Constraining the MSSM Sfermion Mass Matrices with Light Fermion Masses

TTP10-11

Constraining the MSSM sfermion mass matrices with light fermion masses

Andreas Crivellin and Jennifer Girrbach Institut für Theoretische Teilchenphysik Karlsruhe Institute of Technology, Universität Karlsruhe, D-76128 Karlsruhe, Germany (Dated: February 2010) We study the finite supersymmetric loop corrections to fermion masses and mixing matrices in the generic MSSM. In this context the effects of non-decoupling chirally-enhanced self-energies are studied beyond leading order in perturbation theory. These NLO corrections are not only necessary for the renormalization of the CKM matrix to be unitary, they are also numerically important for the light fermion masses. Focusing on the tri-linear A-terms with generic flavor-structure we f LR,RL derive very strong bounds on the chirality-changing mass insertions δIJ by applying ’t Hooft’s naturalness criterion. In particular, the NLO corrections to the up quark mass allow us to constrain u RL u LR the unbounded element δ13 if at the same time δ13 is unequal to zero. Our result is important for single-top production at the LHC.

PACS numbers: 11.10.Gh,12.15.Ff,12.60.Jv,14.80.Ly

I. INTRODUCTION tion as:

2 IJ f XY ∆mF XY A major challenge in particle physics is to understand δIJ = . (1) m2 m 2 the pattern of fermion masses and mixing angles. With f˜IX f˜JY the discovery of neutrino oscillations flavor has become q even more mysterious since the nearly tri-bimaximal mix- In Eq. (1) I and J are flavour indices running from 1 to ing strongly differ from the quark sector. The mini- 2 IJ 3, X, Y denote the chiralities L and R, ∆mF XY with mal supersymmetric standard model (MSSM) does not F = U,D,L is the off-diagonal element of the sfermion 2 provide insight into the flavor problem by contrast the mass matrix (see Appendix A 2) and m ˜2 , m are the fIX f˜JY generic MSSM contains even new sources of flavor and corresponding diagonal ones. In this article we are going chirality violation, stemming from the supersymmetry- to complement the analysis of [14] with respect to three breaking sector which are the sources of the so-called su- important points: persymmetric flavor problem. The origin of these flavor- violating terms is obvious: In the standard model (SM) Electroweak correction are taken into account. the quark and lepton Yukawa matrices are diagonalized • Therefore, we are able to constrain also the flavor- by unitary rotations in flavor space and the resulting ba- violating and chirality-changing terms in the lepton sis defines the mass eigenstates. If the same rotations sector. are carried out on the squark fields of the MSSM, one The constraints on the flavor-diagonal mass inser- obtains the super–CKM/PMNS basis in which no tree– • u,d,l LR level FCNC couplings are present. However, neither the tions δ11,22 are obtained from the requirement 3 3 mass terms m2 , m2 , m2, m2 and m2 of the left– that the corrections should not exceed the mea- × Q˜ u˜ d˜ L˜ e˜ handed and right–handed sfermions nor the tri-linear sured masses. This has already been done in the Higgs–sfermion–sfermion couplings are necessarily diag- seminal paper of Gabbiani et al. [2]. We improve d u this calculation by taking into account QCD cor- onal in this basis. The tri-linear QHdA dR, QHuA uR and LH Ale terms induce mixing between left–handed rections and by using the up-to-date values of the d R fermion masses. and right–handed sfermions after the Higgs doublets Hd and Hu acquire their vacuum expectation values (vevs) The leading chirally-enhanced two-loop corrections vd and vu, respectively. In the current era of precision • are calculated. As we will see, this allows us to flavor physics stringent bounds on these parameters have f RL d RL constrain the elements δ13 (and δ23 ), if at the been derived from FCNC processes in the quark and in f LR same time, also δ (δd LR) is different from zero. the lepton sector, by requiring that the gluino–squark 13 23 loops and chargino–sneutrinos/neutralino–slepton loops Our paper is organized as follows: In Sec. II we study do not exceed the measured values of the considered ob- the impact of chirally enhanced parts of the self-energies servables [1–13]. for quarks and leptons on the fermion masses and mix- However, in [14, 15] it is shown that all flavor viola- ing matrices (CKM matrix and PMNS matrix). First, tion in the quark sector can solely originate from trilinear we introduce the general formalism in Sec. II A and then SUSY breaking terms because all FCNC bounds are sat- specify to the MSSM with non-minimal sources of flavor isfied for MSUSY 500GeV. Dimensionless quantities violation in Sec. II B where we compute the chirally en- are commonly defined≥ in the mass insertion parametriza- hanced parts of the self-energies for quarks and leptons 2

fJ fK fI

Σf (1) f (1) i KJ iΣ − − IK

fJ fI FIG. 2: One-particle irreducible two-loop self-energy constructed out of two one-loop self energies with I 6= J 6= K. iΣf − IJ

FIG. 1: Flavor-valued wave-function renormalization. A. General formalism taking into account also the leading two-loop corrections. Sec. III is devoted to the numerical analysis. Finally we In this section we consider the general effect of one- conclude in Sec. IV. particle irreducible self-energies. It is possible to de- compose any self-energy in its chirality-changing and its chirality-flipping parts in the following way: II. FINITE RENORMALIZATION OF FERMION MASSES AND MIXING MATRICES f f LR 2 f RR 2 ΣIJ (p)= ΣIJ (p )+ p/ΣIJ (p ) PR (2) f RL 2 f LL 2 We have computed the finite renormalization of the + ΣIJ (p )+ p/ΣIJ (p ) PL . CKM matrix by SQCD effects in Ref. [14, 16] and of the PMNS matrix in Ref. [17]. In this section we com- f LR f RL pute the finite renormalization of fermion masses and Note that chirality-changing parts ΣIJ and ΣIJ have f LL f RR mixing angles induced through one-particle irreducible mass dimension 1, while ΣIJ and ΣIJ are dimen- flavor-valued self-energies beyond leading-order. We first sionless. With this convention the renormalization of the consider the general case and then specify to the MSSM. fermion masses is given by:

(0) (0) f LR 2 1 f LL 2 f RR 2 phys m m +Σ (m )+ m Σ (m )+Σ (m ) + δmf = m . (3) fI → fI II fI 2 fI II fI II fI I fI

If the self-energies are finite, the counter-term δmfI in ements are generated radiatively. In this limit it would Eq. (3) is zero in a minimal renormalization scheme like be unnatural to have tree-level Yukawa couplings and MS. In the following we choose this minimal scheme CKM elements in the Lagrangian which are canceled by for two reasons: First, A-terms are theoretical quantities counter-terms as in the on-shell scheme. which are not directly related to physical observables. For such quantities it is always easier to use a minimal The self-energies in Eq. (2) do not only renormalize the f L scheme which allows for a direct relation between theo- fermion masses. Also a rotation 1+∆UIJ in flavor-space retical quantities and observables. Second, we consider which has to be applied to all external fields is induced the limit in which the light fermion masses and CKM el- through the diagram in Fig. 1:

f L 1 2 f LL 2 f RR 2 f LR 2 f RL 2 ∆U = m Σ m + mf mf Σ m + mf Σ m + mf Σ m for I = J, IJ m2 m2 fJ IJ fI J I IJ fI J IJ fI I IJ fI 6 fJ − fI f L 1 f LL 2 f LR 2 2 f LL 2 f RR 2 ∆U = Re Σ m +2mf Σ ′ m + m Σ ′ m +Σ ′ m . (4) II 2 II fI I II fI fI II fI II fI h i

The prime denotes differentiation with respect to the ar- cation of flavor-conserving self-energies. Eq. (3) and gument. The flavor-diagonal part arises from the trun- 3

Eq. (4) are valid for arbitrary one-particle irreducible Again, we denote the sum of all contribution as: self-energies. 0 ± Σℓ LR =Σχ˜ +Σχ˜ . (12) IJ ℓIL ℓJR ℓIL ℓJR − − B. Self-energies in the MSSM With I = J we arrive at the flavor-conserving case. This can lead to significant quantum corrections to fermion Self-energies with supersymmetric virtual particles are masses, but except for the gluino, the pure bino ( g2) ∝ 1 of special importance because of a possible chiral en- and the negligible small bino-wino mixing ( g1g2) con- hancement which can lead to order-one corrections. In tribution, they are proportional to tree-level∝ Yukawa cou- this section we calculate the chirally enhanced (by a fac- plings. However, if the light fermion masses are gener- AIJ f ated radiatively from chiral flavor-violation in the soft tor M Y IJ or tan β) parts of the fermion self-energies SUSY f SUSY-breaking terms, then the Yukawa couplings of the in the MSSM. Therefore it is only necessary to evaluate first and second generation even vanish and the latter ef- the diagrams at vanishing external momentum. fect is absent at all. Radiatively generated fermion mass We choose the sign of the self-energies Σ to be equal terms via soft tri-linear A-terms corresponds to the up- to the sign of the mass, e.g. calculating a self-energy per bound found from the fine-tuning argument where diagram yields iΣ. Then, with the conventions given − the correction to the mass is as large as the physical mass in the Appendix A, the gluino contribution to the quark itself. This fine-tuning argument is based on ’t Hooft’s self-energies is given by: naturalness principle: A theory with small parameters is 6 natural if the symmetry is enlarged when these param- g mg˜ ˜ g˜q˜i ∗ g˜q˜i 2 2 eters vanish. The smallness of the parameters is then ΣqIL qJR = ΓqJR ΓqIL B0(mg˜,mq˜i ) (5) − − 16π2 i=1 protected against large radiative corrections by the con- X 6 cerned symmetry. If such a small parameter, e.g. a αs (J+3)i Ii 2 2 = CF mgW ∗W B (m ,m ) (6) fermion mass, is composed of several different terms there 2π ˜ Q Q 0 g˜ q˜i i=1 should be no accidental large cancellation between them. X We will derive our upper bounds from the condition that and for the neutralino and chargino contribution to the the SUSY corrections should not exceed the measured quark self-energy we receive: value.

6 4 If we restrict ourself to the case with vanishing first 0 mχ˜0 0 ˜ 0 ˜ χ˜ j χ˜j di χ˜j di 2 2 and second generation tree-level Yukawa couplings, the Σ = Γ ∗Γ B (m 0 ,m ) dIL dJR 2 dJR dIL 0 χ˜ d˜ − − 16π j i off-diagonal entries in the sfermion mass matrices stem i=1 j=1 X X f LR (7) from the soft tri-linear terms. Thus we are left with δIJ only. In the mass insertion approximation with only LR 6 2 ± ± mχ˜ ± ± χ˜ j χ˜j u˜i χ˜j u˜i 2 2 insertion the flavor violating self-energies simplifies. For Σ = Γ ∗Γ B (m ± ,m ) dIL dJR 2 dJR dIL 0 χ˜ u˜i − − 16π j the gluino (neutralino) self-energies which are relevant i=1 j=1 X X for our following discussion for the quark (lepton) case (8) we get: The self-energies in the up-sector are easily obtained by 2α interchanging u and d. We denote the sum of all contri- g˜ s q XY 2 2 2 ΣqIX qJY = Mg˜mq˜JY mq˜IX δIJ C0 M1 ,mq˜JY ,mq˜IX , bution as: − 3π (13) 0 q LR g˜ χ˜ χ˜± Σ =Σ +Σ +Σ (9) B˜ α1 ℓ XY 2 2 2 IJ qIL qJR qIL qJR qIL qJR Σ = M m m δ C M ,m ,m . ℓIX ℓJY 1 ℓ˜JY ℓ˜IX IJ 0 1 ℓ˜ ℓ˜ − − − − 4π JY IX Note that the gluino contribution are dominant in the (14) case of non-vanishing A-terms, since they involve the strong coupling constant. In the lepton case, neutralino– Since the sneutrino mass matrix consists only of a LL slepton and chargino–sneutrino loops contribute the non- block, there are no chargino diagrams in the lepton case ℓ LR decoupling self-energy ΣIJ . With the convention in the with LR insertions at all. Appendix A the self-energies are given by: Since the SUSY particles are known to be much heav- ier than the five lightest quarks it is possible to evaluate 2 3 ± ± mχ˜ ± ± χ˜ j χ˜j ν˜k χ˜j ν˜k 2 2 the one-loop self-energies at vanishing external momen- Σ = Γ ∗Γ B (m ± ,m ), ℓIL ℓJR 2 ℓJR ℓIL 0 χ˜ ν˜k tum and to neglect higher terms which are suppressed by − − 16π j j=1 k=1 2 2 X X powers of mfI /MSUSY . The only possibly sizable decou- (10) pling effect concerning the W vertex renormalization is a 6 4 0 mχ˜0 0 ˜ 0 ˜ loop-induced right-handed W coupling (see [16]). There- χ˜ j χ˜j ℓi χ˜j ℓi 2 2 Σ = Γ ∗Γ B (m 0 ,m ). ℓIL ℓJR 2 ℓJR ℓIL 0 χ˜ ℓ˜ fore Eq. (2) simplifies to − − 16π j i i=1 j=1 X X (11) f (1) f LR (1) f RL (1) ΣIJ =ΣIJ PR +ΣIJ PL (15) 4 at the one-loop level (indicated by the superscript (1)). for the mass renormalization in the MS scheme. Accord- In this approximation the self-energies are always chiral- ing to Eq. (4) the flavor-valued rotation which has to be ity changing and contribute to the finite renormalization applied to all external fermion fields is given by: of the quark masses in Eq. (3) and to the flavor-valued wave-function renormalization in Eq. (4). At the one- loop level we receive the well known result

m(0) m(1) = m(0) +Σf LR (1) (16) fI → fI fI II

m Σf LR (1) + m Σf RL (1) m Σf LR (1) + m Σf RL (1) 0 f2 12 f1 12 f3 13 f1 13 m2 m2 m2 m2  f2 − f1 f3 − f1  f LR (1) f RL (1) f LR (1) f RL (1) f L (1)  mf1 Σ21 + mf2 Σ21 mf3 Σ23 + mf2 Σ23  ∆U =  0  . (17)  m2 m2 m2 m2   f1 − f2 f3 − f2   f LR (1) f RL (1) f LR (1) f RL (1)   mf1 Σ + mf3 Σ mf2 Σ + mf3 Σ   31 31 32 32 0   m2 m2 m2 m2   f1 f3 f2 f3   − − 

The corresponding corrections to the right-handed wave- be constrained from the CKM and PMNS renormaliza- functions are obtained by simply exchanging L with R tion. However, since we treat, in the spirit of Ref. [18], and vice versa in Eq. (17). Note that the contributions all diagrams in which no ﬂavor appears twice on quark f RL (1) lines as one-particle irreducible, chirally-enhanced self- of the self-energies ΣIJ with J > I are suppressed by small mass ratios. Therefore, the corresponding oﬀ- energies can also be constructed at the two-loop level diagonal elements of the sfermion mass matrices cannot (see Fig. (2)):

f RL (1) f LR (1) f LR (1) f RL (1) f RR (2) 2 ΣIK ΣKJ f LL (2) 2 ΣIK ΣKJ ΣIJ p = 2 2 , ΣIJ p = 2 2 , K=I,J p mf K=I,J p mf 6 − K 6 − K (18) P f LR (1) f LR (1) P f RL (1) f RL (1) f LR (2) 2 ΣIK ΣKJ f RL (2) 2 ΣIK ΣKJ ΣIJ p = mfK 2 2 , ΣIJ p = mfK 2 2 . K=I,J p mf K=I,J p mf 6 − K 6 − K P P Therefore, the chiral-enhanced two-loop corrections to the masses and the wave-function renormalization are given by:

f LR (1) f LR (1) f LR (1) f LR (1) f LR Σ Σ Σ Σ (0) m(0) +Σ (1) 12 21 13 31 m f1 11 f1 − mf2 − mf3  f LR (1) f LR (1)   (0)  (0) f LR (1) Σ23 Σ32 , (19) mf2 →  mf2 +Σ22     − mf3   (0)     mf3   (0) f LR (1)     m 3 +Σ33     f    2 2 f LR (1) f LR (1) f LR (1) f LR (1) f LR (1) f RL (1) Σ12 Σ13 Σ13 Σ32 Σ12 Σ23 2 2 2  − 2m − 2m − m 2 m 3 m  f2 f3 f f f3 2 2 f LR (1) f LR (1)  f RL (1) f RL (1) f LR (1) f RL (1)  f (2)  Σ Σ Σ23 Σ12 Σ Σ  ∆UL = 23 31 21 13 , (20)  2 2 2   m 2 m 3 − 2m − 2m m  f f f3 f2 f3  2 2   f LR (1) f LR (1)  f RL (1) f RL (1) f RL (1) f LR (1)  Σ13 Σ23   Σ32 Σ21 Σ31 Σ12   2 2   m 2 m 3 − m 2 m 3 − 2m − 2m   f f f f f3 f3    5

u LR -4 where we have neglected small mass ratios. In the quark ∆11 10 2000 case, we already know about the hierarchy of the self- 1 energies from our ﬁne-tuning argument. In this case Eq. (20) is just necessary to account for the unitarity of (0) the CKM matrix [14]. However, the corrections to mf1 1500 in Eq. (19) can be large. For this reason we can also f LR (1) 1.5 constrain Σ31 with ’t Hooft’s naturalness criterion in GeV ~ f LR (1) q m if at the same time Σ13 is diﬀerent from zero. 1000 2.5 2

3 III. NUMERICAL ANALYSIS 500 3.5

500 1000 1500 2000 In this section we are going to give a complete nu- m~ in GeV g merical evaluation of the all possible constraints on the u LR -1 SUSY breaking sector from ’t Hooft’s naturalness argu- ∆22 10 2000 ment. This criterion is applicable since we gain a flavor 0.4 symmetry [14] if the light fermion masses are generated radiatively. Therefore the situation is different from e.g. the little hierarchy problem, where no additional symme- 1500 try is involved. First of all, it is important to note that all off-diagonal elements of the fermion mass matrices have 0.6 in GeV ~ to be smaller than the average of their assigned diagonal q m 1000 1 elements 0.8

IJ 2 2 2 1.2 ∆mF < m m , (21) XY f˜IX f˜JY 1.4 q 500 since otherwise one sfermion mass eigenvalue is negative. 500 1000 1500 2000 We note that in Ref. [2] this constraint is not imposed. m~ in GeV g All constraints in this section are non-decoupling since d LR -4 ∆11 10 we compute corrections to the Higgs-quark-quark cou- 2000 pling which is of dimension 4. Therefore, our constraints 1.5 on the soft-supersymmetry-breaking parameters do not vanish in the limit of inﬁnitely heavy SUSY masses but 3 1500 rather converge to a constant [14]. However, even though 2 f LR δIJ is a dimensionless parameter it does not only in-

volve SUSY parameter. It is also proportional to a in GeV ~ q vacuum expectation and therefore scales like v/MSUSY. m 1000 f LR Thus, our constraints on δIJ do not approach a con- 4 stant for MSUSY but rather get stronger. Similar effects occur in Higgs-mediated→ ∞ FCNC processes which de- couple like 1/M 2 rather than 1/M 2 [19–21]. How- 500 5 Higgs SUSY 6 6 ever, Higgs-mediated eﬀects can only be induced within 500 1000 1500 2000 m~ in GeV supersymmetry in the presence of non-holomorphic terms g which are not required for our constraints. An example d LR -2 ∆22 10 of a non-decoupling Higgs-mediated FCNC process is the 2000 observable RK = Γ (K eν) /Γ (K µν) that is cur- 0.4 → → rently analyzed by the NA62-experiment. In this case 0.8 Higgs contributions can induce deviations from lepton ﬂavor universality [10, 22, 23]. 1500 0.5

0.6 in GeV ~ q A. Constraints on ﬂavor-diagonal mass insertions m 1000 at one loop

1 The diagonal elements of the A-terms can be con- 500 1.4 strained from the fermion masses by demanding that 500 1000 1500 2000 m~ in GeV g

u,d LR FIG. 3: Constraints on the diagonal mass insertions δ11,22 obtained by applying ’t Hooft’s naturalness criterion. 6

l LR -4 ∆11 10 to measurable contributions to the muon anomalous mag- 1400 1 netic moment. This arises from the same one-loop diagram as Σℓ LR with an external photon attached. There- 1200 22 fore, the SUSY contribution is not suppressed by a loop 1000 factor compared to the case with tree-level Yukawa cou- 1.5 plings. e 800 m 3 2 600 B. Constraints on flavor-off-diagonal mass insertions from CKM and PMNS renormalization 400 4 200 1. CKM matrix 200 400 600 800 1000 1200 1400 M1 A complete analysis of the constraints for the CKM ∆l LR 22 renormalization was already carried out in Ref. [14]. The 1400 0.2 numerical effect of the chargino contributions is negligi- 1200 ble at low tan β and the neutralino contributions amount only to corrections of about 5% of the gluino contribu- 1000 tions. Therefore, we refer to the constraints on the off- 0.3 q LR diagonal elements δIJ given in Ref. [14]. Μ 800 m 0.4

600 0.6 2. Threshold corrections to PMNS matrix 400

1 Up to now, we have only an upper bound for the matrix 200 iδ 200 400 600 800 1000 1200 1400 element Ue3 = sin θ13e− and thus for the mixing angle θ ; the best-fit value is at or close to zero: θ =0.0+7.9 M 13 13 0.0 1 [25]. It might well be that it vanishes at tree level due− to ℓ LR a particular symmetry and obtains a non-zero value due FIG. 4: Contraints on the diagonal mass insertion δ11 22 as a , to corrections. So we can ask the question if threshold function of M1 and me˜, mµ˜. corrections to the PMNS matrix could spoil the prediction θ13 =0◦ at the weak scale. We demand the absence f LR (1) of fine-tuning for these corrections and therefore require ΣII mfI [see Eq. (16)]. The bounds on the flavor- conserving≤A-term for the up, charm, down and strange that the SUSY loop contributions do not exceed the value quarks are shown in Fig. (3) and the constraints from the of Ue3, electron and muon mass are depicted in Fig. (4). The up- phys per bound derived from the fermion mass is roughly given ∆Ue3 U . (25) | |≤ e3 by

The renormalization of the PMNS matrix is described in 3πm (M ) q LR qI SUSY detail in [17], where the on-shell scheme was used. As δII . (22) αs(MSUSY )MSUSY discussed in Sec. (II) we also use the MS scheme in this

section. Then the physical PMNS matrix is given by: for quarks and phys (0) 8πm U = U + ∆U , (26) δℓ LR . ℓI (23) II α M 1 SUSY where ∆U should not be confused with the wave function fL for leptons in the case of equal SUSY masses. In the renormalization ∆U . Then ∆U is given by lepton case Eq. (23) can be further simpliﬁed, since we T can neglect the running of the masses: ∆U = ∆U ℓL U (0). (27)

ℓ LR 500 GeV Note that in contrast to the corrections to the CKM ma- δ11 . 0.0025 MSUSY , | | (24) trix, there is a transpose in ∆U ℓL, because the ﬁrst index ℓ LR . 500 GeV δ22 0.5 MSUSY . of the PMNS matrix corresponds to down-type fermions and not to up-type fermion as in the CKM matrix. Only However, as already pointed out in Ref. [24] a muon the corrections to the small element Ue3 can be sizeable, mass that is solely generated radiatively potentially leads since all other elements are of order one. If we set all 7 oﬀ-diagonal element to zero except for δℓ LR = 0, we get 13 6 200 2 % ∆U ℓLU phys U phys ∆U ℓL in 150 31 τ3 e3 31 e3

∆Ue3 = − U ℓL 2 È

1+ ∆U 13 100

11 l (28) L

U ℓ RL D phys Σ31 È 50 Uτ3 . ≈− mτ 0 0.00 0.05 0.10 0.15 0.20 0.25 Note that here, in contrast to the renormalization of the ∆l 13 CKM matrix, the physical PMNS element appears. This LR 500 is due to the fact that one has to solve the linear system in Eq. (27) as described in [17]. By means of the ﬁne- 400 % tuning argument we can in principle derive upper bounds in

ℓ LR e3 300 for δ . The results depend on the SUSY mass scale U 13 È

13 200 l MSUSY and the assumed value for θ13. L U D

Here, we consider the corrections stemming from È 100 flavor-violating A-terms to the small matrix element Ue3. The δℓLL-contribution was already studied in [17] with 0 13 0 1 2 3 4 5 the result that they are negligible small. We also made a o ℓ LR Θ13 in comment about the δ13 -contribution which is outlined in more detail. Our results depend on the overall SUSY ℓ LR ℓ LR FIG. 5: |∆Ue3|/Ue3 in %. Top: as a function of δ13 for mass scale, the value of θ13 and of δ . In Fig. (5) ◦ 13 MSUSY = 1000 GeV and different values of θ13 (green 1 ; blue: ◦ ◦ you can see the percentage deviation of Ue3 through this 13 ℓ LR 3 ; red: 5 ). Bottom: as a function of θ for MSUSY = 1000 SUSY loop corrections in dependence of δ (top) and ℓ 13 ℓ LR 13 GeV and different values of δLR (red: δ13 = 0.5; blue: ℓ LR ℓ LR θ13 (bottom) for MSUSY = 1000 GeV. The constraints δ13 = 0.3; green: δ13 = 0.1) (both from top to bottom) ℓ LR on δ13 get stronger with smaller θ13 and with larger ℓ LR MSUSY. In Fig. (6) the excluded θ13,δ13 -region is below the curves for different MSUSY scales. The derived 14 bound can be simplified to 12 10 ℓ LR 500 GeV δ13 . 0.2 θ13 in degrees . (29) o 8

MSUSY | | in

13 6 Θ Exemplarily, we get for reasonable SUSY masses of 4 MSUSY = 1000 GeV and θ13 = 3◦ an upper bound of 2 ℓ LR ℓ LR δ13 0.3. The constraints on δ13 from τ eγ are 0 of the order≤ of 0.02 [17] and in general better→ than our 0.0 0.1 0.2 0.3 0.4 0.5 l LR derived bounds if θ13 is non-zero. As an important con- ∆13 sequence, we note that τ eγ impedes any measurable → ℓ LR correction from supersymmetric loops to Ue3 : E.g. for FIG. 6: The excluded `θ13, δ13 ´-region is below the curves 3 sparticle masses of 500 GeV we find ∆Ue 10− cor- for (from bottom to top) MSUSY = 500 GeV (red), 1000 | 3|≤ responding to a correction to the mixing angle θ13 of at GeV (blue), 2000 GeV (green) and 5000 GeV (yellow). The black dashed line denotes the future experimental sensitivity most 0.06◦. That is, if the DOUBLE CHOOZ experiment ◦ to θ13 = 3 . measures Ue3 = 0, one will not be able to ascribe this result to the SUSY6 breaking sector. Stated positively, 3 Ue3 & 10− will imply that at low energies the flavor symmetries imposed on the Yukawa sector to motivate tri-bimaximal mixing are violated. This finding confirms absent in the case of a radiative fermion mass) and the ℓ LL ℓ LR supersymmetric loop corrections we can derive bounds the pattern found in [17] where the product δ13 δ33 f LR f LR ℓ LR has been studied instead of δ . on the products δIK δKI which contain the so far less 13 f LR constrained elements δKI , K > I.

C. Constraints from two-loop corrections to We apply the fine-tuning argument to the two-loop fermion masses contribution originating from flavor-violating A-terms, f LR(2) f LR(2) e.g. Σ mf1 . The bound Σ = mf1 corre- 11 ≤ 11 Combining two flavor-violating self-energies can have sponds to a 100% change in the fermion mass through su- sizable impacts on the light fermion masses according to persymmetric loop corrections which is equivalent to the Eq. (19). Requiring that no large numerical cancella- case that the fermion Yukawa coupling vanishes. The up- tions should occur between the tree-level mass (which is per bound depends on the overall SUSY mass scale and 8

l LR l LR is roughly given as Constraints on ∆13 ∆31 from me 0.5 2 q LR q LR 9π mqI mq3 (MSUSY ) δI3 δ3I . 2 , I = 3 (30) (αs(MSUSY )MSUSY) 6 0.4 for quarks and 2 ℓ LR ℓ LR 64π mℓ1 mℓ3 0.3 δ13 δ31 . 2 (31)

(α1MSUSY) LR l 13 ∆ for leptons. Again, Eq. (31) can be further simpliﬁed 0.2

500 GeV 2 δℓ LRδℓ LR 0.021 . (32) 13 31 ≤ M 0.1 SUSY

The contributions proportional to δf LRδf LR cannot be 13 31 0.0 important, since these elements are already severely con- 0.0 0.1 0.2 0.3 0.4 0.5 strained by FCNC processes [26]. As studied in Ref. [27], ∆l LR single-top production involves the same mass insertion 31 l LR l LR u LR Constraints on ∆ ∆ from me δ31 which can also induce a right-handed W coupling 13 31 d LR if at the same time δ33 = 0 [16]. Therefore our bound 0.10 can be used to place a constraint6 on this cross section. u,ℓ LR u,ℓ LR 0.05 Also the product δ23 δ32 cannot be constrained, 31 l LR since the muon and the charm are too heavy. However, ∆ 0.00 d LR d LR 13 l LR

δ23 δ32 can be constrained as shown in Fig. (10). Our ∆ results for the up, down, and electron mass are depicted -0.05 in Fig. (8),(9) and (7). In the quark case also the bounds -0.10 q LR from the CKM renormalization on δ13,23 are taken into account. 200 400 600 800 1000 MSUSY

IV. CONCLUSIONS FIG. 7: Results of the two-loop contribution to the electron mass. Above: Region compatible with the naturalness principle for (from top to bottom) MSUSY = 200 GeV (yellow), According to ’t Hooft’s naturalness principle, the 500 GeV (green), 800 GeV (blue), 1000 GeV (red). Bottom: smallness of a quantity is linked to a symmetry that is ℓ LR ℓ LR Allowed range for δ13 δ31 as a function of MSUSY. restored if the quantity is zero. The smallness of the Yukawa couplings of the first two generations (as well as the small CKM elements involving the third generation) suggest the idea that Yukawa couplings (except for the third generation) are generated through radiative corrections [14, 15, 24, 28–30]. It might well be that the chiral flavor symmetry is broken by soft SUSY-breaking terms The PMNS renormalization is a bit more involved rather than by the trilinear tree-level Yukawa couplings. since the matrix is not hierarchical. The radiative de- We use ’t Hooft’s naturalness criterion to constrain cay τ eγ severely limits the size of the loop correction u,d,ℓ LR → the chirality-changing mass insertion δIJ from the ∆Ue3 to the PMNS element Ue3. In a previous paper we mass and CKM renormalization. Therefore, we compute have studied this topic for effects triggered by the prod- ℓLL ℓ LR the finite renormalization of fermion masses and mixing uct δ13 δ33 [17]. In this paper we have complemented ℓ LR angles in the MSSM, taking into account the leading two- that analysis by investigating δ13 instead. Assuming loop effects. These corrections are not only important, in reasonable slepton masses and noting that the Daya Bay order to obtain a unitary CKM matrix, they are also nu- neutrino experiment is only sensitive to values of θ13 merically important for light fermion masses. This allows above 3◦, we conclude that the threshold corrections to f LR f LR d LR d LR us to constrain the product δ13 δ31 (and δ23 δ32 ) Ue3 are far below the measurable limit. Consequently, if which is important, especially with respect to the before a symmetry at a high scale imposes tri-bimaximal mix- u RL unconstrained element δ13 . All constraints given in this ing, SUSY loop corrections cannot spoil this prediction paper are non-decoupling. This means they do not van- θ13 = 0 at the weak scale. This is an important result for ish in the limit of infinitely heavy SUSY masses unlike the proper interpretation of a measurement of θ13. Thus the bounds from FCNC processes. Therefore our con- if DOUBLE CHOOZ or Daya Bay neutrino experiment straints are always stronger than the FCNC constraints will measure a non-zero θ13 then this is also true at a for sufficiently heavy SUSY (and Higgs) masses. high energy scale. 9

u LR u LR d LR d LR Constraints on ∆13 ∆31 from mu Constraints on ∆13 ∆31 from md 0.0030 0.14

0.0025 0.12

0.10 0.0020

0.08 LR LR 0.0015 d 13 u 13 ∆ ∆ 0.06 0.0010 0.04 0.0005 0.02 0.0000 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 ∆d LR ∆u LR 31 31 d LR d LR u LR u RL Constraints on ∆13 ∆31 from md Constraints on ∆13 ∆13 from mu 0.02 0.0006 0.0004 0.01 0.0002 LR d 31 ∆

RL 0.0000 u 13 LR ∆ 0.00 d 13 ∆

LR -0.0002 u 13 ∆ -0.0004 -0.01 -0.0006

-0.02 400 600 800 1000 1200 1400 400 600 800 1000 1200 1400 MSUSY

MSUSY FIG. 9: Results of the two-loop contribution to the down FIG. 8: Results of the two-loop contribution to the up quark quark mass. Above: Region compatible with the naturalness mass. Above: Region compatible with the naturalness prin- principle for (from top to bottom) MSUSY = 500 GeV (yel- ciple (100% bound) for (from top to bottom) MSUSY = 500 low), 1000GeV (green), 1500 GeV (blue), 2000 GeV (red). d LR d LR GeV (yellow), 1000GeV (green), 1500 GeV (blue), 2000 GeV Bottom: Allowed range for δ13 δ31 as a function of MSUSY. u LR u LR (red). Bottom: Allowed range for δ13 δ31 as a function of MSUSY. Appendix A: Conventions

1. Loop integrals

For the self-energies, we need the following loop integrals:

Acknowledgments x x y y B (x, y)= ∆ ln ln , (A1) 0 − − x y µ2 − y x µ2 − − 1 with ∆ = γE +ln4π. ǫ − x y z We like to thank Ulrich Nierste for helpful discus- xy ln + yz ln + xz ln C (x,y,z)= y z x . (A2) sions and proofreading the article. This work is sup- 0 (x y)(y z)(z x) ported by BMBF grants 05HT6VKB and 05H09VKF − − − and by the EU Contract No. MRTN-CT-2006-035482, “FLAVIAnet”. Andreas Crivellin and Jennifer Gir- rbach acknowledge the financial support by the State of 2. Diagonalization of mass matrices and Feynman Baden-Württemberg through Strukturiertes Promotion- rules skolleg Elementarteilchenphysik und Astroteilchenphysik and the Studienstiftung des deutschen Volkes, respec- For the vacuum expectation value we choose the nor- tively. malization without the factor √2 and define the Yukawa 10

d LR d LR Constraints on ∆23 ∆32 from ms MN can be diagonalised with an unitary transformation 0.030 such that the eigenvalues are real and positive.

0.025 0 mχ˜1 0 D . ZN⊤MN ZN = MN = . . (A6) 0.020  .  0 m 0  χ˜4 

LR 0.015   d 23 ∆ D 2 For that purpose, ZN† MN† MN ZN = (MN ) can be used. 0.010 ZN consists of the eigenvectors of the Hermitian matrix MN† MN . Then the columns can be multiplied with iφ T D 0.005 phases e , such that ZN MN ZN = MN has positive and real diagonal elements.

0.000 0.0 0.1 0.2 0.3 0.4 0.5 d LR ∆32 ± d LR d LR Charginos χ˜i Constraints on ∆23 ∆32 from ms 0.015

0.010

0.005

LR ˜ + ˜ + ˜ ˜ d 32 Ψ± = W , Hu , W −, Hd− , ∆ 0.000 LR

d 23 1 ∆ -0.005 ± = Ψ± ⊤ MCΨ± + h.c. Lχ˜mass −2 -0.010 0 X⊤ M2 g2vu MC = , X = . (A7) -0.015 X 0 g2vd µ 400 600 800 1000 1200 1400

MSUSY The rotation matrices for the positive and negative FIG. 10: Results of the two-loop contribution to the strange charged fermions diﬀer, such that quark mass. Above: Region compatible with the naturalness principle (100% bound) for (from top to bottom) MSUSY = T mχ˜1 0 500 GeV (yellow), 1000GeV (green), 1500 GeV (blue), 2000 Z XZ+ = . (A8) d LR d LR − 0 mχ˜2 GeV (red). Bottom: Allowed range for δ23 δ32 as a function of MSUSY. couplings in the following way: Sleptons

vu I I + 2 2 The sleptons L˜ =e ˜IL and R˜ =e ˜ mix to six v = vu + vd = 174 GeV, tan β = , (A3) 2 IR vd ˜ q charged mass eigenstates Li, i =1 . . . 6: ml = vdYl, md = vdYd, mu = vuYu. (A4) − − ˜I Ii ˜ ˜I (I+3)i ˜+ L2 = WL ∗ℓi−, R = WL ℓi , 2 2 0 (mL)LL (mL)LR 2 2 Neutralinos χ˜i W † W = diag m ,...,m , L 2 2 L ℓ˜1 ℓ˜6 (mL)RL† (mL)RR In the following we mainly use the convention of [31]. and the slepton mass matrix is composed of 0 ˜ ˜ ˜ 0 ˜ 0 Ψ = B, W , Hd , Hu , e2 v2 v2 1 2c2 2 IJ d − u − W 2 2 1 0 0 (mL)LL = 2 2 δIJ + vdYℓI δIJ χ0 = (Ψ )⊤MN Ψ + h.c. 4s c L ˜mass −2 W W 2 T g1vd g1vu M 0 +(mL˜ )IJ , 1 − √2 √2 0 M g2vd g2vu e2 v2 v2 2 √2 √2 2 IJ d u 2 2 2 M =  1 2 −  . (A5) (m ) = − δIJ + v Y δIJ + m , N g vd g vd 0 µ L RR − 2c2 d ℓI ˜eIJ √2 √2 W −g1vu g2vu −   µ 0  (m2 )IJ = v µY IJ + v AIJ .  √2 − √2 −  L LR u ℓ ∗ d ℓ ∗   11

2 2 2 2 Lepton-slepton-neutralino coupling 2 IJ e vd vu 1 4cW (mU )LL = − 2 2 − δIJ − 12sW cW Feynman rule for incoming lepton ℓI , outgoing neu- 2 2 2 T +vuYui δIJ + (VmQ˜ V †)IJ , tralino and slepton ℓ˜i: 2 2 2 2 IJ e vd vu 2 2 2 0 Ii (mU )RR = − δIJ + vuYuI δIJ + m˜ , χ˜ ℓ˜i W I i 2 uIJ j L 1j 2j ( +3) 3j 3cW iΓℓ =i g1ZN + g2ZN + YℓI WL ZN PL I √2 2 IJ IJ IJ (mU )LR = vdµYu ∗ vuAu ∗. 0 ˜ − − χ˜ ℓi =Γ j ℓIL | {z } (I+3)i 1j Ii 3j + i g √2W Z ∗ + Yℓ W Z ∗ PR. Quark-squark-gluino coupling − 1 L N I L N 0 χ˜ ℓ˜ j i =Γℓ Feynman rule for incoming quark dI , uI , outgoing IR ˜ | {z } (A9) gaugino and squark di, u˜i:

˜ iΓg˜di = ig √2T a W IiP + W (I+3)iP , (A10) dI s D L D R Lepton-sneutrino-chargino coupling − g˜u˜i a Ii (I+3)i iΓ = igs√2T W ∗PL + W ∗PR . (A11) uI − U U Feynman rule for incoming lepton ℓI , outgoing chargino and sneutrinoν ˜J :

± Quark-squark-neutralino coupling ν˜J χ˜i 1i 2i IJ iΓℓ = i g2Z+ PL + YℓI Z ∗PR Wν ∗. I − − Feynman rule for incoming quark dI , uI , outgoing neu- ˜ Down-squarks tralino and squark di, u˜i:

I I The down-squarks Q˜ = d˜IL and D˜ = d˜∗ mix to six 0 Ii 2 IR χ˜ d˜i W g ˜ iΓ j =i D 1 Z1j + g Z2j + Y W (I+3)iZ3j P mass eigenstates d , i =1 . . . 6: dI N 2 N dI D N L i √2 − 3 ˜I Ii ˜ ˜ I (I+3)i ˜+ √2g Q2 = WD ∗di−, D = WD di , 1 (I+3)i 1j Ii 3j + i WD ZN ∗ + YdI WD ZN ∗ PR, 2 2 − 3 (mD)LL (mD)LR 2 2 ! W † , W = diag m ,...,m , D 2 2 D d˜1 d˜6 0 Ii (m )† (m )RR χ˜ u˜i W g D RL D j U ∗ 1 1j 2j (I+3)i 4j iΓuI =i Z g2Z YuI W ∗Z PL √2 − 3 N − N − U N and the downs-squark mass matrix is composed of 2√2g1 (I+3)i 1j Ii 4j 2 2 2 2 ∗ ∗ ∗ e v v 1+2c + i WU ZN YuI WU ∗ZN PR. 2 IJ d u W 3 − ! (mD)LL = − 2 2 δIJ − 12sW cW 2 2 2 T +vdYdI δIJ + (mQ˜ )IJ , 2 2 2 Quark-squark-chargino coupling 2 IJ e vd vu 2 2 2 (m ) = − δIJ + v Y δIJ + m , D RR 2 d dI dIJ˜ − 6cW 2 IJ IJ IJ Feynman rule for incoming quark dI , uI , outgoing (m ) = vuµY ∗ + vdA ∗. D LR d d chargino and squarku ˜i, d˜i:

Up-squarks ± χ˜j u˜i Ji 1j (J+3)i 2j JI iΓ =i g2W ∗Z + Yu W ∗Z V PL dI − U + J U + Finally, one has six up-squarksu ˜i composed from ﬁelds Ji 2j JI ˜I ˜ I I + i YdI WU ∗Z ∗ V PR., Q1 =u ˜IL and U =u ˜IR − − ± χ˜ d˜ iΓ j i =i g W JiZ1j Y W (J+3)iZ2j V JI P ˜I Ii + ˜ I (I+3)i uI 2 D dJ D ∗ L Q1 = WU u˜i , D = WU ∗u˜i−, − − − − 2 2 Ji 2j JI T (mU )LL (mU )LR 2 2 + i YuI WD Z+ ∗ V ∗PR. WU 2 2 , WU∗ = diag mu˜1 ,...,mu˜6 . (m )† (m ) U RL U RR 12

[1] J. S. Hagelin, S. Kelley, and T. Tanaka, Nucl. Phys. [16] A. Crivellin (2009), 0907.2461. B415, 293 (1994). [17] J. Girrbach, S. Mertens, U. Nierste, and S. Wiesenfeldt [2] F. Gabbiani, E. Gabrielli, A. Masiero, and L. Silvestrini, (2009), 0910.2663. Nucl. Phys. B477, 321 (1996), hep-ph/9604387. [18] H. E. Logan and U. Nierste, Nucl. Phys. B586, 39 (2000), [3] M. Ciuchini et al., JHEP 10, 008 (1998), hep- hep-ph/0004139. ph/9808328. [19] L. J. Hall, R. Rattazzi, and U. Sarid, Phys. Rev. D50, [4] F. Borzumati, C. Greub, T. Hurth, and D. Wyler, Phys. 7048 (1994), hep-ph/9306309. Rev. D62, 075005 (2000), hep-ph/9911245. [20] C. Hamzaoui, M. Pospelov, and M. Toharia, Phys. Rev. [5] D. Becirevic et al., Nucl. Phys. B634, 105 (2002), hep- D59, 095005 (1999), hep-ph/9807350. ph/0112303. [21] T. Banks, Nucl. Phys. B303, 172 (1988). [6] E. Arganda and M. J. Herrero, Phys. Rev. D73, 055003 [22] A. Masiero, P. Paradisi, and R. Petronzio, Phys. Rev. (2006), hep-ph/0510405. D74, 011701 (2006), hep-ph/0511289. [7] A. Masiero, S. K. Vempati, and O. Vives, New J. Phys. [23] J. Girrbach and U. Nierste (2010), in preparation. 6, 202 (2004), hep-ph/0407325. [24] F. Borzumati, G. R. Farrar, N. Polonsky, and S. D. [8] J. Foster, K.-i. Okumura, and L. Roszkowski, JHEP 08, Thomas, Nucl. Phys. B555, 53 (1999), hep-ph/9902443. 094 (2005), hep-ph/0506146. [25] G. L. Fogli, E. Lisi, A. Marrone, A. Palazzo, and A. M. [9] M. Ciuchini et al., Phys. Lett. B655, 162 (2007), hep- Rotunno (2008), 0806.2649. ph/0703204. [26] S. Dittmaier, G. Hiller, T. Plehn, and M. Spannowsky, [10] A. Masiero, P. Paradisi, and R. Petronzio, JHEP 11, 042 Phys. Rev. D77, 115001 (2008), 0708.0940. (2008), 0807.4721. [27] T. Plehn, M. Rauch, and M. Spannowsky, Phys. Rev. [11] M. Ciuchini et al., Nucl. Phys. B783, 112 (2007), hep- D80, 114027 (2009), 0906.1803. ph/0702144. [28] S. Weinberg, Phys. Rev. Lett. 29, 388 (1972). [12] A. Crivellin and U. Nierste (2009), 0908.4404. [29] J. F. Donoghue, H. P. Nilles, and D. Wyler, Phys. Lett. [13] W. Altmannshofer, A. J. Buras, S. Gori, P. Paradisi, and B128, 55 (1983). D. M. Straub (2009), 0909.1333. [30] J. Ferrandis and N. Haba, Phys. Rev. D70, 055003 [14] A. Crivellin and U. Nierste, Phys. Rev. D79, 035018 (2004), hep-ph/0404077. (2009), 0810.1613. [31] J. Rosiek (1995), hep-ph/9511250. [15] A. Crivellin (2009), 0905.3130.