PHYSICAL REVIEW D 76, 095017 (2007) Pion leptonic decays and supersymmetry
Michael J. Ramsey-Musolf,1,2 Shufang Su,3 and Sean Tulin1 1California Institute of Technology, Pasadena, California 91125, USA 2University of Wisconsin, Madison, Wisconsin, 53706-1390, USA 3Department of Physics, University of Arizona, Tucson, Arizona 85721, USA (Received 23 July 2007; published 29 November 2007)
We compute supersymmetric contributions to pion leptonic ( l2) decays in the minimal supersym- metric standard model (MSSM). When R-parity is conserved, the largest contributions to the ratio Re= ! e e = ! arise from one-loop V A V A corrections. These con- tributions can be potentially as large as the sensitivities of upcoming experiments; if measured, they would imply significant bounds on the chargino and slepton sectors complementary to current collider limits. We also analyze R-parity-violating interactions, which may produce a detectable deviation in Re= while remaining consistent with all other precision observables.
DOI: 10.1103/PhysRevD.76.095017 PACS numbers: 11.30.Pb, 12.15.Lk, 13.20.Cz
I. INTRODUCTION ory, Re= has been calculated with even better precision [14]: RSM 1:2352 0:0001 10 4. Experimentally, Low-energy precision tests provide important probes of e= new physics that are complementary to collider experi- the most precise measurements of Re= have been obtained ments [1–3]. In particular, effects of weak-scale supersym- at TRIUMF [15] and PSI [16]. Taking the average of these metry (SUSY)—one of the most popular extensions of the results gives [17] standard model (SM)—can be searched for in a wide EXPT 4 R 1:230 0:004 10 ; (3) variety of low-energy tests: muon g 2 [4], - and e= -decay [5,6], parity-violating electron scattering [7], in agreement with the SM. Future experiments at these electric dipole moment searches [8], and SM-forbidden facilities will make more precise measurements of Re= , transitions like ! e [9], etc. (for a recent review, see aiming for precision at the level of <1 10 3 (TRIUMF Ref. [10]). In this paper, we compute the SUSY contribu- [18]) and 5 10 4 (PSI [19]). These projected uncertain- tions to pion leptonic ( l2) decays and analyze the con- ties are close to the conservative estimate of theoretical ditions under which they can be large enough to produce uncertainties given in Ref. [12]. observable effects in the next generation of experiments. Deviations Re= from the SM predictions in Eq. (2) In particular, we consider the ratio Re= , defined by would signal the presence of new, lepton flavor-dependent ! e e physics. In the minimal supersymmetric standard model e e SUSY R : (1) e= (MSSM), a nonvanishing Re= may arise from either ! tree-level or one-loop corrections. In Sec. II, we consider The key advantage of R is that a variety of QCD effects SUSY e= contributions to Re= arising from R-parity conserving that bring large theoretical uncertainties—such as the pion interactions (Fig. 1). Although tree-level charged Higgs decay constant F and lepton flavor-independent QCD exchange can contribute to the rate ! ‘ , radiative corrections—cancel from this ratio. Indeed, this correction is flavor independent and cancels from R is one of a few electroweak observables that involve e= Re= . One-loop corrections induce both scalar and vector hadrons and yet are precisely calculable (see [11] for semileptonic dimension six four-fermion operators. Such discussion and Refs. [12,13] for explicit computations). interactions contribute via pseudoscalar and axial vector Moreover, measurements of this quantity provide unique pion decay amplitudes, respectively. We show that the probes of deviations from lepton universality of the pseudoscalar contributions are negligible unless the ratio charged current (CC) weak interaction in the SM that are of the up- and down-type Higgs vacuum expectation values induced by loop corrections and possible extensions of the 3 (vevs) is huge (vu=vd tan * 10 ). For smaller tan SM. In the present case, we focus on contributions from the most important effects arise from one-loop contribu- SUSY that can lead to deviations from lepton universality. tions to the axial vector amplitude, which we analyze in Until recently, the two most precise theoretical calcula- detail by performing a numerical scan over MSSM pa- tions of Re= in the SM were [12,13] rameter space. We find that experimental observation of 4 SUSY loop-induced deviations at a significant level would SM 1:2352 0:0005 10 ; R (2) require further reductions in both the experimental error e= 1:2354 0:0002 10 4; and theoretical SM uncertainty. Such improvements could where the theoretical uncertainty comes from pion struc- lead to stringent tests of ‘‘slepton universality’’ of the ture effects. Recently, by utilizing chiral perturbation the- charged current sector of the MSSM, for which it is often
1550-7998=2007=76(9)=095017(12) 095017-1 © 2007 The American Physical Society MICHAEL J. RAMSEY-MUSOLF, SHUFANG SU, AND SEAN TULIN PHYSICAL REVIEW D 76, 095017 (2007)
SUSY FIG. 1. Representative contributions to Re= : (a) tree-level charged Higgs boson exchange, (b) external leg diagrams, (c) vertex diagrams, (d) box diagrams. Graph (a) contributes to the pseudoscalar amplitude, graphs (b),(c) contribute to the axial vector amplitude, and graph (d) contributes to both amplitudes. assumed that the left-handed first and second generation independent contributions to the semileptonic radiative sleptons e~L and ~ L are degenerate (see e.g. [20]) and thus corrections cancel from Re= . SUSY Re= ’ 0. Now consider the contribution from an induced pseudo- In Sec. III, we consider corrections to Re= from R- scalar four-fermion effective operator of the form parity-violating (RPV) processes. These corrections enter G V PS ud 5 5 at tree level, but are suppressed by couplings whose LPS p 1 ‘b u: (6) 2 strength is constrained by other measurements. In order to analyze these constraints, we perform a fit to the current Contributions to Re= from operators of this form were low-energy precision observables. We find that, at 95% considered in a model-independent operator framework in SUSY C.L., the magnitude of RPV contributions to Re= could Ref. [21] and in the MSSM in Ref. [22]. In the MSSM, be several times larger than the combined theoretical and such an operator can arise at tree level [Fig. 1(a)] through anticipated experimental errors for the future Re= experi- charged Higgs exchange and at one-loop through box ments. We summarize the main results and provide con- graphs [Fig. 1(d)]. These amplitudes determine the value clusions in Sec. IV. Details regarding the calculation of of GPS. The total amplitude is one-loop corrections are given in the appendix. iM 0 iM V F G m u 1 5 AV PS ud ‘ G II. R-PARITY CONSERVING INTERACTIONS v 1 PS ! (7) ‘ G ‘ A. Pseudoscalar contributions The tree-level amplitude for ! ‘ that arises where ‘ from the V A V A four-fermion operator is 2 3 m 5 10 ‘ e p !‘ ’ (8) 0 m m m 20 ‘ iMAV i2 2G Vudh0jd PLuj iu PLv‘ ‘ u d
2VudF G m‘u PRv‘; (4) is an enhancement factor reflecting the absence of helicity suppression in pseudoscalar contributions as compared to where PL;R are the left- and right-handed projection opera- V A V A contributions [23]. Pseudoscalar con- tors, tributions will be relevant to the interpretation of Re= if F 92:4 0:07 0:25 MeV (5) GPS
!‘ * 0:0005; (9) is the pion decay constant, G is the Fermi constant G extracted from the muon lifetime, and Vud is the (1, 1) and if GPS!‘ is lepton flavor dependent. component of the Cabibbo-Kobayashi-Maskawa (CKM) The tree-level pseudoscalar contribution [Fig. 1(a)] matrix. The first error in Eq. (5) is experimental while gives the second arises from uncertainties associated with QCD effects in the one-loop SM electroweak radiative correc- 0 m‘ tan mu cot md tan GPS p ; (10) tions to the 2 decay rate. The superscript ‘‘(0)’’ and 2 2 2mH v subscript ‘‘AV’’ in Eq. (4) denote a tree-level, axial vector contribution. At one-loop order, one must subtract the where mH is the mass of the charged Higgs boson. Thus, radiative corrections to the muon-decay amplitude—since we have G is obtained from the muon lifetime—while adding the 0 2 GPS m tan mu cot md tan corrections to the semileptonic CC amplitude. The correc- ! : (11) ‘ m m m2 tions to the muon-decay amplitude as well as lepton-flavor- G u d H
095017-2 PION LEPTONIC DECAYS AND SUPERSYMMETRY PHYSICAL REVIEW D 76, 095017 (2007) It is indeed possible to satisfy (9) for These extreme values of tan can be problematic, leading yb and y to become nonperturbatively large. To avoid this mH scenario, we need roughly tan & 65 (see [20] and refer- tan * 20 : (12) 100 GeV ences therein).
0 Pseudoscalar contributions can also arise through mix- Note that the combination GPS=G !‘ entering Eq. (7) ing of left- and right-handed scalar superpartners. Since is independent of lepton flavor and will cancel from Re= . each left-right mixing insertion introduces a factor of , the In principle, however, the extraction of F from 2 decay 1 2 leading contributions to GPS will still be O . However, could be affected by tree-level charged Higgs exchange if the triscalar SUSY-breaking parameters af are not sup- if the correction in Eq. (9)is* 0:003 in magnitude, pressed by yf as normally assumed, it is possible to have corresponding to a shift comparable to the theoretical O 1 , potentially leading to significant contributions. SM uncertainty as estimated in Ref. [12]. In the case of This possibility, although not experimentally excluded, is charged Higgs exchange, one would require tan * considered theoretically ‘‘unnatural’’ as it requires some 120 mH =100 GeV to generate such an effect. fine-tuning to avoid spontaneous color and charge breaking One-loop contributions to GPS are generated by box (see Ref. [6] for discussion). Neglecting this possibility and graphs [Fig. 1(d)]. The magnitude of these contributions extremely large values of tan , we conclude that loop- is governed by the strength of chiral symmetry breaking in induced pseudoscalar contributions are much too small to both the quark and lepton sectors. Letting generically be detected at upcoming experiments. These conclusions denote either a Yukawa coupling yf or a ratio mf=MSUSY are consistent with an earlier, similar analysis in Ref. [22]. (where f e; ; u; or d), we find that 1 2 B. Axial vector contributions GPS mW 2 ; (13) 2 One-loop radiative corrections also contribute to the G 8 sW MSUSY axial vector amplitude. The total amplitude can be written where the superscript ‘‘(1)’’ denotes one loop-induced as pseudoscalar interaction. We have verified by explicit 5
computation that the O contributions vanish. The reason iMAV Vudf G m‘u 1 v‘ 1 r^ r^ ; is that in each pair of incoming quarks or outgoing leptons (15) the two fermions must have opposite chirality in order to where r^ and r^ denote one-loop contributions to the contribute to G 1 . Since CC interactions in the MSSM are PS semileptonic and -decay amplitudes, respectively, and purely left-handed, the chirality must change at least twice where the hat indicates quantities renormalized in the in each graph, with each flip generating a factor of .For modified dimensional reduction (DR) scheme. Since r^ example, we show one pseudoscalar contribution in Fig. 2 2 cancels from Re= , we concentrate on the SUSY contribu- that is proportional to y yd. Here, the chirality tions to r^ that do not cancel from R . It is helpful to changes at the ud~H~ and ~ H~ vertices. Potentially, this e= particular contribution can be enhanced for large tan ; distinguish various classes of contributions however, to satisfy (9), we need SUSY ‘ ‘ q q r^ ; (16) L V L V B GB 3 3 MSUSY ‘ q ‘ q tan * 10 : (14) where L ( L), V ( V ), B, and GB denote leptonic 100 GeV (hadronic) external leg [Fig. 1(b)], leptonic (hadronic) vertex [Fig. 1(c)], box graph [Fig. 1(d)], and gauge boson propagator contributions, respectively. The corrections q L;V and GB cancel from Re= , so we do not discuss them further (we henceforth omit the ‘‘‘’’ superscript). The explicit general formulas for L;V;B, calculated in DR, are given in the appendix. We have verified that L and V agree with Ref. [24] for the case of a pure SU 2 L chargino/ neutralino sector. At face value, it appears from Eqs. (A7)–(A9) that SUSY Re= carries a nontrivial dependence on MSSM pa- rameters since the SUSY masses enter both explicitly in the loop functions and implicitly in the mixing matrices Z, defined in Eqs. (A1)–(A6). Nevertheless, we are able to identify a relatively simple dependence on the SUSY 1 2 FIG. 2. This contribution to GPS is suppressed by y yd. spectrum.
095017-3 MICHAEL J. RAMSEY-MUSOLF, SHUFANG SU, AND SEAN TULIN PHYSICAL REVIEW D 76, 095017 (2007) SUSY We first consider Re= in a limiting case obtained scheme cancels in Re= . [In addition, this scheme- with three simplifying assumptions: (1) no flavor mixing dependent constant enters into the extraction of G ; hence, among scalar superpartners; (2) no mixing between left- the individual decay widths ! ‘ ‘ are also indepen- and right-handed scalar superpartners; and (3) degeneracy dent of the renormalization scheme.] ~ between ‘L and ~‘ and no gaugino-Higgsino mixing. Our The reason for this simplification is that under our first assumption is well justified; experimental bounds on assumptions, we have effectively taken a limit that is flavor violating processes constrain the contributions to equivalent to computing the one-loop corrections in the Re= from lepton flavor violation in the slepton soft- absence of electroweak symmetry breaking. In the limit of breaking sector to be less than the sensitivities at upcoming unbroken SU 2 L U 1 Y, the one-loop SUSY vertex and experiments by a factor of 10 20 [22]. external leg corrections sum to a universal constant which Our second assumption has minimal impact. In the is renormalization scheme dependent, but renormalization absence of flavor mixing, the charged slepton mass matrix scale independent [24]. [For unbroken SU 2 L, the SM decomposes into three 2 2 blocks; thus, for flavor ‘, the vertex and external leg corrections yield an additional ~ ~ logarithmic scale dependence; hence, the SU 2 mass matrix in the f‘L; ‘Rg basis is L ! -function receives contributions from both charge and M2 s2 1 m2 cos2 m a‘ tan L W 2 Z ‘ y‘ wave function renormalization.] In addition, virtual ; a‘ 2 2 2 Higgsino contributions are negligible, since their interac- m‘ y tan MR sWmZ cos2 ‘ tions are suppressed by small first and second generation 2 2 where ML (MR) is the SUSY-breaking mass parameter for Yukawa couplings. Setting all external momenta to zero left-handed (right-handed) sleptons, a‘ is the coefficient and working in the limit of unbroken SU 2 L symmetry, we for the SUSY-breaking triscalar interaction, y‘ is the find that the Higgsino contributions to L V are Yukawa coupling, and is the Higgsino mass parameter. 2 2 y‘=32 . Under particular models of SUSY-breaking mediation, it is In this illustrative limit, the only nonzero contributions usually assumed that a =y M , and thus left-right SUSY ‘ ‘ SUSY to Re= come from two classes of box graphs mixing is negligible for the first two generations due to the [Fig. 1(d)]—one involving purely wino-like interactions smallness of me and m . Of course, a‘ could be signifi- and the other with both a virtual wino and bino. The sum of cantly larger and induce significant left-right mixing [6]. these graphs is For reasons discussed above, we neglect this possibility. 2 We have adopted the third assumption for purely illus- ‘ mW 2 F x ;x t F x ;x ;x trative purposes; we will relax it shortly. Clearly, fermions B 2 2 1 L Q W 2 B L Q 12 sW M2 of the same weak isospin doublet are not degenerate; their (19) masses obey 2 2 2 2 where we have defined m~ m ~ mW cos2 m‘ (17) ‘L ‘ 2 2 2 2 2 3 xL xL 2 lnxL
m m m cos2 m m : (18) F x ;x d~ u~L W d u 1 L Q 2 L 2 xL xQ 1 xL In addition, gaugino mixing is certainly always present, as x x 2 lnx 1 the gaugino mass matrices contain off-diagonal elements Q Q Q x x 1 x 2 1 x 1 x proportional to mZ [see Eqs. (A2) and (A4)]. However, the Q L Q L Q third assumption becomes valid for MSUSY mZ. (20) Under our three assumptions, the SUSY vertex and external leg corrections sum to a constant that is indepen- and dent of the superpartner masses, leading to considerable p 1 xB xB 2 xB lnxB simplifications. The bino [U 1 gaugino] vertex and ex- F x ;x ;x Y 2 B L Q 2 1 xB xB xL xB xQ ternal leg corrections exactly cancel. The wino [SU 2 L p gaugino] vertex and leg corrections do not cancel; rather, xL xL 2 xB lnxL =4 s2 , a constant that carries no depen- V L W 1 xL xL xB xL xQ dence on the slepton, gaugino, or Higgsino mass parame- p x x 2 x lnx ters. The occurrence of this constant is merely an artifact of Q Q B Q ; (21) our use of the DR renormalization scheme. (In comparison, 1 xQ xQ xL xQ xB in modified minimal subtraction, we find V L 0 in 2 2 2 2 2 2 1 where x M =M , x m =M , and x m =M , this same limit. ) This dependence on the renormalization B 1 2 L ‘~ 2 Q Q~ 2 with masses M1, M2, m‘~, and mQ~ of the bino, wino, left- 1Technically, since MS breaks SUSY, it is not the preferred handed ‘-flavored slepton, and left-handed 1st generation renormalization scheme for the MSSM. However, this aspect is squark, respectively. Numerically, we find that always not important in the present calculation. F1 F2; the reason is that the sum of bino-wino graphs
095017-4 PION LEPTONIC DECAYS AND SUPERSYMMETRY PHYSICAL REVIEW D 76, 095017 (2007)
3.5 f ’ 0:001f: (24) V L 4 s2 8 s2 3 W W 2.5 The factor f measures the departure of V L from Q universality. If the SUSY spectrum is such that our third ,x
L 2 assumption is valid, we expect f ! 0. For realistic values x xQ 1 0.2 of the SUSY parameters, two effects lead to a nonvanishing
F 1.5 f xQ : (a) splitting between the masses of the charged and 1 1.0 neutral left-handed sleptons that results from breaking of xQ 5.0 0.5 SU 2 L, and (b) gaugino-Higgsino mixing. The former effect is typically negligible. To see why, we recall from 0 1 2 3 4 5 Eq. (18) that xL 2 mW m~ m 1 O ; (25) FIG. 3. The box graph loop function F x ;x as a function ‘ ~‘ m2 1 L Q ‘~ x m2 =M2 x m2 =M2 x of L L~ 2 for several values of Q Q~ 2. For L where we have neglected the small nondegeneracy propor- xQ 1 (i.e. SUSY masses degenerate), F1 xL;xQ 1.For xL 1 or xQ 1 (i.e. very massive sleptons or squarks), tional to the square of the lepton Yukawa coupling. We find F1 xL;xQ !0. that the leading contribution to f from this nondegeneracy is at least m4 =m4 , which is & 0:1 for m * 2M . O W ‘~ ‘~ W Significant gaugino mixing can induce f O 1 . The tend to cancel, while the sum of pure wino graphs all add crucial point is that the size of f from gaugino mixing is coherently. Hence, bino exchange (through which the term governed by the size of M2.IfM2 mZ, then the wino proportional to F2 arises) does not significantly contribute SUSY decouples from the bino and Higgsino, and contributions to to Re= . V L approach the case of unbroken SU 2 L. On the In Fig. 3, we show F x ;x as a function of x for 1 L Q L other hand, if M2 mZ, then V L can differ substan- fixed x . Since F is symmetric under x $ x , Fig. 3 also 2 Q 1 L Q tially from =4 sW. shows F1 as a function of xQ, and hence how B depends In the limit that m~ M (‘ e, ), we also have a ‘L 2 on mu~L .ForxL, xQ 1, we have F1 O 1 , while if either decoupling scenario where B 0, V L 4 s2 , and x 1 or x 1, then F ! 0, which corresponds to the W L Q 1 thus f 0. Hence, a significant contribution to R decoupling of heavy sleptons or squarks. There is no e= requires at least one light slepton. However, regardless of enhancement of for x 1 or x 1 (i.e. if M is B L Q 2 the magnitude of f,ifm m , then these corrections very heavy) due to the overall 1=M2 suppression in (19). e~L ~L 2 will cancel from R . The total box graph contribution is e= It is instructive to consider the dependence of individual SUSY SUSY contributions B and V L to Re= , as shown in Re= e
2Re RSM B B e= 4 2 2 2 2 m 2 m 4 mW me~ Q~ m ~ Q~ ’ F1 ; F1 ; : 2
2 2 2 2 2 10 6 sW M2 M2 M2 M2 M2 0 (22) µ e
R 2 SUSY Clearly Re= vanishes if both sleptons are degenerate µ 4 SUSY and is largest when they are far from degeneracy, such that e Total
R Vert. Leg me~L m ~ L or me~L m ~ L . In the latter case, we have 6 Box only SUSY 8 Re= 100 GeV 2 100 150 200 300 500 700 1000
& 0:001 (23) SM µ GeV Re= MSUSY SUSY FIG. 4. Re= versus , with fixed parameters M1 for e.g. MSUSY M2 mu~ me~ m ~ . L L L 100 GeV, M2 150 GeV, me~L 100 GeV, m ~L 500 GeV, We now relax our third assumption to allow for gaugino- mu~ 200 GeV. The thin solid line denotes contributions from ~ L Higgsino mixing and nondegeneracy of ‘ and ~‘. Both of ( V L) only; the dashed line denotes contributions from B these effects tend to spoil the universality of V L, only; the thick solid line shows the sum of both contributions to SUSY giving Re= .
095017-5 MICHAEL J. RAMSEY-MUSOLF, SHUFANG SU, AND SEAN TULIN PHYSICAL REVIEW D 76, 095017 (2007)
4 0:001.
4 (ii) Both the vertex leg and box contributions are 2 largest if M m and vanish if M m .If 10 2 O Z 2 Z M2 O mZ , then at least one chargino must be µ 0
e light. R (iii) The contributions to RSUSY vanish if m m 2 e= e~L ~L
µ and are largest if either m ~L me~L or m ~ L me~L . SUSY e 4 Total SUSY R (iv) The contributions to R are largest if e~ or Vert. Leg e= L ~L 6 Box only is O mZ . (v) If mZ, then the lack of gaugino-Higgsino mix- 100 150 200 300 500 700 1000 ing suppresses the V L contributions to M GeV SUSY 2 Re= . (vi) If mu~ , m~ mZ, then the B contributions to RSUSY=RSM M L dL FIG. 5. e= e= as a function of 2, with 200 GeV SUSY and all other parameters fixed as in Fig. 4. Each line shows the Re= are suppressed due to squark decoupling. ~ contribution indicated as in the caption of Fig. 4. (vii) If u~L, dL, and are all O mZ , then there may be cancellations between the V L and B con- tributions. RSUSY is largest if it is dominated by Figs. 4 and 5. In Fig. 4, we plot the various contributions e= either or contributions. as a function of the supersymmetric mass parameter , V L B We now study RSUSY quantitatively by making a nu- e= with M1 100 GeV, M2 150 GeV, me~L 100 GeV, m 500 GeV, m 200 GeV. We see that the merical scan over MSSM parameter space, using the fol- ~ L u~L V lowing ranges: L contributions (thin solid line) vanish for large , since
in this regime gaugino-Higgsino mixing is suppressed and f j j j j g SUSY mZ=2 < M1; M2 ; ;mu~L < 1 TeV there is no V L contribution to Re= . However, the m =2 < fm ;m g < 5 TeV B contribution (dashed line) is nearly independent, Z ~e ~ (26) since box graphs with Higgsino exchange (which depend 1 < tan <50 on ) are suppressed in comparison to those with only gaugino exchange. In Fig. 5, we plot these contributions as sign ; sign M2 1; a function of M2, with 200 GeV and all other pa- where me~ , m ~ , and m~ are determined from Eqs. (17) rameters fixed as above. We see that both V L and B L L dL and (18). contributions vanish for large M2. One general feature observed from these plots is that Direct collider searches impose some constraints on the parameter space. Although the detailed nature of these V L and B contributions tend to cancel one another; therefore, the largest total contribution to RSUSY occurs constraints depends on the adoption of various assumptions e= and on interdependencies on the nature of the MSSM and when either or is suppressed in comparison to V L B its spectrum [17], we implement them in a coarse way in the other. This can occur in the following ways: (1) if order to identify the general trends in corrections to R . m , then may be large, while is suppressed, e= Z B V L First, we include only parameter points in which there are and (2) if mu~ , m~ mZ, then V L may be large, L dL no SUSY masses lighter than m =2. (However, the current while is suppressed. In Fig. 5, we have chosen parame- Z B bound on the mass of lightest neutralino is even weaker ters for which there is a large cancellation between V than this.) Second, parameter points which have no L and B. However, by taking the limits !1 or SUSY charged SUSY particles lighter than 103 GeV are said to mu~ ;m~ !1, R would coincide with the B or L dL e= satisfy the ‘‘LEP II bound.’’ (This bound may also be V L contributions, respectively. weaker, in particular, regions of parameter space.) Because the V L and B contributions tend to Additional constraints arise from precision electroweak cancel, it is impossible to determine whether e~L or ~ L is data. We consider only MSSM parameter points whose heavier from Re= measurements alone. For example, a contributions to the oblique parameters S, T, and U [25] positive deviation in Re= can result from two scenarios: agree with electroweak precision observables (EWPO). A SUSY (1) Re= is dominated by box graph contributions with recent fit to both high- and low-energy EWPO using the SUSY value of m 170:9 1:8 GeV [26] has been reported in me~
095017-6 PION LEPTONIC DECAYS AND SUPERSYMMETRY PHYSICAL REVIEW D 76, 095017 (2007) where the errors quoted are 1 standard deviation and where 0.005 the value of the standard model Higgs boson mass has been set to the LEP lower bound mh 114:4 GeV. Using the correlation matrix given in Ref. [27] and the computation 0.002 of superpartner contributions to the oblique parameters
Μ
reported in Ref. [7], we determine the points in the e MSSM parameter space that are consistent with EWPO R 0.001 at 95% confidence. Because we have neglected the 3rd generation and right-handed scalar sectors in our analysis Μ SUSY e and parameter scan, we do not calculate the entire MSSM R 0.0005 contributions to S, T, and U. 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 0.0002 contributions to S, T, and U from the remaining scalar superpartners (that we neglect) only cause further devia- 100 150 200 300 500 700 1000 tions from the measured values of the oblique parameters. Min m ,mµ GeV In addition, we have assumed that the lightest CP-even eL L Higgs mass is the same as the SM Higgs mass reference FIG. 7. RSUSY as a function of Min[m , m ], the mass of e= e~L ~L point: mh 114:4 GeV, neglecting the corrections due to the lightest first or second generation charged slepton. The dark the small mass difference, and the typically small contri- and light regions denote the regions of MSSM parameter space butions from the remaining heavier Higgs bosons. consistent and inconsistent, respectively, with the LEP II bound. 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 points within our scan consistent with the LEP II bound, measurement of the muon anomalous magnetic moment while the light regions contain all MSSM points inconsis- [28] tend to favor a positive sign for the parameter and tent with the LEP II bound, but with no superpartners relatively large values of tan . Eliminating the points with lighter than mZ=2. In effect, the dark (light) regions show how large RSUSY=R can be, assuming consistency sign 1 will exclude half the parameter space in our e= e= scan, but the general trends are unaffected. (inconsistency) with the LEP II bound, as a function of a SUSY We show the results of our numerical scan in Figs. 6–9. given parameter. In Fig. 6, we show Re= =Re= as a In Figs. 6–8, the dark regions contain all MSSM parameter function of the ratio of slepton masses me~L =m ~ L . If both
0.005 0.005
0.002 0.002 µ
µ e e R R 0.001 0.001
µ
µ SUSY e SUSY e
R 0.0005 R 0.0005
0.0002 0.0002
100 150 200 300 500 700 1000 0.1 1 10 mχ1 GeV µ meL m L SUSY FIG. 8. Re= versus m 1, the mass of the lightest chargino. RSUSY m =m FIG. 6. e= as a function of the ratio e~L ~L . The dark The dark and light regions denote the regions of MSSM parame- and light regions denote the regions of MSSM parameter space ter space consistent and inconsistent, respectively, with the consistent and inconsistent, respectively, with the LEP II bound. LEP II bound.
095017-7 MICHAEL J. RAMSEY-MUSOLF, SHUFANG SU, AND SEAN TULIN PHYSICAL REVIEW D 76, 095017 (2007) parameter points which satisfy the LEP II bound. The solid shaded areas correspond to regions of the j j-m plane 1000 u~L SUSY where the largest value of Re= lies within the indicated SUSY 700 ranges. It is clear that Re= can be largest in the regions
where either (1) is small, mu~L is large, and the largest 500 4 contributions to RSUSY are from , or (2) is 10 e= V L 2 m RSUSY µ large, u~L is small, and the largest contribution to e= δ Re 4 GeV 10 is from . If both and m are light, then RSUSY can 300 5 B u~L e= e µ µ 4 R 10 4 still be very small due to cancellations, even though both 2 10 µ 7 V L and B contributions are large individually. 200 4 Re 5 10 3 More precisely, to satisfy (28), we need either & 1 10 250 GeV,or * 300 GeV and m & 200 GeV. 150 4 Re µ u~L 10 7 3 1 10 100 δRe µ III. CONTRIBUTIONS FROM R-PARITY- VIOLATING PROCESSES 100 150 200 300 500 700 1000 In the presence of RPV interactions, tree-level ex- mu L GeV changes of sfermions (shown in Fig. 10), lead to violations SUSY of lepton universality and nonvanishing effects in Re= . FIG. 9. Contours indicate the largest values of Re= ob- tained by our numerical parameter scan (26), as a function of The magnitude of these tree-level contributions is gov- j j erned by both the sfermion masses and by the parameters and mu~L . The solid shaded regions correspond to the largest SUSY 0 and 0 that are the coefficients in RPV interactions: values of Re= within the ranges indicated. All values of 11k 21k RSUSY e= correspond to parameter points which satisfy the 0 ~ y L L Q d ... (29) LEP II bound. RPV; L 1 ijk i j k Defining [29,30] sleptons are degenerate, then RSUSY vanishes. Assuming e= j 0 j2 the LEP II bound, in order for a deviation in R to match 0 ~ ijk e= f p 0; (30) ijk G m2 the target precision at upcoming experiments, we must 4 2 f~ have
SUSY contributions to Re= from RPV interactions are Re= j Re= =Re= j * 0:0005; (28) RPV Re= and thus me~L =m ~ L * 2 or m ~L =me~L * 2. (This result is 0 0 SM 2 11k 2 21k: (31) consistent with an earlier analysis [22], where the authors Re= RSUSY m conclude that e= would be unobservably small if eL and m L differ by less than 10%.) Note that RPV contribution to the muon lifetime (and, thus, SUSY In Fig. 7, we show Re= =Re= as a function of the Fermi constant G ) cancels in Re= , therefore does not Min[me~ , m ~ ], the mass lightest first or second generation enter the expression. L L 0 slepton. If the lighter slepton is extremely heavy, then both The quantities ijk etc. are constrained by existing SUSY heavy sleptons decouple, causing Re= to vanish. precision measurements and rare decays. A summary of Assuming the LEP II bound, to satisfy (28), we must the low-energy constraints is given in Table III of Ref. [10], which includes tests of CKM unitarity (primarily through have me~L & 300 GeV or m ~ L & 300 GeV. SUSY RPV effects in superallowed nuclear -decay that yields a In Fig. 8, we show Re= =Re= as a function of m 1, SUSY precise value of jVudj [31]), atomic parity-violating (PV) the lightest chargino mass. If m 1 is large, Re= van- Cs measurements of the cesium weak charge QW [32], the ishes because M2 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 (28), we must have m 1 & 250 GeV. Finally, we illustrate the interplay between V L SUSY and B by considering Re= as a function of j j and m RSUSY u~L . In Fig. 9, we show the largest values of e= obtained in our numerical parameter scan, restricting to FIG. 10. Tree-level RPV contributions to Re= .
095017-8 PION LEPTONIC DECAYS AND SUPERSYMMETRY PHYSICAL REVIEW D 76, 095017 (2007) ratio Re= itself [15,16], a comparison of the Fermi con- loop corrections discussed above. On the other hand, stant G with the appropriate combination of , mZ, and agreement of Re= with the SM would lead to considerable 2 sin W [33], and the electron weak charge determined from tightening of the constraints on this scenario, particularly 0 SLAC E158 measurement of parity-violating Møller scat- in the case of 21k, which is currently constrained only by tering [34]. Re= and deep inelastic ( ) scattering [36]. In Fig. 11 we show the present 95% C.L. constraints on The presence of RPV interactions would have significant 0 0 the quantities 11k and 21k obtained from the aforemen- implications for both neutrino physics and cosmology. It tioned observables [interior of the blue (dark gray) curve]. has long been known, for example, that the existence of 0 Since the ijk are positive semidefinite quantities, only the L 1 interactions—such as those that could enter region of the contour in the upper right-hand quadrant are Re= —will induce a Majorana neutrino mass [37], while shown. The green (light gray) curve indicates the possible the presence of nonvanishing RPV couplings would imply implication of a future measurement of the proton weak that the lightest supersymmetric particle is unstable and, charge planned at Jefferson Lab [35], assuming agreement therefore, not a viable candidate for cold dark matter. The with the standard model prediction for this quantity and the future measurements of Re= could lead to substantially anticipated experimental uncertainty. The dashed red tighter constraints on these possibilities or uncover a pos- (gray) curve shows the possible impact of future measure- sible indication of RPV effects. In addition, we note that ments of Re= , assuming agreement with the present cen- the present uncertainty associated with RPV effects enter- tral value but an overall error reduced to the level ing the 2 decay rate would affect the value of F at a anticipated in Ref. [18]; with the error anticipated in level of about half the theoretical SM uncertainty as esti- Ref. [19] the width of the band would be a factor of 2 mated by Ref. [12]. smaller than shown. Two general observations emerge from Fig. 11. First, 0 0 IV. CONCLUSIONS given the present constraints, values of 21k and 11k differing substantially from zero are allowed. For values Given the prospect of two new studies of lepton univer- of these quantities inside the blue (dark gray) contour, sality in ‘2 decays [18,19] with experimental errors that SUSY Re= could differ from zero by up to 5 standard devia- are substantially smaller than for existing measurements 4 tions for the error anticipated in Ref. [18]. Such RPV and possibly approaching the 5 10 level, an analysis effects could, thus, be considerably larger than the SUSY of the possible implications for supersymmetry is a timely exercise. In this study, we have considered SUSYeffects on the ratio Re= in the MSSM both with and without R-parity 0.01 violation. Our results indicate that in the R-parity conserv- ing case, effects from SUSY loops can be of order the planned experimental error, in particular, limited regions of 0.008 the MSSM parameter space. Specifically, we find that a deviation in Re= due to the MSSM at the level of 0.006 SUSY Re=
′ 21k 0:0005 & & 0:001; (32) ∆ Re= 0.004 implies (1) the lightest chargino 1 is sufficiently light
0.002 m 1 & 250 GeV;
(2) the left-handed selectron e~L and smuon ~L are highly 0 nondegenerate: 0246′ ∆ −3 me~ 11k x 10 L me~L 1
* 2 or & ; m ~ m 2 FIG. 11 (color online). Present 95% C.L. constraints on RPV L ~L parameters 0 , j 1, 2 that enter R obtained from a fit to j1k e= (3) at least one of e~L or ~ L must be light, such that precision electroweak observables. Interior of the dark blue (dark gray) contour corresponds to the fit using the current value of me~L & 300 GeV or m ~L & 300 GeV; SM Re= =Re= [15,16], while the dashed red (gray) contour cor- responds to the fit using the future expected experimental pre- and (4) the Higgsino mass parameter and left-handed up cision [18], assuming the same central value. The light green squark mass mu~L satisfy either (light gray) curve indicates prospective impact of a future measurement of the proton weak charge at Jefferson Lab [35]. j j & 250 GeV
095017-9 MICHAEL J. RAMSEY-MUSOLF, SHUFANG SU, AND SEAN TULIN PHYSICAL REVIEW D 76, 095017 (2007) SM or measurements with Re= could yield significant new con- straints on these possibilities.
j j * 300 GeV;mu~L & 200 GeV: ACKNOWLEDGMENTS Under these conditions, the magnitude RSUSY may fall e= We would like to thank M. Wise for useful discussions. within the sensitivity of the new R measurements. e= M. R. M. and S. T. are supported under U.S Department of In conventional scenarios for SUSY-breaking mediation, Energy Contract No. DE-FG02-05ER41361 and NSF one expects the left-handed slepton masses to be compa- No. PHY-0555674. S. S. is supported under U.S. rable, implying no substantial corrections to SM predic- Department of Energy Contract No. DE-FG02-04ER- tions for Re= . Significant reductions in both experimental 41298. error and theoretical, hadronic physics uncertainties in SM Re= would be needed to make this ratio an effective probe APPENDIX: GENERAL RADIATIVE of the superpartner spectrum. CORRECTIONS IN THE MSSM On the other hand, constraints from existing precision electroweak measurements leave considerable latitude for The MSSM Lagrangian and Feynman rules [38] are observable effects from tree-level superpartner exchange in expressed in terms of chargino and neutralino mixing the presence of RPV interactions. The existence of such matrices Z and ZN, respectively, which diagonalize the superpartner mass matrices, defined as follows. The four effects would have important consequences for both neu- 0 trino physics and cosmology, as the presence of the L neutralino mass eigenstates i are related to the gauge 0 ~ ~ 3 ~0 ~0 0 RPV interactions would induce a Majorana mass term for eigenstates B; W ; Hd; Hu by the neutrino and allow the lightest superpartner to decay to 0 ij 0 Z ; (A1) SM particles too rapidly to make it a viable dark matter i N j candidate. Agreement between the results of the new Re= where
0 1 0 1 M1 0 c sWmZ s sWmZ m 0 000 B C B 1 C B 0 M c c m s c m C B 0 m 0 00C T B 2 W Z W Z C B 2 C ZN ZN (A2) @ c s m c c m 0 A @ 00m 0 0 A W Z W Z 3 s s m s c m 0 000m 0 W Z W Z 4 is the diagonalized neutralino mass matrix. The chargino is the diagonalized slepton mass matrix. There are two 2 mass eigenstates i are related to the gauge eigenstates classes of off-diagonal elements in M~ which can contrib- ~ ~ ~ ~ ‘ W ; Hu and W ; Hd by ute to slepton mixing: mixing between flavors and mixing ij between left- and right-handed components of a given
Z ; (A3) i j flavor, both of which arise through SUSY-breaking terms. where (Left-right mixing due to SUSY-preserving terms will be p ! suppressed by m‘=m‘~ and is irrelevant for the first two M 2s m m 0 T 2 W 1 generations.) Z p Z 0 m 2c mW 2 Similarly, up-type squarks, down-type squarks, and Z Z Z (A4) sneutrinos have mixing matrices U, D, and , respec- tively, defined identically to ZL —except for the fact that is the diagonalized chargino mass matrix. We note that the there are no right-handed sneutrinos in the MSSM and thus off-diagonal elements which mix gauginos and Higgsinos there are only three sneutrino mass eigenstates. stem solely from electroweak symmetry breaking. There are three types of contributions to RSUSY in the ~ e= The charged slepton mass eigenstates Li are related to MSSM: external leg, vertex, and box graph radiative cor- ~ the gauge eigenstates ‘ e~L; ~ L; ~L; e~R; ~R; ~R by rections. The leptonic external leg corrections [Fig. 1(b)] ~ ij are ‘ i Z L~j; (A5) L i 1j 2j 2 L jZN tW ZN j B m 0 ;m ~ where 16 s2 j i 0 1 W 2 1k 2 1j 2j 2 m 0 j j j j 0 L~ 2 Z B m k ;mL~ ZN tW ZN B m ;mL~ B 1 C i j i y 2 B . C 1k 2 ZLM~ZL . (A6) j j ‘ @ . A 2 Z B m k ;m ~i ; (A7) 2 0 m ~ L6 where the loop function is [39]
095017-10 PION LEPTONIC DECAYS AND SUPERSYMMETRY PHYSICAL REVIEW D 76, 095017 (2007) Z 1 M2 B m1;m2 dxx ln 2 2 : 0 m1 1 x m2x The leptonic vertex corrections [Fig. 1(c)] are
I 1j 2j 1j 2j V ZN tW ZN ZN tW ZN C2 m ~ ;m 0 ;mL~ 8 s2 i j i W 2j 1j 1k 2j 1k 1 4j 2k 2 Z t Z Z Z Z p Z Z C m 0 ;m ;m N W N N N 2 j ~i k 2 2j 1k 1 3j 2k Z Z p Z Z m 0 m C m 0 ;m ;m N N j k 1 j ~i k 2 2j 1j 1k 2j 1k 1 3j 2k 2 Z t Z Z Z Z p Z Z C m ;m~ ;m 0 N W N N N 2 k Li j 2 1 2j 1k 4j 2k ZN Z p ZN Z m 0 m C1 m ;mL~ ;m 0 ; (A8) 2 j k k i j with loop functions Z 1 1 C1 m1;m2;m3 dxdy 2 2 2 0 m1x m2y m3 1 x y Z 1 M2 C2 m1;m2;m3 dxdy ln 2 2 2 : 0 m1x m2y m3 1 x y The corrections from box graphs [Fig. 1(d)] are
2 I mW 1k 2 2m 1m 2m 1 1m jZ j Z tWZ Z tWZ D1 m 0 ;m~ ;m ;m~ B 8 s2 N N N 3 N m dL k Li W 1j 2 2m 1m 2m 1 1m jZ j Z t Z Z t Z D m ;m ;m 0 ;m N W N N W N 1 j u~L m ~i 3 1j 1j 2m 1m 2m 1 1m Z Z Z t Z Z t Z m 0 m D m 0 ;m~ ;m ;m N W N N W N m j 2 m dL j ~i 3 1k 1k 2m 1m 2m 1 1m Z Z Z t Z Z t Z m 0 m D m ;m ;m 0 ;m~ ; (A9) N W N N 3 W N m k 2 k u~L m Li with loop functions Z 1 1 Dn m1;m2;m3;m4 dxdydz 2 2 2 2 n : 0 m1x m2y m3z m4 1 x y z
In formulas (A7)–(A9), I 1 corresponds to ! e e and I 2 corresponds to ! . All other indices are summed over. We use DR renormalization at scale M. We have defined tW tan W and sW sin W. We have neglected terms proportional to either Yukawa couplings or external momenta [which will be suppressed by O m =MSUSY ]. Finally, the SUSY contribution to Re= is
SUSY Re= 1 2 1 2 1 2 2Re V V L L B B : (A10) Re=
In addition, the following are some useful formulas needed to show the cancellations of vertex and leg corrections in the limit of no superpartner mixing:
2 C2 m1;m2;m1 B m2;m1 2m1C1 m1;m2;m1 2B m1;m2 2B m2;m1 1:
095017-11 MICHAEL J. RAMSEY-MUSOLF, SHUFANG SU, AND SEAN TULIN PHYSICAL REVIEW D 76, 095017 (2007) [1] J. Erler and M. J. Ramsey-Musolf, Prog. Part. Nucl. Phys. [21] B. A. Campbell and D. W. Maybury, Nucl. Phys. B709, 54, 351 (2005). 419 (2005). [2] M. J. Ramsey-Musolf, AIP Conf. Proc. 842, 661 [22] A. Masiero, P. Paradisi, and R. Petronzio, Phys. Rev. D 74, (2006). 011701 (2006). [3] M. J. Ramsey-Musolf, arXiv:nucl-th/0608035. [23] P. Herczeg, Phys. Rev. D 52, 3949 (1995). [4] S. P. Martin and J. D. Wells, Phys. Rev. D 64, 035003 [24] E. Katz, L. Randall, and S. f. Su, Nucl. Phys. B536,3 (2001). (1998). [5] A. Kurylov and M. J. Ramsey-Musolf, Phys. Rev. Lett. 88, [25] M. E. Peskin and T. Takeuchi, Phys. Rev. Lett. 65, 964 071804 (2002). (1990). [6] S. Profumo, M. J. Ramsey-Musolf, and S. Tulin, [26] CDF Collaboration, arXiv:hep-ex/0703034. arXiv:hep-ph/0608064 [Phys. Rev. D (to be published)]. [27] S. Profumo, M. J. Ramsey-Musolf, and G. Shaughnessy, [7] A. Kurylov, M. J. Ramsey-Musolf, and S. Su, Phys. Rev. D J. High Energy Phys. 08 (2007) 010. 68, 035008 (2003). [28] G. W. Bennett et al. (Muon G-2 Collaboration), Phys. Rev. [8] M. Pospelov and A. Ritz, Ann. Phys. (N.Y.) 318, 119 D 73, 072003 (2006). (2005). [29] M. J. Ramsey-Musolf, Phys. Rev. D 62, 056009 (2000). [9] J. Hisano, T. Moroi, K. Tobe, and M. Yamaguchi, Phys. [30] V.D. Barger and K. m. Cheung, Phys. Lett. B 480, 149 Rev. D 53, 2442 (1996). (2000). [10] M. J. Ramsey-Musolf and S. Su, arXiv:hep-ph/0612057. [31] J. C. Hardy and I. S. Towner, Phys. Rev. C 71, 055501 [11] D. A. Bryman, Comments Nucl. Part. Phys. 21, 101 (2005). (1993). [32] S. C. Bennett and C. E. Wieman, Phys. Rev. Lett. 82, 2484 [12] W. J. Marciano and A. Sirlin, Phys. Rev. Lett. 71, 3629 (1999); C. S. Wood et al., Science 275, 1759 (1997). (1993). [33] W. J. Marciano, Phys. Rev. D 60, 093006 (1999). [13] M. Finkemeier, Phys. Lett. B 387, 391 (1996). [34] P.L. Anthony et al. (SLAC E158 Collaboration), Phys. [14] V. Cirigliano and I. Rosell, arXiv:0707.3439; J. High Rev. Lett. 95, 081601 (2005). Energy Phys. 10 (2007) 005. [35] R. Carlini, J. M. Finn, S. Kowalski, and S. Page, spokes- [15] D. I. Britton et al., Phys. Rev. D 46, R885 (1992). persons, JLab Experiment E-02-020. [16] G. Czapek et al., Phys. Rev. Lett. 70, 17 (1993). [36] G. P. Zeller et al. (NuTeV Collaboration), Phys. Rev. Lett. [17] W. M. Yao et al. (Particle Data Group), J. Phys. G 33,1 88, 091802 (2002); 90, 239902(E) (2003). (2006). [37] J. Schechter and J. W. F. Valle, Phys. Rev. D 22, 2227 [18] T. Numao and D. A. Bryman (PIENU Collaboration), (1980). future experiment, https://www.pienu.triumpf.ca. [38] J. Rosiek, Phys. Rev. D 41, 3464 (1990); J. Rosiek, [19] D. Pocˇanic´ and A. van der Shaaf (PEN Collaboration), arXiv:hep-ph/9511250. future experiment, http://pen.phys.virginia.edu/. [39] G. Passarino and M. J. G. Veltman, Nucl. Phys. B160, 151 [20] S. P. Martin, arXiv:hep-ph/9709356. (1979).
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