mechanism [103] at which the ¯q is uneventfully zero by introducing a new global symmetry U(1)PQ. This symmetry is spontaneously broken by the VEV of an effective scalar field which leads to a Nambu-Goldstone boson, the axion. The axion could be a dark matter candidate but hasn’t been observed in current experiments. In the framework of mirror fermion model, as in a recent study in [105], with the presence of global symmetry U(1)SM⇥U(1)MF one could rotate the q_QCD to zero. Therefore the total effective ¯q angle now is ¯q = q_QCD+q_Weak=q_Weak, whereq_Weak =ArgDet M is the contribution of the electroweak sector toq term through quark mass matrices. An estimation in [105] shown that ArgDet M < 10 ¹⁰ (or ¯q < 10 ¹⁰) and this can happen due to the smallness of neutrino masses and some reasonable values of the Yukawa couplings. In fact, to achieve this new solution to the strong CP problem, the Yukawa couplings gSqbetween the SM quarks, the mirror quarks, and the Higgs singlets have set to be approximately equal 10 ¹of the Yukawa couplings gSl between the SM leptons, the mirror leptons, and the Higgs singlets at maximal new CP violation phases. If one takes into account the constraints from low-energy searches forµ ! eg [43, 37] and µ e conversion in nuclei [44], which indicating gSl<10 ⁴, then gSq<10 ⁵. Note that in contrast with the Peccei-Quinn solution, this solution of the strong CP problem does not require the existence of axion.

In this work, we will ignore the contribution from theq term to the nEDM. The four fermion interaction also play a minor role, so its contribution will be suppressed in our calculation. So the main contributions to the nEDM are from the quark (chromo-) EDMs and the Weinberg operator term. Therefore, one can write down the nEDM as a linear combination of the coefficients of dominant contribution terms as follows

dn = dn(dq,d_q^c) + dn(CW) . (4.99) There are many methods in the literature to estimate dn, such as naive dimensional analysis [106, 104], chiral perturbation theory [107, 108] and lattice QCD [109], etc. In this work, we will use QCD sum rules which is more systematically than others. The contribution from the quark (chromo-) EDMs by using QCD sum rules [110, 111] is

dn(dq,d_q^c) = (1.0^+0.5_0.5)h

1.4 (dd 0.25du) + 1.1e(d_d^c +0.5d_u^c)i

, (4.100)

84 Low-energy Constraints in the EW-n_R Model

kS

qi qi

q_m^M q_m^M

(a)

kS

q_i q_i

q_m^M q_m^M

g

(b)

kS

g

g

g qi, q^Mm

q^Mm, qi

(c)

Fig. 4.28The diagrams contributing to a) the quark EDM, b) the quark chromo-EDM and c) gluon chromo-EDM in the mirror fermion model.

The contribution to the nEDM from the Weinberg operator [112] is

|dn(CW)/e| = (1.0^+1.0_0.5)20(MeV)CW (4.101) with the sign left undetermined.

Neutron EDM in EW-nR Model

The amplitude for extracting the quark EDM, qi(p) ! qi(p⁰)g(q) (Fig. 4.28a) and the quark chromo-EDM, qi(p) ! qi(p⁰)g(q) (Fig. 4.28b) are written as follows

M q_i ! q_i g = e_µ^⇤(q) ¯uqi(p⁰)h

is^µnq_n C_L^qiPL+C^q_RⁱPR

iuqi(p), (4.102)

M q_i ! q_i g = e_µ^a⇤(q)g_sT^a ¯uqi(p⁰)h

is^µnq_n⇣

Ce_L^qiP_L+ eC_R^qiP_R⌘i

u_qi(p),

(4.103)

4.4 Neutron EDM 85

where the coefficients C^q_L,Rⁱ and eC_L,R^qi are

C_L^qi = e 16p²

Â

3 k=0

Â

3 m=1

(m_qiQ_qM m

m²_qM m

h

V_im^Rqik⇣

V_im^Rqik⌘_⇤

+ V_im^Lqik⇣

V_im^Lqik⌘_⇤i

I m²_fkS m²_qM m

!

+Q_q^M_m m_qM m

V_im^Rqik⇣

V_im^Lqik⌘_⇤

J m²_fkS m²_qM m

!)

, (4.104)

C_R^qi = e 16p²

Â

3 k=0

Â

3 m=1

(m_qiQ_qM m

m²_qM

m

h

V_im^Lqik⇣

V_im^Lqik⌘_⇤

+ V_im^Rqik⇣

V_im^Rqik⌘_⇤i

I m²_fkS m²_qM

m

!

+Q_q^M_m

m_q^M_mV_im^Lqik⇣

V_im^Rqik⌘_⇤

J m²_fkS m²_qM m

!)

, (4.105)

Ce_L^qi = 1 16p²

Â

3 k=0

Â

3 m=1

( m_qi m²_qM

m

h

V_im^Rqik⇣

V_im^Rqik⌘_⇤

+ V_im^Lqik⇣

V_im^Lqik⌘_⇤i

I m²_fkS m²_qM m

!

+ 1

m_qM m

V_im^Rqik⇣

V_im^Lqik⌘_⇤

J m²_fkS m²_qM m

!)

, (4.106)

Ce_R^qi = 1 16p²

Â

3 k=0

Â

3 m=1

( mqi

m²_qM

m

h

V_im^Lqik⇣

V_im^Lqik⌘_⇤

+ V_im^Rqik⇣

V_im^Rqik⌘_⇤i

I m²_fkS m²_qM

m

!

+ 1

m_q^M_mV_im^Lqik⇣

V_im^Rqik⌘_⇤

J m²_fkS m²_qM m

!)

, (4.107)

with mqiand m_q^M_m denote the masses of SM quarks and the mirror quarks, respectively; Q_q^M_m denotes the charge of the mirror quark q^M_m. The coupling coefficients V ^Lqk and V^Rqk are defined in Eq. (3.27). Note that in above calculations we have assumed the mirror quark masses are much heavier than the SM quarks (m_q^M_m mqi) and since we are only considering light quarks (u,d) so mqi! 0 in the loop functions I (r) and J (r), which can be found in Appendix A with r = m²_fkS/m²_qM

m. The amplitudes in Eqs.(4.102, 4.103) can be reproduced by the quark (chromo-) EDM effective Lagrangian and compare it with one shown in Eq. (4.97).

Then the coefficients dqi and d_q^c_i can be extracted as follows

dqi = i

2 C_L^qi C_R^qi

= e

16p²Im

Â

³

k=0

Â

3 m=1

Q_qM m

m_q^M_m V_im^Lqik⇣

V_im^Rqik⌘_⇤

J m²_fkS m²_qM m

!

, (4.108)

86 Low-energy Constraints in the EW-n_R Model

and

d_q^c_i = i 2

⇣Ce_L^qi Ce_R^qi⌘

= 1

16p²Im

Â

³

k=0

Â

3 m=1

1 m_qM

m

V_im^Lqik⇣

V_im^Rqik⌘_⇤

J m²_fkS m²_qM

m

!

. (4.109)

The expression for the coefficient CW of the Weinberg’s gluonic operator has been studied in [104, 113, 114]. In the mirror fermion model, as shown in Fig. (4.28c), the interaction between the scalar singlets, the SM quarks, and the mirror quarks will contribute to the Weinberg’s gluonic operator. A straightforward calculation, including all possible permutations of the external gluon lines, gives us the following amplitude of this three gluon vertex

M_W = 1

3 g³_s

(4p)⁴Im

Â

³

k=0

Â

3 i,m=1

V_im^Lqik⇣

V_im^Rqik⌘_⇤

F (m_qi,m_q^M_m,m_fkS)

⇥h

(p₁ p₂)^re^µnabp_1ap_2b+2(pⁿ₁e^µrabp_1ap_2b+p^µ₂e^nrabp_1ap_2b)i (4.110)

⇥ f^abce_µ^a(p₁)e_n^b(p₂)e_r^c( p₁ p₂) ,

(4.111) where the p^1,2,3ande^a,b,care the momenta and polarizations of the external gluons and the function F (mqi,m_q^M_m,m_fkS)is given by

F (mqi,m_qM

m,m_fkS) = Z ₁

0 dx^{Z 1}

0 dy y³x³(1 x)mqim_qM

h m

x(1 x)ym²_qM

m+x(1 y)m²_qi+ (1 x)(1 y)m²_fkSi2

+ (mqi$ mq^M_m) . (4.112)

The amplitude in Eq. (4.110) can be reproduced by the effective Lagrangian

L_{e f f}^W = 1 3

g³_s

(4p)⁴Im

Â

³

k=0

Â

3 i,m=1

V_im^Lqik⇣

V_im^Rqik⌘_⇤

F (mqi,m_qM

m,m_fkS)f^abcG^a_µne^nbrsG^b_rsG^µ,c_b . (4.113) By comparing the above expression with the gluon chromo-EDM term in Eq. (4.97), one can extract the coefficient CW as

CW = 1 2

g³_s

(4p)⁴Im

Â

³

k=0

Â

3 i,m=1

V_im^Lqik⇣

V_im^Rqik⌘_⇤

F (mqi,m_q^M_m,m_fkS) . (4.114)

4.4 Neutron EDM 87

Inserting Eqs. (4.108, 4.109, 4.114) into Eqs. (4.99, 4.100, 4.101), we have

d_n=Im

Â

³

k=0

Â

3 m=1

1 m_qM

m

nD₁h

V_1m^Ldk⇣

V_1m^Rdk⌘_⇤ +1

2V_1m^Luk⇣

V_1m^Ruk⌘_⇤i

+D₂V_3m^Luk⇣

V_3m^Ruk⌘_⇤o , (4.115) where the dimensionless coefficients D1,D₂are

D₁ = 1^+0.5_0.5( e 16p²)⇣

1.4( 1

3) +1.1⌘

J m²_fkS m²_qM

m

! ,

D₂ = 1^+1.0_0.520(MeV)e⇣ 1 2

⌘⇣m_q^M_mg³_s (4p)⁴

⌘

F (mt,m_q^M_m,m_fkS) . (4.116) Note that the light quarks involved in the loop of Weinberg’s gluonic operator are suppressed in Eq. 4.115.

4.4.3 Numerical Analysis

In this section, we show our numerical results of the nEDM in the model. First of all, let us set up the relevant parameters space.

• The Yukawa couplings, in general, are complex so that one can parametrize them as follows: g^u_iS=|g^u_iS|e^iqui, g^d_iS=|g^d_iS|e^iqdi and g^Q_iS=|g^Q_iS|e^iqQi , where i = 0,1,2 andq_uⁱ,q_dⁱ andq_Qⁱ are the phases in Yukawa couplings which would be new sources of the CP violation. For simplicity, we will assume |g^u_iS| = |g^d_iS| = |g^Q_iS| = |gS|, quⁱ =q_dⁱ =q and q_Qⁱ =q⁰. Moreover, for later purpose, we define a phasea = q q⁰. Forq and q⁰ 2 (0,2p), implying that a 2 ( 2p,2p).

• For the mass of the singlet scalars fkS, we will take them to be degenerate and set m_fkS =M_S =1 GeV for k = 0,1,2,3. Similarly for the mirror quark masses, we also assume they are degenerate, i.e. m_q^M_m =m_M. In addition, since the scale of symmetry breaking is at EW scale (⇠ 246 GeV), the mass of mirror quarks are set to be less than ⇠ 900 GeV. Thus, we will vary the common value mM from 100 GeV to 800 GeV. With this set up, one easily see that mM M_S, thus, the loop function J (m²_fkS/m²_qM

m)⇡ J (0) = 1/2.

• We will study the following two scenarios:

Scenario 1:

V_L^u=V_CKM^† and V_R^u=V_L^uM =V_R^uM=1. (4.117)

88 Low-energy Constraints in the EW-n_R Model

Scenario 2:

V_L^u=V_L^uM =V_CKM^† and V_R^u=V_R^uM =1. (4.118) Here VCKM=V_L^u(V_L^d)^†is the well-known Cabibbo-Kobayashi-Maskawa matrix. The Wolfenstein parameterization of this matrix, which approximates the standard parame-terization very well, can be written as

V_CKM = 0 B@

1 l²/2 l Al³(r ih)

l 1 l²/2 Al²

Al³(1 r ih) Al² 1

1

CA + O(l⁴)

wherel,A,r and h are the four Wolfenstein parameters. The CP violation can be determined by measuring (r ih). In the literature, one usually defines r +ih = (( ¯r + i ¯h)p

1 A²l⁴)/(p

1 l²[1 A²l⁴( ¯r + i ¯h)]). The current global fit results (±1s) for these Wolfenstein parameters [115] arel = 0.22509^+0.00029_0.00028, A = 0.8250^+0.0071_0.0111, r = 0.1598¯ ^+0.0076_0.0072 and ¯h = 0.3499^+0.0063_0.0061.

With the above assumptions, the nEDM in Eq. (4.115) can be simplified according to which scenario we want to study:

• For Scenario 1:

d_n^S1 = 3g²_S mM

h˜D1⇣1

4(l² 2) 1⌘

˜D2

isin(a). (4.119)

• For Scenario 2:

d_n^S2 = g²_S mM

(

˜D1h 1 4

⇣2A²l⁶(2h²+r) + l⁴ 5l²+18⌘

sin(a) + 2A²hl⁶(1 2r)cos(a)i

+ ˜D₂h⇣

A²l⁶( 2h²+r 1) + l² 3⌘

sin(a) + A²hl⁶(1 2r)cos(a)i) , (4.120) where ˜D1,2=D_1,2(m_fkS ! MS,m_qM

m ! mM).

In Fig. 4.29, we show the contour of 0.15, 0.5, 1 mm and 1cm for the decay length of q^M₁ ! qi+fkS on the (log₁₀|gS|,mM) plane. Note that the decay length of this process is sensitive to neither the scenario 1 or 2 in the limit of MS⌧ mM. One can see that the decay length in the range of few mm to cm implies the Yukawa couplings are of order ⇠ 10 ⁶.

In Fig. 4.30 and 4.31, we plot the logarithm of the nEDM versus log₁₀|gS| in the scenarios 1 (Fig. 4.30) and scenario 2 (Fig. 4.31) with CP phasea = 0, p/4 and 3p/2. Note that

4.4 Neutron EDM 89

Decay length 0.15mm 0.5mm 1mm 1cm

-8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 200

300 400 500 600 700 800

Log10(gS) Excluded

-5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0

Log10(|dn|)e.cm

Scenario 1

mM(GeV)

200 400 600 800

Excluded

-5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0

Log10(|dn|)e.cm

Scenario 1

mM(GeV)

200 400 600 800

Fig. 4.30The logarithm of the nEDM versus log₁₀|gS| in the Scenarios 1 with CP phase a = p/4 and 3p/2. The pink line is the current upper limit of the nEDM from ILL. The color pattern represents various values of the mirror quark mass mMas shown at the right palette.

90 Low-energy Constraints in the EW-n_R Model Excluded

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

Log10(|dn|)e.cm

Scenario 2

mM(GeV)

200 400 600 800

Excluded

-5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0

Log10(|dn|)e.cm

Scenario 2

mM(GeV)

200 400 600 800

Excluded

-5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0

Log10(|dn|)e.cm

Scenario 2

mM(GeV)

200 400 600 800

Fig. 4.31Same as Fig. (4.30) for Scenario 2 with CP phasea = 0, a = p/4 and 3p/2.

in the case ofa = 0 the nEDM in the scenario 1 vanishes, which can obviously see from Eq. (4.119), while in the scenario 2 it does not but less constraint by about three orders of magnitude on the common Yukawa coupling |gS| as compared to other two cases. The pink line is the current upper limit of the nEDM from ILL. The light grey region represents the exclusive region. The color pattern represents the various values of the mirror quark mass m_M .

In Fig. 4.32 we show the contours for the nEDM which is set to be the current upper limit from ILL (solid pink line) and the decay length of q^M₁ ! qi+f_kS(red dashed line) on the plane of the phasea and the logarithm of the magnitude of Yukawa coupling. The left figure is for scenario 1, while the right one is for scenario 2. One can see that except at sin(a) = 0 where the nEDM vanishes in scenario 1, our results are not sensitive to the two scenarios that we are studying.

4.4.4 Summary

Even though the nEDM has not been observed in the current experiments, its search is significant for the new physics models, especially for models incorporate additional sources of CP violation which may be important for baryogenesis. In this work, we have presented

4.4 Neutron EDM 91

Decay length = 0.15 mm 1 mm

1 cm Excluded

0.0 0.5 1.0 1.5 2.0

-7 -6 -5 -4 -3 -2 -1

α/π Log10|gS|

Scenario 1

Decay length = 0.15 mm 1 mm 1 cm Excluded

0.0 0.5 1.0 1.5 2.0

-7 -6 -5 -4 -3 -2 -1

α/π Log10|gS|

Scenario 2

Fig. 4.32The contour plot for the nEDM set to be the current upper limit at ILL (solid pink line) and decay length of q^M₁ ! qi+fkS(red dashed line) on the plane of the phasea and the logarithm of the magnitude of Yukawa coupling.

the nEDM in the mirror fermion model with (1) electroweak scale non-sterile right-handed neutrinos and (2) the discrete A4symmetry assigned for the quark sector of the model. Within this model, the nEDM is enhanced by the new Yukawa couplings between the scalar singlets, the SM quarks and the mirror quarks, which contain new sources of CP violation. The one-loop coefficients for the EDM and chromo-EDM quark operators and the two-loop coefficient of the Weinberg 3-gluon operator, which dominantly contribute to the nEDM, were computed.

The current upper limit on the nEDM from ILL imposes important constraints on the parameter space of the model. In particular, at maximal new CP violation phases, the Yukawa couplings between the SM quarks, the mirror quarks, and the Higgs singlets was constrained to be less than ⇠ 10 ⁴. This result is almost comparable with the constraint from the axionless solution of the strong CP problem proposed in [105] at which the Yukawa couplings have been estimated to be less than ⇠ 10 ⁵. Note that the estimation in [105] was obtained by assuming that the Yukawa couplings between the SM quarks, the mirror quarks, and the Higgs singlets are approximately equal 10 ¹times the Yukawa couplings between the SM leptons, the mirror leptons, and the Higgs singlets.

It is interesting to note that within this region of parameter space, the decay lengths of the mirror quarks are in the range of 1 mm to 1 cm. A recent study in [22] suggested that if one designs new search algorithms with displaced vertices located at this range, the mirror

92 Low-energy Constraints in the EW-n_R Model

quarks may be detectable at LHC. Further detailed simulation is needed for searching mirror quarks at the LHC.

在文檔中 新粒子物理模型中的帶電輕子變味反應與雙希格斯粒子生成反應 (頁 97-107)