The large CP phase in B −B mixing and Unparticle Physics s s J. K. Parry Center for High Energy Physics, Tsinghua University, Beijing 100084, China∗ InthisworkweinvestigatethecontributiontoB mixingfrombothscalarandvectorunparticles d,s inanumberofscenarios. Theemphasisofthisworkistoshowtheimpactoftherecentlydiscovered 3σ evidence for new physics found in the CP phase of B mixing. Here we show that the inclusion s of the CP phase constraints for both B and B mixing improves the bounds set on the unparticle d s couplings by a factor of 2 ∼ 4, and one particular scenario of scalar unparticles is found to be excluded by the 3σ measurement of φ . s PACS: 13.20.He 13.25.Hw 12.60.-i [arXiv:0806.4350] 9 0 0 2 I. INTRODUCTION n a It has recently been suggested [1, 2] that there may exist a nontrivial scale invariant sector at high energies, known J as unparticle stuff. These new fields with an infrared fixed point are called Banks-Zaks fields [3], interacting with 2 standard model fields via heavy particle exchange, 1 1 h] MkOSMOBZ. (1) U p - Here O is a standard model (SM) operator of mass dimension d , O is a Banks-Zaks(BZ) operator of mass SM SM BZ p dimension d , with k =d +d −4, and M is the mass of the heavy particles mediating the interaction. At a e BZ SM BZ U h scale denoted by ΛU the BZ operators match onto unparticle operators with a new set of interactions, [ 3 C ΛUdBZ−dUO O (2) U Mk SM U v U 0 5 where OU is an unparticle operator with scaling dimension dU and CU is the coefficient of the low-energy theory. 3 Unparticle stuff of scaling dimension dU looks like a nonintegral number dU of invisible massless particles. In this 4 work we shall study both scalar unparticles (O ) and vector unparticles (Oµ ) with couplings to the SM quarks as U U . follows, 6 0 8 cS,q(cid:48)q cS,q(cid:48)q Scalarunparticles: L q(cid:48)γ (1−γ )q∂µO + R q(cid:48)γ (1+γ )q∂µO (3) 0 ΛdU µ 5 U ΛdU µ 5 U : U U v Xi Vectorunparticles: ΛcVLd,Uq−(cid:48)q1 q(cid:48)γµ(1−γ5)qOUµ + ΛcVRd,Uq−(cid:48)q1 q(cid:48)γµ(1+γ5)qOUµ (4) r U U a Here we assume that the left-handed and right-handed flavour-dependent dimensionless couplings cS, cS, cV and cV L R L R areindependentparameters. Weanalyzeanumberofscenarios,ineachcasedeterminingtheallowedparameterspace and placing bounds on the unparticle couplings. The propagators for scalar and vector unparticle fields are as follows [2, 4], (cid:90) A 1 d4xeiP.x(cid:104)0|TO (x)O (0)|0(cid:105) = i dU e−iφU (5) U U 2sindUπ(P2+i(cid:15))2−dU (cid:90) A (−gµν +PµPν/P2) d4xeiP.x(cid:104)0|TOµ(x)Oν(0)|0(cid:105) = i dU e−iφU (6) U U 2sindUπ (P2+i(cid:15))2−dU ∗Electronicaddress: [email protected] 2 where 16π5/2 Γ(d +1/2) A = U , φ =(d −2)π (7) dU (2π)2dU Γ(dU −1)Γ(2dU) U U Since the publication of the first theoretical papers on this new subject [1, 2] there has been a huge interest in unparticle phenomenology, for example, Collider signatures [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], CP violation [36, 37, 38, 39, 40, 41], meson mixing [5, 36, 42, 43, 44, 45, 46, 47], lepton flavour violation [48, 49, 50, 51, 52, 53, 54], consequences in astrophysics [55, 56, 57, 58, 59, 60, 61, 62], in neutrino physics [63, 64, 65, 66, 67] and in supersymmetry [68, 69, 70]. In this work we shall study the constraints coming from the measurements of B meson mass differences ∆M and also their s,d s,d CP violating phases φ . In the B system these quantities have been well measured for some time and show only s,d d smalldeviationsfromtheSMexpectation. IntheB systemrecentmeasurementshavealsofoundsmalldiscrepancies s between the SM expectation for ∆M [71], but now the CP violating phase φ , measured by the D∅[72] and CDF[73] s s collaborations reveals a deviation of 3σ[74, 75] [89]. This is the first evidence for new physics in b ↔ s transitions. Studying both vector and scalar unparticles, we study the constraints imposed by these latest measurements on the coupling between SM fields and unparticles, with particular interest on the impact of φ . s In Sec. II we discuss unparticle contributions to general meson-antimeson mixing and in particular, to the case of B mixing. Section III contains the results of our numerical analysis where we first consider scalar and then vector unparticle effects in B mixing. In each case we set bounds on the allowed parameter space using constraints from d,s measurements of ∆M and from φ . Finally, Sec. IV concludes our results. d,s d,s II. MESON-ANTIMESON MIXING FROM UNPARTICLES With unparticle operators coupling to standard sodel operators as in Eq. (3,4) it is possible for unparticle physics to contribute to meson-antimeson mixing via the s- and t-channel processes shown in Fig. 1. FIG. 1: s- and t-channel unparticle contributions to meson mixing. Using the interactions listed in Eq. (3,4) to evaluate the s- and t-channel contributions to meson mixing as shown in Fig. 1 we obtain the effective Hamiltonian for scalar unparticles as, HS,q(cid:48)q = AdU e−iφU (cid:18) 1 + 1 (cid:19) eff 2sindUπ Λ2UdU t2−dU s2−dU (cid:104) (cid:16) (cid:17)(cid:16) (cid:17) × Q m cS,q(cid:48)q−m cS,q(cid:48)q m cS,q(cid:48)q−m cS,q(cid:48)q 2 q(cid:48) L q R q R q(cid:48) L (cid:16) (cid:17) +Q˜ m cS,q(cid:48)q−m cS,q(cid:48)q)(m cS,q(cid:48)q−m cS,q(cid:48)q (8) 2 q(cid:48) R q L q L q(cid:48) R (cid:16) (cid:17)(cid:16) (cid:17)(cid:105) +2Q m cS,q(cid:48)q−m cS,q(cid:48)q m cS,q(cid:48)q−m cS,q(cid:48)q 4 q(cid:48) L q R q L q(cid:48) R 3 and for vector unparticles we obtain, HV,q(cid:48)q = AdU e−iφU eff 2sindUπΛ2UdU−2 ×(cid:26)(cid:18) 1 + 1 (cid:19) (cid:104)Q (cid:16)m cV,q(cid:48)q−m cV,q(cid:48)q(cid:17)(cid:16)m cV,q(cid:48)q−m cV,q(cid:48)q(cid:17) t3−dU s3−dU 2 q(cid:48) L q R q R q(cid:48) L (cid:16) (cid:17)(cid:16) (cid:17) + Q˜ m cV,q(cid:48)q−m cV,q(cid:48)q m cV,q(cid:48)q−m cV,q(cid:48)q 2 q(cid:48) R q L q L q(cid:48) R (cid:16) (cid:17)(cid:16) (cid:17)(cid:105) + 2Q m cV,q(cid:48)q−m cV,q(cid:48)q m cV,q(cid:48)q−m cV,q(cid:48)q (9) 4 q(cid:48) L q R q L q(cid:48) R − (cid:18) 1 + 1 (cid:19)(cid:20)Q (cid:16)cV,q(cid:48)q(cid:17)2+Q˜ (cid:16)cV,q(cid:48)q(cid:17)2−4Q (cid:16)cV,q(cid:48)qcS,q(cid:48)q(cid:17)(cid:21)(cid:27) t2−dU s2−dU 1 L 1 R 5 L R In the final line we have used a Fierz identity to rearrange the operator (V −A)⊗(V +A) into the scalar operator Q . Here HS describes the effective Hamiltonian for the case of scalar unparticles and HV for vector unparticles, 5 eff eff hence the total effective Hamiltonian is simply, Hq(cid:48)q =HS,q(cid:48)q+HV,q(cid:48)q (10) eff eff eff Above we have defined the quark operators Q −Q as follows, 1 5 Q = q(cid:48)αγ qα q(cid:48)βγµqβ (11) 1 L µ L L L Q = q(cid:48)αqα q(cid:48)βqβ (12) 2 R L R L Q = q(cid:48)αqβ q(cid:48)βqα (13) 3 R L R L Q = q(cid:48)αqα q(cid:48)βqβ (14) 4 R L L R Q = q(cid:48)αqβ q(cid:48)βqα (15) 5 R L L R Writing the effective Hamiltonian in terms of these operators we have, 5 3 Hq(cid:48)q =(cid:88)(CS +CV)Q +(cid:88)(C˜S +C˜V)Q˜ (16) eff i i i i i i i=1 i=1 Here the operators Q˜ are obtained from Q by the exchange L↔R. 1,2,3 1,2,3 The hadronic matrix elements, taking into account for renormalization effects, are defined as follows, (cid:104)M0|Q (µ)|M0(cid:105) = 1M f2 B (µ) (17) 1 3 M M 1 (cid:104)M0|Q (µ)|M0(cid:105) = − 5 M f2 R(µ)B (µ) (18) 2 24 M M 2 (cid:104)M0|Q (µ)|M0(cid:105) = 1 M f2 R(µ)B (µ) (19) 3 24 M M 3 (cid:104)M0|Q (µ)|M0(cid:105) = 1M f2 R(µ)B (µ) (20) 4 4 M M 4 (cid:104)M0|Q (µ)|M0(cid:105) = 1 M f2 R(µ)B (µ) (21) 5 12 M M 5 (cid:18) M (cid:19)2 where, R(µ) = M (22) m (µ)+m (µ) q q(cid:48) From Eq. (8-9), it is straightforward to calculate the Wilson coefficients for scalar unparticles which are as follows, CS = AdU e−iφU (cid:18)MM2 (cid:19)dU (cid:16)m cS,q(cid:48)q−m cS,q(cid:48)q(cid:17)(cid:16)m cS,q(cid:48)q−m cS,q(cid:48)q(cid:17) (23) 2 sind π M4 Λ2 q(cid:48) L q R q R q(cid:48) L U M U CS = 2 AdU e−iφU (cid:18)MM2 (cid:19)dU (cid:16)m cS,q(cid:48)q−m cS,q(cid:48)q(cid:17)(cid:16)m cS,q(cid:48)q−m cS,q(cid:48)q(cid:17) (24) 4 sind π M4 Λ2 q(cid:48) L q R q L q(cid:48) R U M U C˜S = AdU e−iφU (cid:18)MM2 (cid:19)dU (cid:16)m cS,q(cid:48)q−m cS,q(cid:48)q(cid:17)(cid:16)m cS,q(cid:48)q−m cS,q(cid:48)q(cid:17) (25) 2 sind π M4 Λ2 q(cid:48) R q L q L q(cid:48) R U M U CS = CS =CS =C˜S =C˜S =0 (26) 1 3 5 1 3 4 and for vector unparticles we find the following Wilson coefficients, CV = − AdU e−iφU (cid:18)MM2 (cid:19)dU−1(cid:16)cV,q(cid:48)q(cid:17)2 (27) 1 sind π M2 Λ2 L U M U CV = AdU e−iφU (cid:18)MM2 (cid:19)dU−1(cid:16)m cV,q(cid:48)q−m cV,q(cid:48)q(cid:17)(cid:16)m cV,q(cid:48)q−m cV,q(cid:48)q(cid:17) (28) 2 sind π M4 Λ2 q(cid:48) L q R q R q(cid:48) L U M U CV = 2 AdU e−iφU (cid:18)MM2 (cid:19)dU−1(cid:16)m cV,q(cid:48)q−m cV,q(cid:48)q(cid:17)(cid:16)m cV,q(cid:48)q−m cV,q(cid:48)q(cid:17) (29) 4 sind π M4 Λ2 q(cid:48) L q R q L q(cid:48) R U M U CV = 4 AdU e−iφU (cid:18)MM2 (cid:19)dU−1(cid:16)cV,q(cid:48)qcV,q(cid:48)q(cid:17) (30) 5 sind π M2 Λ2 L R U M U C˜V = − AdU e−iφU (cid:18)MM2 (cid:19)dU−1(cid:16)cV,q(cid:48)q(cid:17)2 (31) 1 sind π M2 Λ2 R U M U C˜V = AdU e−iφU (cid:18)MM2 (cid:19)dU−1(cid:16)m cV,q(cid:48)q−m cV,q(cid:48)q(cid:17)(cid:16)m cV,q(cid:48)q−m cV,q(cid:48)q(cid:17) (32) 2 sind π M4 Λ2 q(cid:48) R q L q L q(cid:48) R U M U CV = C˜V =0 (33) 3 3 where we have approximated t = s ∼ M2 . From these two sets of Wilson coefficients it is clear that the case of M vector unparticles not only includes more contributions as CV (cid:54)= 0, CV (cid:54)= 0, but also that their Wilson coefficients 1 5 areenhancedbyafactorΛ /M2 comparedtothescalarunparticlecase. Asaresultthevectorunparticleparameter U M space shall be suppressed by the same factor. These Wilson coefficients will mix with each other as a result of renormalization group(RG) running down to the scale of M . For the B system, with a scale of new physics Λ = 1 TeV, these Wilson coefficients at the scale M U µ =m may be approximated as [77], b b C (µ ) ≈ 0.805C (Λ ) (34) 1 b 1 U C (µ ) ≈ 1.988C (Λ )−0.417C (Λ ) (35) 2 b 2 U 3 U C (µ ) ≈ −0.024C (Λ )+0.496C (Λ ) (36) 3 b 2 U 3 U C (µ ) ≈ 3.095C (Λ )+0.725C (Λ ) (37) 4 b 4 U 5 U C (µ ) ≈ 0.086C (Λ )+0.884C (Λ ) (38) 5 b 4 U 5 U The ∆F =2 transitions are defined as, (cid:104)M0|H∆F=2|M0(cid:105)=M (39) eff 12 with the meson mass eigenstate difference defined as, ∆M ≡M −M =2|M | (40) H L 12 We can define in a model independent way the contribution to meson mixings in the presence of new physics (NP) as, M =MSM(1+R) (41) 12 12 where MSM denotes the SM contribution and R≡reiσ =MNP/MSM parameterizes the NP contribution. 12 12 12 The associated CP phase may then be defined as, φ≡arg(M )=φSM+φNP (42) 12 where φSM =arg(MSM) and φNP =arg(1+reiσ). 12 At this point it is important to make a further remark regarding the unparticle like model derived from a scale invariant theory of continuous mass fields as discussed in [78]. Although this model contains no fixed point or dimensionaltransmutationandthereforedoesnotcorrespondtounparticlesasdefinedin[2],itdoescontainunparticle- like local operators coupling to the SM. In this model it can be shown that the scale invariance will be broken by interactions with SM fields, resulting in a mass gap which could be rather large [79]. In this setting we may then expectthatthereareunparticlestateslyingbelowthescaleof1TeV.Thereforetheaboveapproachofintegratingout 5 unparticleeffectsatthescaleof1TeVwouldnotbevalidinthisspecialcase,rathertheWilsoncoefficientsshouldbe insteaddirectlycalculatedatthelow-energyscale. Suchacalculationwouldofferitsownchallenges,inparticularthe t-channel exchange would involve a nonlocal hadronic matrix element. In this work we shall not consider such a case andinsteadassumethatourunparticlesdonotderivefromascaleinvarianttheoryofcontinuousmassfields. Rather, we follow closely along the general framework set out by Georgi’s original work in [2] where there is dimensional transmutation at the scale Λ . U A. B mixing and unparticle physics d,s InthisworkweshallfocusontheconstraintsimposedonunparticlephysicscouplingsfromB mixing. Therefore d,s we set q(cid:48) =b, q =s, d and M0 =B0, B0. s d In the B system, the standard model contribution to Mq is given by, 12 G2M2 Mq,SM = F WM ηˆBf2 Bˆ (V∗V )2S (x ) (43) 12 12π2 Bq Bq Bq tq tb 0 t where G is Fermi’s constant, M the mass of the W boson, ηˆB = 0.551 [80] is a short-distance QCD correction F W identical for both the B and B systems. The bag parameter Bˆ and decay constant f are nonperturbative s d Bq Bq quantitiesandcontainthemajorityofthetheoreticaluncertainty. V andV areelementsoftheCabibbo-Kobayashi- tq tb Maskawa (CKM) matrix [81, 82], and S (x ≡m¯2/M2 )=2.34±0.03, with m¯ (m )=164.5±1.1 GeV [83], is one of 0 t t W t t the Inami-Lim functions [84]. We can now constrain both the magnitude and phase of the NP contribution, r and σ , through the comparison q q of the experimental measurements with SM expectations. From the definition of Eq. (41) we have the constraint, ∆M (cid:113) ρ ≡ q = 1+2r cosσ +r2 (44) q ∆MSM q q q q The values for ρ given by the UTfit analysis [74, 76] at the 95% C.L. are, q ρ = [0.53, 2.05] (45) d ρ = [0.62, 1.93] (46) s These constraints on ρ encode the CP conserving measurements of ∆M . The phase associated with NP can also q d,s be written in terms of r and σ , q q r sinσ sinφNP = q q , q (cid:113) 1+2r cosσ +r2 q q q 1+r cosσ cosφNP = q q (47) q (cid:113) 1+2r cosσ +r2 q q q Here [74, 76] gives the 95% C.L. constraints, φNP = [−16.6, 3.2]o (48) d φNP = [−156.90, −106.40]o∪[−60.9, −18.58]o (49) s these constraint represent those of the CP phase measurements of φ . d,s In order to consistently apply the above constraints all input parameters are chosen to match those used in the analysis of the UTfit group [74, 76] with the nonperturbative parameters, (cid:113) f Bˆ = 262±35MeV (50) Bs Bs ξ = 1.23±0.06MeV (51) f = 230±30MeV (52) Bs f = 189±27MeV (53) Bd 6 III. NUMERICAL ANALYSIS Inouranalysisweshallfirstconsiderthecaseofscalarandvectorunparticlesseparatelyandfurthersubdivideeach into four classes as follows, • One real coupling; c (cid:54)=0 and c =0, with c ∈R L R L • Two real couplings; c (cid:54)=0 and c (cid:54)=0, with {c ,c }∈R L R L R • One complex coupling; c (cid:54)=0 and c =0, with c ∈C L R L • Two complex couplings; c (cid:54)=0 and c (cid:54)=0, with {c ,c }∈C L R L R Hence we have a total of eight scenarios to consider in the following analysis. For our analysis we shall take the unparticle scale Λ = 1 TeV. In the literature [5, 36, 42, 43, 44, 45, 46, 47] it U has been common to perform an analysis with a fixed value of the scaling dimension, d = 3. It was recently shown U 2 [85,86]thatfromunitarityconsiderations[87]thereexistsaboundonthescalingdimension,d ≥j +j +2−δ . U 1 2 j1j2,0 Here (j ,j ) are the operators Lorentz spins. In particular this leads to the bounds d ≥ 2(3) for our scalar(vector) 1 1 U unparticle operators, which conflicts with the choice 3. First it should be mentioned that these bounds are derived 2 from a conformal field theory(CFT) point of view. Here we are considering a scale invariant sector at high energies, and in general scale invariant theories need not also be invariant under the conformal group. The specific example of Banks-Zaks fields though, is in fact invariant under the full conformal group [85, 86]. FIG. 2: Constraints on the d versus cS parameter space from B mixing(left) and B mixing(right) for the case of a single U L d s real coupling cS (cid:54)=0 and cS =0. Black points indicate the ∆M allowed regions, while grey points indicate the regions are L R d,s in agreement with both ∆M and the CP phase φ . q q In order to make a connection with the literature, and to quantify the impact of the recent measurement of the CP phase φ , we choose to fix the scaling dimension at d = 3. Through simple scaling of the coefficient S , it is also s U 2 dU possible to make a connection to a value of the scaling dimension in the region suggested by the above mentioned unitarity bound. In general the bounds on unparticle couplings with the scaling dimension in the region d ≥ 2(3) U shall be much weaker than for d = 3, unless the value is close to an integer, where there is a pole in the unparticle U 2 propagator. For example, if we write the scaling dimension for scalar unparticles as d = 2+(cid:15) , with (cid:15) a small U S S positivenumber,thenwehave|S |=|S |foravalueof(cid:15) ≈0.00021. Hencetheboundsderivedherefrom dU=3/2 dU=2+(cid:15)S S theCPconservingquantities∆M shallalsoapplytothecased ≈2.00021. Thesituationisnotsostraightforward d,s U for the bounds derived from CP violating constraints of φ . For vector unparticles we have a similar situation for d,s d =3+(cid:15) , with (cid:15) ≈2×10−11. The above unitarity bound shall be considered in more detail elsewhere [88]. U V V A. Scalar unparticles InthissectionweshallanalyzethecontributiontoB mixingfromscalarunparticlestakinganumberofexample d,s scenarios. In each case we shall use the measurements of ∆M as well as the CP phases φ to constrain the d,s d,s unparticle parameter space. 7 1. One real coupling: cS (cid:54)=0, cS =0 L R In this case the Wilson coefficients at the sale Λ simplify to, U CS = − AdU e−iφU (cid:18)MM2 (cid:19)dU m2 (cid:16)cS,bq(cid:17)2 (54) 2 sind π M4 Λ2 b L U M U CS = 2 AdU e−iφU (cid:18)MM2 (cid:19)dU m m (cid:16)cS,bq(cid:17)2 (55) 4 sind π M4 Λ2 b q L U M U C˜S = − AdU e−iφU (cid:18)MM2 (cid:19)dU m2 (cid:16)cS,bq(cid:17)2 (56) 2 sind π M4 Λ2 q L U M U CS = CS =CS =C˜S =C˜S =0 (57) 1 3 5 1 3 where q =d,s correspond to B or B mixing. d s In this first case, in order to have two free parameters, we shall allow the scaling dimension d to vary. In general U the unparticle effects shall be smaller for larger d . In this case the scaling dimension d also uniquely determines U U the NP CP phase and as such the small CP phase allowed by φ and the large CP phase required by the recent d measurement of φ will act as a stringent constraint on the d −cS parameter space. s U L Fig. 2 displays the constraint on the d −cS parameter space from the B mixing(left panel) and B mixing(right U L d s panel), including CP phase constraints. As expected the general feature of these plots is that for small d the mixing U contribution from unparticles is large and so the couplings cS,bq are strongly constrained. For larger values of d the L U opposite is true with d >2 resulting in no bound on cS,bq from either ∆M or ∆M . U L d s FIG. 3: Plot of the variation of the phase and magnitude of the quantity S with the scaling dimension d . dU U In the case of scalar unparticles the Wilson coefficients have the common quantity, S = AdU e−iφU (cid:18)MM2 (cid:19)dU (58) dU sind π M4 Λ2 U M U Here the NP CP phase is determined solely by arg(S ) such that σ = arg(S /MSM). The magnitude of S also dU q dU 12 dU determines the size of the unparticle contribution to B mixing and hence determines the level of constraint on the coupling cS. Fig. 3 shows the variation with d of the phase and magnitude of S , from which we can see that the L U dU phase of S is constrained to be in the range (−π,0). The magnitude of S is generally decreasing for increasing dU dU d except for when d approaches 2 where there is a singularity due to the factor 1/sind π. The general features of U U U Fig. 2 are now easy to understand. The large value of S for small d restricts the allowed values of cS to a very dU U L narrow region. For example, for d = 3 we have the bounds, U 2 |cS,bd| < 0.067 (59) L |cS,bs| < 0.23 (60) L 8 ThedifferenceintheleveloftheaboveconstraintsonthecouplingscS,bd andcS,bs areduetotheratioofCKMmatrix L L elements V /V ≈λ. Increasing d causes S to decrease sharply, resulting in an increase in the allowed size of the td ts U dU couplings cS as seen in Fig. 2. Approaching d =2 causes a rapid increase in S and thus the allowed values of cS L U dU L are again heavily constrained. For d >2 the quantity S becomes increasingly small, resulting in no bound on the U dU couplings cS. L FIG.4: PlotoftheallowedcS-cS parameterspaceforB mixing(left)andB mixing(right)inthecaseoftworealcouplings. L R d s Black points show regions which agree with the measurement of ∆M while grey points show additional agreement with the d,s measurement of the CP phases φ . d,s We can see from Fig. 2 that there are windows in the allowed parameter space of d −cS. From Eq. (44) we can U L determine r in terms of σ as, q q (cid:113) r =−cosσ ± ρ2−sin2σ (61) q q q q and for ρ <1 the phase σ is constrained to be in the region, q q −π−arcsinρ ≤σ ≤−π+arcsinρ (62) q q q Taking the minimum allowed value of ρ we can determine these windows as corresponding to, q −π−arcsinρmin+φSM ≤arg(S )≤−π+arcsinρmin+φSM (63) q q dU q q with φSM =−2β =−0.0409 and φSM =2β =0.781, determines these windows to be, s s d B : 1.56< d <1.92 (64) d U B : 1.80< d <2.00 (65) s U To the inside of these windows, with smaller values of the coupling cS, the unparticle contribution is small enough to L satisfy the mixing constraint. To the outside of these windows, with larger values of cS, the unparticle contribution L is larger than the SM contribution. When the phase σ is in the range (−π,−π) the quantity cosσ in Eq. (44) is q 2 q negative, allowing larger values of r . This solution corresponds to the case when the unparticle contribution carries q the opposite sign to the SM and turns over the sign of M . 12 The black points of Fig. 2 correspond to the region of parameter space allowed by the measurement of the B mixing parameters ∆M , while grey points also satisfy the constraint from the measurement of the CP phases φ respec- d,s d,s tively. The left panel of Fig. 2 shows that the allowed region is strongly constrained by the measurement of φ in d addition to the constraint of ∆M . This additional constraint restricts the parameter space of cs,bd considerably. For d L example, for d = 3 and including the CP constraint we have the improved bound, U 2 |cS,bd| < 0.024 (66) L 9 FIG. 5: Variation of allowed parameter space of the real coupling cS =cS with scaling dimension d for B mixing (left) and L R U d B mixing (right). Black plotted points agree with the CP conserving mixing quantities ∆M , while grey points also agree s d,s with the CP phases φ . d,s which is almost three times smaller than the bound from ∆M alone. As shown in Eq. (49), in the B system there d s isstillatwofoldambiguityinthemeasurementoftheCPphaseφ . Thistwofoldambiguityisthenseenintheright s panelofFig.2astwodistinctpairsofgreyregions. ThepairofgreyregionswithlargercS,bs,andconsequentlylarger L r , corresponds to φNP = [−156.90, −106.40]o. This region is clearly disfavoured as it corresponds to the coupling s s cS,bs being outside of the perturbative region. On the other hand the pair of grey regions with smaller cS,bs and r , L L s corresponds to φNP =[−60.9, −18.58]o. In this case the coupling cS is constrained to be in a narrow band away from s L zero. For example with d = 3 we have a two-sided bound, U 2 0.106<|cS,bs|<0.23 (67) L The small cS,bs region also indicates that a value of the scaling dimension in the range 1.22<d <1.87 is preferred. L U Despitethetwofoldambiguityinφ itisclearthatvaluesofthescalingdimensionintheranged <2arepreferred s U by the measurement of φ . Combining these φ preferred values for d with the constraints from φ implies a general s s U d bound on cS,bd as, L |cS,bd|<0.13 (68) L 2. Two real coupling: cS (cid:54)=0, cS (cid:54)=0 L R InthissecondcaseweshallallowtheleftandrightunparticlecouplingscS andcS tobothberealandnonzero. For L R this analysis we shall fix the scaling dimension d = 3. The allowed cS−cS parameter space is plotted in Fig. 4 with U 2 L R black points constrained by ∆M and grey points further constrained by the CP phase φ . The allowed region d,s d,s extends out along two lines where there is a cancellation between cS and cS. As a result no direct bound can be set L R on the unparticle couplings. Starting from the definition of R = Mq,NP/Mq,SM and taking the limit m → 0 we arrive at an approximate q 12 12 q relation, (cid:104) (cid:105) 6 (cS,bq)2+(cS,bq)2 −32cS,bqcS,bq ≈r (cid:15) (69) L R L R q q where the small quantity (cid:15) is defined as, q |Mq,SM| (cid:15) = 12 (70) q M f2 |S | Bq Bq dU Here it is clear that a large cancellation between the couplings cS and cS is possible. From Eq. (61) we find that L R (ρ −1)2 ≤r2 ≤(ρ +1)2 which leads to, q q q −0.1x ≤r (cid:15) ≤x (71) q q q q 10 withx =1andx =λ2. ThesolutiontotheseequationsisthenaparabolainthecS−cS parameterspacedescribed s d L R by, (cid:114) 16 1 (cid:16) (cid:17)2 cS,bq = cS,bq± 220 cS,bq +6(cid:15) (72) L 6 R 6 R q If we set (cid:15) ≈0, then these allowed regions follow along the lines, q 1 cS,bq ≈5cS,bq, cS,bq ≈ cS,bq (73) L R L 5 R in good agreement with the results of the full calculation shown in Fig. 4. FIG.6: Plotoftheallowed|cS|-φS parameterspaceforB mixing(left)andB mixing(right)forthecaseofasinglecomplex L L d s coupling cS (cid:54)= 0 and cS = 0. Black plotted points agree with the CP conserving mixing quantities ∆M , while grey points L R d,s also agree with φ . d,s In this case it is possible to have very large values of the couplings, as long as they lie approximately on the lines shown in Eq. (73). These solutions with large cancellations are clearly highly fine-tuned. If we consider the case of cS =cS then we have no such fine-tuning and we can extract the bounds, L R cS,bd ≡cS,bd < 2.5×10−2 (74) L R cS,bs ≡cS,bs < 0.14 (75) L R from the constraints of ρ . From the grey regions of Fig. 4 we again see that the measurement of the CP phases d,s φ further constrains the allowed parameter space. In the B system the allowed region is reduced by more than a d,s d half. For the case cS =cS the CP constraint provides a much improved bound, L R cS,bd ≡cS,bd < 6.7×10−3 (76) L R In the B system the new CP measurement of φ prefers points away from the origin. It is also interesting to notice s s that the case, cS,bs =cS,bs is disfavoured by the latest measurement of φ at the 3σ level. L R s The case of equal left and right couplings is shown in more detail in Fig. 5 where the allowed parameter space is plotted as a function of the scaling dimension d . Fig. 5 shows the allowed parameter space for both B (left panel) U d andB (rightpanel). Fromtheleftpanelwecanseethattheallowedparameterspaceisrathersimilartothatshown s in Fig. 2. This time the windows in the plots have disappeared for the case of B and have moved in the case of B . d s When we have cS = cS we get a new contribution from CS not present in the case of Sec. IIIA1 with cS = 0. The L R 4 R introduction of CS changes the overall sign of MNP and so in this case these holes obey a modified relation similar 4 12 to Eq. (63) except there is no factor of −π. This means that for the case of B there is no such window and for B d s the window is now at 2 < d < 2.23, as shown in Fig. 5. In this case the impact of the CP phase measurement φ U d is to again reduce the allowed parameter space. The impact of the recently measured φ is much more profound, as s can be seen from the right panel of Fig. 5 where there are no grey points and only black regions. The lack of any grey regions in this plot indicates that this case cannot satisfy the 3σ measurement of φ for any choice of the scaling s dimension d . U