Axial vector contributions

8.2 R-parity conserving interactions

8.2.2 Axial vector contributions

uL νL

dR µR

He⁺

Be

de⁻_R µe⁻R

Figure 8.2:This contribution toG⁽¹⁾_{P S}is suppressed by²² =y_µy_d.

doscalar contributions are much too small to be detected at upcoming experiments. These conclusions are consistent with an earlier, similar analysis in Ref. [72].

At face value, it appears from equations (8.37-8.39) that∆R_e/µ^SUSYcarries a non-trivial dependence on MSSM parameters since the SUSY masses enter both explicitly in the loop functions and implicitly in the mixing matricesZ, defined in equations (8.32-8.36). Nev- ertheless, we are able to identify a relatively simple dependence on the SUSY spectrum.

We first consider ∆R^{SU SY}_e/µ in a limiting case obtained with three simplifying assump- tions: (1) no flavor mixing among scalar superpartners; (2) no mixing between left- and right-handed scalar superpartners; and (3) degeneracy between`eLandeν`and no gaugino- Higgsino mixing. Our first assumption is well justified; experimental bounds on flavor violating processes constrain the contributions to R_e/µ from lepton flavor violation in the slepton soft-breaking sector to be less than the sensitivies at upcoming experiments by a factor of10−20[72].

Our second assumption has minimal impact. In the absence of flavor mixing, the charged slepton mass matrix decomposes into three 2× 2 blocks; thus, for flavor `, the mass matrix in the{`e_L,`e_R}basis is



 M_L² +¡

s²_W − ¹₂¢

m²_Zcos 2β m_`

³a`

y` −µtanβ

´

m_`

³a`

y` −µtanβ

´

M_R² −s²_Wm²_Zcos 2β



 ,

whereM_L²(M_R²) is the SUSY-breaking mass parameter for left-handed (right-handed) slep- tons,a_{`</sub>is the coefficient for the SUSY-breaking triscalar interaction,y<sub>`}is the Yukawa cou- pling, andµis the Higgsino mass parameter. Under particular models of SUSY-breaking mediation, it is usually assumed thata_{`</sub>/y<sub>`} ∼ M_{SU SY}, and thus left-right mixing is negli- gible for the first two generations due to the smallness ofme andmµ. Of course,a` could be significantly larger and induce significant left-right mixing [58]. For reasons discussed above, we neglect this possibility.

We have adopted the third assumption for purely illustrative purposes; we will relax it shortly. Clearly, fermions of the same weak isospin doublet are not degenerate; their

masses obey

m²_`e

L = m²_{eν`</sub>−m<sup>2</sup><sub>W</sub> cos 2β+m<sup>2</sup><sub>`} (8.17) m²_de

L = m²_euL−m²_W cos 2β+m²_d−m²_u . (8.18) In addition, gaugino mixing is certainly always present, as the gaugino mass matrices con- tain off-diagonal elements proportional tom_Z [see Eqs. (8.33, 8.34)]. However, the third assumption becomes valid forM_{SU SY} Àm_Z.

Under our three assumptions, the SUSY vertex and external leg corrections sum to a constant that is independent of the superpartner masses, leading to considerable simpli- fications. The Bino [U(1)_Y gaugino] vertex and external leg corrections exactly cancel.

The Wino [SU(2)_Lgaugino] vertex and leg corrections do not cancel; rather,∆_V + ∆_L = α/4πs²_W, a constant that carries no dependence on the slepton, gaugino, or Higgsino mass parameters. The occurrence of this constant is merely an artifact of our use of the DR renor- malization scheme. (In comparison, in modified minimal subtraction, we find∆_V+∆_L = 0 in this same limit.¹) This dependence on renormalization scheme cancels inRe/µ. (In addi- tion, this scheme-dependent constant enters into the extraction ofG_µ; hence, the individual decay widthsΓ(π →`ν`)are also independent of renormalization scheme.)

The reason for this simplification is that under our assumptions, we have effectively taken a limit that is equivalent to computing the one-loop corrections in the absence of elec- troweak symmetry breaking. In the limit of unbroken SU(2)_L×U(1)_Y, the one-loop SUSY vertex and external leg corrections sum to a universal constant which is renormalization scheme-dependent, but renormalization scale-independent [75]. (For unbroken SU(2)_L, the SM vertex and external leg corrections yield an additional logarithmic scale dependence;

hence, the SU(2)_L β-function receives contributions from both charge and wavefunction renormalization.) In addition, virtual Higgsino contributions are negligible, since their in- teractions are suppressed by small first and second generation Yukawa couplings. Setting all external momenta to zero and working in the limit of unbroken SU(2)L symmetry, we

1Technically, since M S breaks SUSY, it is not the preferred renormalization scheme for the MSSM.

However, this aspect is not important in the present calculation.

find that the Higgsino contributions to∆L+ ∆V arey_`²/32π².

In this illustrative limit, the only non-zero contributions to ∆R^SUSY_e/µ come from two classes of box graphs (Fig. 8.1d) — one involving purely Wino-like interactions and the other with both a virtual Wino and Bino. The sum of these graphs is

∆^(`)_B = α 12πs²_W

µm²_W M₂²

¶£

F1(xL, xQ) +t²_WF2(xB, xL, xQ)¤

(8.19) where we have defined

F₁(x_L, x_Q)≡ 3 2

· x_L(x_L−2) lnx_L

(xL−xQ)(1−xL)² (8.20)

+ xQ(xQ−2) lnxQ

(x_Q−x_L)(1−x_Q)² − 1

(1−x_L)(1−x_Q)

¸

and

F2(xB, xL, xQ)≡ 1 2

· x_B(x_B+ 2√

x_B) lnx_B (1−x_B)(x_B−x_L)(x_B−x_Q) + x_L(x_L+ 2√

x_B) lnx_L

(1−x_L)(x_L−x_B)(x_L−x_Q) (8.21) + xQ(xQ+ 2√

xB) lnxQ

(1−xQ)(xQ−xL)(xQ−xB)

¸ ,

where xB ≡ M₁²/M₂², xL ≡ m²<sub>`e</sub>/M<sub>2</sub><sup>2</sup>, and xQ ≡ m<sup>2</sup><sub>Qe</sub>/M<sub>2</sub><sup>2</sup>, with masses M1, M2, me`, andm_Qe of the Bino, Wino, left-handed`-flavored slepton, and left-handed 1st generation squark, respectively. Numerically, we find that alwaysF1 ÀF2; the reason is that the sum of Bino-Wino graphs tend to cancel, while the sum of pure Wino graphs all add coher- ently. Hence, Bino exchange (through which the term proportional toF2 arises) does not significantly contribute to∆R_e/µ^SUSY.

In Fig. 8.3, we showF1(xL, xQ)as a function ofxLfor fixedxQ. SinceF1is symmetric underx_L ↔ x_Q, Fig. 8.3 also showsF₁ as a function of x_Q, and hence how∆_B depends onmeuL. For xL, xQ ∼ 1, we haveF1 ∼ O(1), while if either xL À 1orxQ À 1, then F₁ → 0, which corresponds to the decoupling of heavy sleptons or squarks. There is no enhancement of∆B for xL ¿ 1orxQ ¿ 1(i.e. if M2 is very heavy) due to the overall

0 1 2 3 4 5 x_L

0.5 1 1.5 2 2.5 3 3.5

F1HxL,xQL

x_Q=5.0 x_Q

=1.0 x_Q

=0.2

Figure 8.3: The box graph loop functionF₁(x_L, x_Q)as a function ofx_L ≡ m²_e

L/M₂² for several values ofx_Q ≡m²_e

Q/M₂². Forx_L ∼x_Q ∼1(i.e. SUSY masses degenerate),F₁(x_L, x_Q)∼1. For x_LÀ1orx_Q À1(i.e. very massive sleptons or squarks),F₁(x_L, x_Q)→0.

1/M₂² suppression in (8.19).

The total box graph contribution is

∆R^SUSY_e/µ

R^SM_e/µ = 2Re[∆^(e)_B −∆^(µ)_B ]

' α

6πs²_W

µm_W M₂

¶₂

(8.22)

×

"

F1

Ãm²_ee M₂², m²_Qe

M₂²

!

−F1

Ãm²_µe M₂², m²_Qe

M₂²

!#

.

Clearly∆R^SUSY_e/µ vanishes if both sleptons are degenerate and is largest when they are far from degeneracy, such thatmeeL ÀmeµL ormeeL ¿meµL. In the latter case, we have

¯¯

¯¯

¯

∆R^SUSY_e/µ R^SM_e/µ

¯¯

¯¯

¯ . 0.001×

µ100GeV MSU SY

¶₂

(8.23)

for e.g. M_{SU SY} ≡M₂ ∼m_euL ∼m_eeL ¿m_eµL.

We now relax our third assumption to allow for gaugino-Higgsino mixing and non- degeneracy of`eand eν<sub>`</sub>. Both of these effects tend to spoil the universality of ∆_V + ∆_L, giving

∆_V + ∆_L− α

4πs²_W ≡ α

8πs²_W f '0.001f . (8.24)

100 150 200 300 500 700 1000 ΜHGeVL

-8 -6 -4 -2 0 2 4

DReΜSUSY



ReΜ´10-4

Total Box only Vert.+Leg

Figure 8.4: ∆R^SUSY_e/µ versusµ, with fixed parametersM₁ = 100GeV,M₂ = 150GeV,m_eeL = 100 GeV,m_eµL = 500GeV,m_euL = 200GeV. Thin solid line denotes contributions from(∆_V+∆_L)only;

dashed line denotes contributions from∆_Bonly; thick solid line shows the sum of both contributions to∆R^SUSY_e/µ .

The factorf measures the departure of∆_V + ∆_Lfrom universality. If the SUSY spectrum is such that our third assumption is valid, we expect f → 0 . For realistic values of the SUSY parameters, two effects lead to a non-vanishingf: (a) splitting between the masses of the charged and neutral left-handed sleptons that results from breaking of SU(2)_L, and (b) gaugino-Higgsino mixing. The former effect is typically negligible. To see why, we recall from Eq. (8.17) that

m_{`e</sub>=m<sub>eν`}

"

1 +O Ãm²_W

m²_`e

!#

, (8.25)

where we have neglected the small non-degeneracy proportional to the square of the lepton Yukawa coupling. We find that the leading contribution tof from this non-degeneracy is at leastO(m⁴_W/m⁴_e

`), which is.0.1form<sub>e`</sub> &2m_W.

Significant gaugino mixing can induce f ∼ O(1). The crucial point is that the size of f from gaugino mixing is governed by the size of M₂. If M₂ À m_Z, then the Wino decouples from the Bino and Higgsino, and contributions to∆_V + ∆_Lapproach the case of unbroken SU(2)_L. On the other hand, ifM₂ ∼m_Z, then∆_V + ∆_Lcan differ substantially fromα/4πs²_W.

In the limit thatm_e`

L ÀM₂(`=e,µ), we also have a decoupling scenario where∆_B =

100 150 200 300 500 700 1000 M₂HGeVL

-6 -4 -2 0 2 4

DReΜSUSY



ReΜ´10-4

Total Box only Vert.+Leg

Figure 8.5: ∆R^SUSY_e/µ /R^SM_e/µ as a function ofM₂, withµ= 200GeV and all other parameters fixed as in Fig. 8.4. Each line shows the contribution indicated as in the caption of Fig. 8.4.

0,∆_V + ∆_L = _4πs^α2

W, and thusf = 0. Hence, a significant contribution to∆R_e/µrequires at least one light slepton. However, regardless of the magnitude off, ifm_eeL = m_µeL, then these corrections will cancel fromR_e/µ.

It is instructive to consider the dependence of individual contributions∆_Band∆_V+∆_L to∆R^{SU SY}_e/µ , as shown in Figs. 8.4 and 8.5. In Fig. 8.4, we plot the various contributions as a function of the supersymmetric mass parameterµ, with M₁ = 100GeV, M₂ = 150 GeV, m_eeL = 100 GeV, m_eµL = 500 GeV, m_ueL = 200 GeV. We see that the ∆_V + ∆_L contributions (thin solid line) vanish for large µ, since in this regime gaugino-Higgsino mixing is suppressed and there is no ∆_V + ∆_L contribution to ∆R^{SU SY}_e/µ . However, the

∆_B contribution (dashed line) is nearly µ-independent, since box graphs with Higgsino exchange (which depend onµ) are suppressed in comparison to those with only gaugino exchange. In Fig. 8.5, we plot these contributions as a function ofM₂, withµ= 200GeV and all other parameters fixed as above. We see that both∆_V + ∆_Land∆_B contributions vanish for largeM₂.

One general feature observed from these plots is that ∆_V + ∆_Land∆_B contributions tend to cancel one another; therefore, the largest total contribution to∆R^{SU SY}_e/µ occurs when either ∆_V + ∆_L or ∆_B is suppressed in comparison to the other. This can occur in the following ways: (1) ifµÀm_Z, then∆_Bmay be large, while∆_V + ∆_Lis suppressed, and (2) ifm_ueL, mdeL Àm_Z, then∆_V + ∆_Lmay be large, while∆_B is suppressed. In Fig. 8.5,

we have chosen parameters for which there is a large cancellation between∆V + ∆Land

∆_B. However, by taking the limitsµ→ ∞orm_euL, m_deL → ∞,∆R_e/µ^{SU SY} would coincide with the∆B or∆V + ∆Lcontributions, respectively.

Because the ∆_V + ∆_L and∆_B contributions tend to cancel, it is impossible to deter- mine whethereeL orµeLis heavier fromR_e/µ measurements alone. For example, a positive deviation inR_e/µ can result from two scenarios: (1) ∆R^{SU SY}_e/µ is dominated by box graph contributions withmeeL < meµL, or (2) ∆R^{SU SY}_e/µ is dominated by ∆V + ∆L contributions withm_eeL > m_eµL.

Guided by the preceding analysis, we expect for∆R^{SU SY}_e/µ :

• The maximum contribution is

¯¯

¯∆R^{SU SY}_e/µ /R_e/µ

¯¯

¯∼0.001.

• Both the vertex + leg and box contributions are largest ifM2 ∼ O(mZ)and vanish if M₂ Àm_Z. IfM₂ ∼ O(m_Z), then at least one chargino must be light.

• The contributions to∆R^{SU SY}_e/µ vanish ifmeeL =mµeLand are largest if eithermµeL ¿ m_eeL orm_µeL Àm_eeL.

• The contributions to∆R_e/µ^{SU SY} are largest ifeeLorµeLisO(mZ).

• If µ À m_Z, then the lack of gaugino-Higgsino mixing suppresses the ∆_V + ∆_L contributions to∆R^{SU SY}_e/µ .

• Ifm_euL, mdeL À m_Z, then the ∆_B contributions to∆R_e/µ^{SU SY} are suppressed due to squark decoupling.

• Ifeu_L,de_L, andµare allO(m_Z), then there may be cancellations between the∆_V+∆_L and∆_B contributions. ∆R^{SU SY}_e/µ is largest if it is dominated byeither∆_V + ∆_L or

∆_Bcontributions.

We now study∆R^{SU SY}_e/µ quantitatively by making a numerical scan over MSSM param-

0.1 1. 10.

m_e^Ž_Lm̎

L

0.0002 0.0005 0.001 0.002 0.005

È

DReΜSUSY

È 

ReΜ

0.1 1. 10.

m_e^Ž_Lm̎

L

0.0002 0.0005 0.001 0.002 0.005

È

DReΜSUSY

È 

ReΜ

Figure 8.6: ∆R^SUSY_e/µ as a function of the ratiom_eeL/m_µeL. The dark and light regions denote the regions of MSSM parameter space consistent and inconsistent, respectively, with the LEP II bound.

eter space, using the following ranges:

m_Z/2 < {M₁, |M₂|,|µ|, m_euL} < 1TeV

m_Z/2 < {m_νee, m_eνµ} < 5TeV (8.26) 1 < tanβ < 50

sign(µ), sign(M₂) = ±1, wherem_eeL,m_µeL, andm_deLare determined from Eqs. (8.17,8.18).

Direct collider searches impose some constraints on the parameter space. Although the detailed nature of these constraints depends on the adoption of various assumptions and on interdependencies on the nature of the MSSM and its spectrum [1], we implement them in a coarse way in order to identify the general trends in corrections to R_e/µ. First, we include only parameter points in which there are no SUSY masses lighter thanmZ/2.

(However, the current bound on the mass of lightest neutralino is even weaker than this.) Second, parameter points which have no charged SUSY particles lighter than 103 GeV are

100. 150. 200. 300. 500. 700. 1000.

Min

H

m^Ž_eL, m̎

L

L

HGeVL 0.0002

0.0005 0.001 0.002 0.005

È

DReΜSUSY

Ȑ

ReΜ

100. 150. 200. 300. 500. 700. 1000.

Min

H

m^Ž_eL, m̎

L

L

HGeVL 0.0002

0.0005 0.001 0.002 0.005

È

DReΜSUSY

Ȑ

ReΜ

Figure 8.7: ∆R^SUSY_e/µ as a function of Min[m_eeL, m_µeL], the mass of the lightest first or second generation charged slepton. The dark and light regions denote the regions of MSSM parameter space consistent and inconsistent, respectively, with the LEP II bound.

said to satisfy the “LEP II bound.” (This bound may also be weaker in particular regions of parameter space.)

Additional constraints arise from precision electroweak data. We consider only MSSM parameter points whose contributions to the oblique parameters S, T, and U [76] agree with electroweak precision observables (EWPO). A recent fit to both high- and low-energy EWPO using the value of m_t = 170.9± 1.8 GeV [77] has been reported in Ref. [78], yielding

T = −0.111±0.109

S = −0.126±0.096 (8.27) U = 0.164±0.115

where the errors quoted are one standard deviation and where the value of the Standard Model Higgs boson mass has been set to the LEP lower bound m_h = 114.4GeV. Using the correlation matrix given in Ref. [78] and the computation of superpartner contributions

to the oblique parameters reported in Ref. [59], we determine the points in the MSSM parameter space that are consistent with EWPO at 95% confidence. Because we have neglected the 3rd generation and right-handed scalar sectors in our analysis and parameter scan, we do not calculate the entire MSSM contributions toS,T, andU. Rather, we only include those from charginos, neutralinos, and the first two generation left-handed scalar superpartners. Although incomplete, this serves as a conservative lower bound; in general, the contributions toS,T, andU from the remaining scalar superpartners (that we neglect) only causes further deviations from the measured values of the oblique parameters. In addition, we have assumed that the lightest CP-even Higgs mass is the same as the SM Higgs mass reference point: m_h = 114.4GeV, neglecting the corrections due to the small mass difference, and the typically small contributions from the remaining heavier Higgs bosons.

We do not impose other electroweak constraints in the present study, but note that they will generally lead to further restrictions. For example, the results of the E821 measure- ment of the muon anomalous magnetic moment [79] tend to favor a positive sign for theµ parameter and relatively large values oftanβ. Eliminating the points with sign(µ) = −1 will exclude half the parameter space in our scan, but the general trends are unaffected.

We show the results of our numerical scan in Figs. 8.6–8.9. In Figs. 8.6–8.8, the dark regions contain all MSSM parameter points within our scan consistent with the LEP II bound, while the light regions contain all MSSM points inconsistent with the LEP II bound, but with no superpartners lighter than m_Z/2. In effect, the dark (light) regions show how large∆R^{SU SY}_e/µ /R_e/µcan be, assuming consistency (inconsistency) with the LEP II bound, as a function of a given parameter. In Fig. 8.6, we show∆R^{SU SY}_e/µ /R_e/µas a function of the ratio of slepton massesm_eeL/m_µeL. If both sleptons are degenerate, then∆R^{SU SY}_e/µ vanishes.

Assuming the LEP II bound, in order for a deviation inR_e/µto match the target precision at upcoming experiments, we must have

δR_e/µ ≡¯

¯∆R^{SU SY}_e/µ /R_e/µ¯

¯&0.0005, (8.28)

and thusm_eeL/m_µeL & 2orm_µeL/m_eeL & 2. (This result is consistent with an earlier anal-

100. 150. 200. 300. 500. 700. 1000.

m_Χ1HGeVL 0.0002

0.0005 0.001 0.002 0.005

È

DReΜSUSY

Ȑ

ReΜ

100. 150. 200. 300. 500. 700. 1000.

m_Χ1HGeVL 0.0002

0.0005 0.001 0.002 0.005

È

DReΜSUSY

Ȑ

ReΜ

Figure 8.8: ∆R^SUSY_e/µ versusm_χ1, the mass of the lightest chargino. The dark and light regions denote the regions of MSSM parameter space consistent and inconsistent, respectively, with the LEP II bound.

ysis [72], where the authors conclude that∆R^{SU SY}_e/µ would be unobservably small ifmeL

andm_µL differ by less than 10%.)

In Fig. 8.7, we show∆R_e/µ^{SU SY}/Re/µas a function of Min[meeL,meµL], the mass lightest first or second generation slepton. If the lighter slepton is extremely heavy, then both heavy sleptons decouple, causing ∆R^{SU SY}_e/µ to vanish. Assuming the LEP II bound, to satisfy (8.28), we must havem_eeL .300GeV orm_µeL .300GeV.

In Fig. 8.8, we show ∆R^{SU SY}_e/µ /Re/µas a function of mχ1, the lightest chargino mass.

Ifm_χ1 is large,∆R^{SU SY}_e/µ vanishes becauseM₂ must be large as well, suppressing∆_B and forcing∆V and∆L to sum to the flavor independent constant discussed above. Assuming the LEP II bound, to satisfy (8.28), we must havem_χ1 .250GeV.

Finally, we illustrate the interplay between∆V + ∆Land∆B by consideringδR^{SU SY}_e/µ as a function of|µ|andm_euL. In Fig. 8.9, we show the largest values of δR_e/µ^{SU SY} obtained in our numerical parameter scan, restricting to parameter points which satisfy the LEP II bound. The solid shaded areas correspond to regions of the|µ|-m_euLplane where the largest value of δR^{SU SY}_e/µ lies within the indicated ranges. It is clear that δR^{SU SY}_e/µ can be largest

100 150 200 300 500 700 1000 muŽ

LHGeVL 100

150 200 300 500 700 1000

ÈΜÈHGeVL

∆R^eΜ<2´10^-

4

2´10^-4 < ∆R^eΜ<5´10^-4 5´10^-4< ∆R^eΜ<7´10^-4

7´10^-4< ∆R^eΜ<1´10^-3

∆R^eΜ>1´10^-3

Figure 8.9: Contours indicate the largest values ofδR^SUSY_e/µ obtained by our numerical parameter scan (8.26), as a function of|µ|andm_euL. The solid shaded regions correspond to the largest values ofδR^SUSY_e/µ within the ranges indicated. All values ofδR^SUSY_e/µ correspond to parameter points which satisfy the LEP II bound.

in the regions where either (1) µ is small, m_euL is large, and the largest contributions to

∆R^{SU SY}_e/µ are from∆_V + ∆_L, or (2)µis large,m_euLis small, and the largest contribution to

∆R^{SU SY}_e/µ is from∆_B. If bothµandm_ueLare light, then∆R^{SU SY}_e/µ can still be very small due to cancellations, even though both ∆_V + ∆_L and∆_B contributions are large individually.

More precisely, to satisfy (8.28), we need either µ . 250 GeV, or µ & 300 GeV and m_ueL .200GeV.