SMR/1847-9
Summer School on Particle Physics
11 - 22 June 2007
SPECIAL LECTURE: Unparticle Physics
A. S. Cornell Institut de Physique Nucleaire de Lyon, France UnparticleUnparticle Physics Physics
presented by Alan S. Cornell (Institut de Physique Nucléaire de Lyon) for the ICTP Summer School on Particle Physics, June 15 2007 Outline of lecture
– Motivation – Review the basic model – Example 1: t ĺ u + U – The Unparticle propagator – Example 2: e+ + e- ĺ P+ + P- – Example 3: LFV decay Pĺ e- + e+ + e- – Example 4: B ĺ K + missing energy – Example 5: Lepton anomalous magnetic moments – Example 6: Mono-photon events in e+e- collisions – Summary Unparticle articles out there to date[4-6-2007] • Howard Georgi, “Unparticle Physics”, arXiv: hep- ph/0703260, • Howard Georgi, “Another Odd Thing About Unparticle Physics”, arXiv:0704.2457, • K. Cheung, W-Y. Keung and T-C. Yuan, “Novel signals in unparticle physics”, arXiv:0704.2588, • M. Luo and G. Zhu, “Some Phenomenologies of Unparticle Physics”, arXiv:0704.3532, • C-H. Chen and C-Q. Geng, “Unparticle physics on CP violation”, arXiv:0705.0689, • G-J. Ding and M-L. Yan, “Unparticle Physics in DIS”, arXiv:0705.0794, • Yi Liao, “Bounds on Unparticles Couplings to Electrons: from Electron g-2 to Positronium Decays”, arXiv:0705.0837, • T. M. Aliev, A. S. Cornell and Naveen Gaur, “Lepton Flavour Violation in Unparticle Physics”, arXiv:0705.1326, • X-Q. Li and Z-T. Wei, “Unparticle Physics Effects on D- anti-D Mixing”, arXiv:0705.1821, • C-D. Lu, W. Wang, and Y-M. Wang, “Lepton flavor violating processes in unparticle physics”, arXiv:0705.2909, • M. A. Stephanov, “Deconstruction of Unparticles”, arXiv:0705.3049, • P. J. Fox, A. Rajaraman and Y. Shirman, “Bounds on Unparticles from the Higgs Sector”, arXiv:0705.3092, • N. Greiner, “Constraints On Unparticle Physics In Electroweak Gauge Boson Scattering”, arXiv:0705.3518, • H. Davoudiasl, “Constraining Unparticle Physics with Cosmology and Astrophysics”, arXiv:0705.3636, • D. Choudhury, D. K. Ghosh and Mamta, “Unparticles and Muon Decay”, arXiv:0705.3637, • S-L. Chen and X-G. He, “Interactions of Unparticles with Standard Model Particles”, arXiv:0705.3946, • T. M. Aliev, A. S. Cornell and N. Gaur, “B ĺ K(K*) missing energy in Unparticle physics, arXiv:0705.4542, • P. Mathews and V. Ravindran, “Unparticle physics at hadron collider via dilepton production”, arXiv:0705.4599. Motivation for this model – Scale invariance is a very powerful concept with wide applications. In particle physics it is very predictive in analysing the asymptotic behaviour of correlation functions at high energies – Scale invariance at low energies is broken by the masses of particles – Our quantum mechanical world seems well described by particles – But why can’t we have a scale invariant sector in our theory? – An interacting scale invariant theory would have no particles, it is made of unparticles – So what would unparticles look like? Some definitions of use
Scale invariance: a feature of objects or laws that do not change if the length of energy scales are multiplied by a common factor
An Effective theory is an approximate theory that contains the appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, but ignores the substructure and the degrees of freedom at shorter distances.
That is, we shall use the degrees of freedom appropriate to the scale of the problem, for example, we don’t use quantum gravity to calculate projectile motion! More generally, any theory at momentum p< Ci LLeff 0 Oi ¦ ni i M where the Ci’s are the short distance quantities (in QCD these are perturbatively calculable if M>>ȁQCD) and the Oi’s are the long distance quantities The Basic Set-up • Imagine that at very high energies our theory contains the SM fields and a conformal sector due to fields of a theory with a nontrivial IR fixed point (call them BZ fields) • These two sets interact through the exchange of particles with a mass scale MU • Below the scale MU, there are nonrenormalizable couplings involving both SM and BZ fields suppressed by powers of MU 1 k OOSM BZ for k > 1 MU • As in massless non-Abelian gauge theories, renormalization affects the scale invariant BZ sector inducing dimensional transmutation at an energy scale /U • In the effective theory below the scale /U the BZ operators must match onto the new (unparticle) operators, which have the following form in the effective interaction ddBZ U CUU/ k OOSM U MU where dU is the scaling dimension of the unparticle operator CU is a coefficient determined by the matching condition • As a nontrivial scale invariant IR fixed point theory is thoroughly nonlinear and complicated, the matching of the BZ physics to the unparticle will be complicated cf. high-energy QCD and low-energy hadron states As such, we shall estimate these constants only roughly. • First note though, that this scaling dimension dU can be a non-integer number The reason for this is (by analogy): • This is a CFT defined in terms of an OPE • When one does an OPE and then tries to evaluate the overlap of two operators we get a Taylor-like series (basically the perturbative expansion of the diagrams) in terms of the coupling constants • The first term of the expansion is like the correlator function • Higher order terms give us the anomalous dimension (higher order diagrams changing the classical dimension, hence anomalous) • This anomalous dimension need not be an integer one Typical of scale invariant theories where one needs an OPE. Note that this also happens for B-decays, as we use an OPE there also. Note that in the previous equation, the unparticle operator was a Lorentz scalar, but there are other possible Lorentz structures, some of which we shall present later So what physics do we have below /U? Note that we shall focus on the production of the unparticle stuff • The most important effects will be those involving only one factor of ddBZ U CUU/ k MU The result will be the production of unparticle stuff, contributing to missing energy and momentum Density of final states • To calculate probability distributions we need to know the density of final states Note that this is essentially determined by the scale invariance 4 2 †2dP 0()()0OxOx eiPx 0(0)() O PU P UU³ U (2 )4 From scale invariance the matrix element scalesS with dimension 2dU, requiring 2 20222 0OP (0)UTT () PAPPP ()()()dU UdU Lets compare this to the phase space for n massless particles 4 n n 44§· 2 0dp 0 2 2 2 (2)SG G T T TPp ( pp )( )j APPP ( )( )()n ¨¸¦ jjjn (2 )3 ©¹j 1 j 1 where 5/2 4(1/2)S *n An 2n (2S )** (nnS 1) (2 ) Thus we could say that Unparticle stuff with scale dimension dU looks like a non-integral number dU of invisible particles Let us then identify An above with the AdU from earlier Note that any other choice could always be absorbed into our choice for CU earlier The Operators Re-writing our parameters into the form O i OO dU SM U /U we note that below the scale /U our effective theory will give us four-Fermi like interactions, with powers of dU from dimensional analysis. For example c c S PPPPP AAJJJ''PPwwOO A5 A ddUUUU //UU c c V ''P A P 11AAOO A5 A ddUUUU //UUJJJ As my examples will focus on lepton based processes I will only use the above interactions. However, other interactions are just as easily constructed Example 1: t ĺ u + U H. Georgi, arXiv:hep-ph/0703260 To illustrate the procedure in a realistic situation, consider the decay t ĺ u + U, from the coupling P O 1..P iu w5 tOhcU /dU JJ dBZ CUU/ where the constant O k MU Ignoring the u mass, the final state densities are 2 02 dp)uu ST G p u p u 2 022dU dpAppp)UU d U U U U TT The phase space factor is then 4 4 dp 2 4 §· j dP) SG P pjj dp ) 4 ³ ¨¸¦ 2 ©¹j j and the differential decay rate as 2 M ddP* ) 2m S 22 2 AmEO mE2 d* dtuU tu S2 2 T 2 dE 2 / dU 2 2 dU uU mmEttu 1 d 222d 2 * 4112//U or dUU d E u m t E u m t * dEu 8 7 dU 1.3 6 d 1.6 dln*/dE versus E in U u u ĺ 5 d 1.9 u U units of mt with dU 2.2 /dE 4 * d = 1.3 to 2.8. d 2.5 U ln 3 U d t d 2.8 2 U The dashes get m smaller as j increases 1 0.1 0.2 0.3 0.4 0.5 Eu/mt ĺ As dU ĺ 1 from above, dln*/dEU becomes more peaked at Eu = mt/2, matching smoothly onto the 2-particle decay limit, as expected. For higher dU the shape depends sensitively on dU The kind of peculiar distributions of missing energy that we see in this figure may allow us to discover unparticles experimentally! The Unparticle Propagator We shall now consider some virtual effects of unparticles. Note that interference between SM and virtual unparticle amplitudes can be a very sensitive probe of high-energy processes (as we shall see) Working, as before, to lowest trivial order in the small couplings of unparticles to SM fields, we will require our unparticle operators to be Hermitian and transverse P 0 w POU Note that unparticle operators with different tensor structures can be dealt with in a similar way PQ The transverse 4-vector unparticle propagator is given by PQ P Q 2 42Ad f 2 gPPP/ 2 iPx 0 ( () ())0U ( )dU eTOxOxdxiM 22 dM ³³UU 2SH0 PQ P Q PMi 2 2 Ad gPPP/ 2 dU iPiU 2sin() SSdU Where the tensor structure reflects our requirement on the H operator And the powers of dU such that we maintain scale invariance Note that there will be an imaginary part to the propagator when P2 > 0 (spacelike) and none when P2 < 0 This can be checked by finding the discontinuity across the cut for P2 > 0 SS AAdd112222 2 2(2)(2) iPiiiPeeU ()dUU (1)HH d (1) d U U () d UUU idSS id 2sin( ) 2sin( ) ddUU A 1 dU 22dd22 iPidAP()UU 2sin() () 2sin( ) UdU dU S S Note that while the discontinuity is not singular for integer dU > 1, the propagator is singular in the denominator. This would be a real effect As such we shall look to see what virtual effects, from unparticles, this imaginary part will have Example 2: e+ + e- ĺ P+ + P- H. Georgi, arXiv:0704.2457[hep-ph] PP Let us compute the cross-section for e+ + e- ĺ P+ + P- in the presence of the interactions 11 cckdUU kd VU//''P AU P kkAAJJJOOUU A5 A MMUU ignoring lepton mass where q2 = s, the total CM energy, and T is the angle between the P- and the e- 2 2[2 2222 1cos2 Ms 'VV s ' AA s ' VA s ' AV s T Re** Re 4cos ] ' VV ss ' AA '' VA ss AV T e P* sddsJ where '{xy ¦ xj yj ' j and x,y= V or A jZU ,, The d’s and '’s are jZJ U TT 2 c e 1 sin V deVj 4 T 1 sin cos dU M Z /4 c 0 e TT A d Aj 1 sin cos dU M Z 11 Ad 2 id 2 S ' U sedU U j 2 2sin ssMiM*ZZZ d U S Note that we have assumed that unparticles are lepton-flavour blind, but our earlier expression is entirely general Consider now the total cross-section in the LEP region, where we are used to thinking the Z pole is a poor place to look for interference effects This prejudice is not warranted for unparticle interactions as this can interfere with both real and imaginary parts of the SM Beginning with cV =0 photon and U do not 2 directly contribute. 5|cA| Expect only interference with Z Shown is the 25 50 75 100 125 150 175 200 fractional change in total * for small cA 2 -5|cA| sGeV o for various dU between 1 and 2. This is extremely sensitive to dU. We can understand this by thinking about the phase of the U propagator along the cut IdU = – (dU –1) S The real part is positive for 1 < dU < 3/2 and negative for 3/2 < dU < 2. Thus away from the Z pole we expect destructive (constructive) interference below (above) the pole for 1 < dU < 3/2 and vice-versa for 3/2 < dU < 1. 2 |cA| A closer view of the previous plot around the Z pole 90 91 92 93 sGeV o As can be seen from this plot, things are much more complicated near the Z pole, as both real and imaginary parts contribute to the interference 2 |cA| /2 Here our values of dU are closer to 1.5 (that is, 1.48, 1.49, 1.51 and 1.52) 25 50 75 100 125 150 175 200 sGeV o The situation simplifies in an interesting way for dU = 3/2. In this case the unparticle amplitude interferes only with the imaginary part of the Z exchange. This is a smaller effect as it 2 is proportional to the Z width, rather than s – MZ . It gives constructive interference that peaks on the Z pole and goes to zero far from the pole. 2 10|cV| Now we consider the case of purely vector coupling, where we expect interference with the photon, and only 25 50 75 100 125 150 175 200 weakly with the Z sGeV o We now expect constructive interference for 1 < dU < 3/2 and destructive for 3/2 < dU < 2. Note that the dip at the Z pole arises from our plotting the fractional change, and the large contribution from the pole is in the denominator. Note that the unparticle interference in the matrix element also gives rise to a complicated pattern of changes in the FB asymmetry, which we won’t cover here. However, this does point to some very interesting and detailed interference effects, unique to unparticle stuff, even though we lack a truly detailed picture of what it looks like! Example 3: LFV decay Pĺ e- + e+ + e- T. M. Aliev, A. S. Cornell and N. Gaur, arXiv:0705.1326[hep-ph] For our study we will consider the following set of effective interactions for the unparticlePP operators which have couplings to leptons: c V ''P cA P 11AAJJJOO A5 A ddUUUU //UU The scalar operator, in principle, couples with SM fermions, however, their contributions are proportional to fermion mass and are suppressede here µ−(p ) − − − 1 (p3) µ (p1) e (p2) U U − + (ep2) e(p4) −(p ) e +(p ) e 3 4 (a) (b) The decay is described by the Feynman diagrams above, with thePQ matrix element: MMM 12 with PQ PQ P Q 2 / 2 A gPPP dU dU 2 Mu12 JJJJ aa 125133454 uuaa H v22 Pi dU 2sin / dU 2 / 2 A gPPPPQ P Q dU dU 2 Muaauuaav23125123454 22 Pi dU 2sin JJJJ/ H dU where P = p1 –p2 and Q = p1 –p3. If we take the particles as massless, then S 2222Re * MMM 12 MM 12 S 2232 [2222 . . . . M11123414231324 F aaaa ^ pppppppp ` 4....] aaaa1234^` pp 1 4 pp 2 3 pp 1 3 pp 2 4 2232 [2222 . . . . M2 F 2 aaaa 1 2 3 4^` pppppppp 14 23 12 34 4....] aaaa1234^` pp 1 4 pp 2 3 pp 1 2 pp 3 4 Re* 32Re * 2222 4 . . MM12 FFaaaa 12^` 1 2 3 4 aaaapp 12341423 pp Our calculation will be done in the CM frame of the outgoing electron and positron, as denoted by the momenta p and p SSS 3 4 respectively. 11133GG3 G dp24dp3 dp 2 4 2 *dppppM 333 SG 1234 22222 22 22 EP EEE234 S 2 2 d* 11§·m 4m 11 2 11 e M 22293 ¨¸ dsdxs ©¹ s s P 0.01 Λ = 1000 GeV Λ 0.0001 = 5000 GeV Note that the present Λ = 10000 GeV experimental limits are 1e-06 BR < 1 x 10-12 1e-08 1e-10 BR 1e-12 Here we have 1e-14 presented variation of 1e-16 BR against dU for 1e-18 various values of /U 1e-20 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 du As can be seen, the branching ratio is very sensitive to the scaling dimension 1e-09 1e-10 1e-10 1e-11 1e-11 1e-12 1e-12 1e-13 BR BR 1e-13 1e-14 1e-14 1e-15 Λ = 1000 GeV Λ = 1000 GeV Λ = 2000 GeV Λ = 5000 GeV Λ = 5000 GeV 1e-15 Λ = 10000 GeV 1e-16 1e-16 1e-17 0 2 4 6 8 10 0 2 4 6 8 10 × -6 × -5 a1 (= a2) 10 a1 (= a2) 10 Here we show the dependence of BR against the various ai’s, where a1 and a2 correspond to the LFV interactions Again this is done for different values of /U This shows how sensitive the BR is dU and LFV couplings Note that the same LFV couplings will be involved in other LFV processes, such as Pĺ e J. As such, an exploration of the phenomenology of LFV unparticle operators on radiative LFV processes and their possible correlation with this decay, would be interesting. Compare this with the well known strong correlation of these LFV processes in SUSY, and how these correlations tend to change substantially in LHT1 1 M. Blanke, A. J. Buras, B. Duling, A. Poschenrieder and C. Tarantino, arXiv:hep-ph/0702136 Example 4: B ĺ K + missing energy T. M. Aliev, A. S. Cornell and N. Gaur, arXiv:0705.4542[hep-ph] Decays of the form b ĺ s + missing energy have been the focus of much investigation at B factories, with measured 5 results of Br Bo u KQQ 1.4 10 In the SM the decay B ĺ K QQ is described at quark level by the Hamiltonian S G * D F 11 HVVCsb 10J JQJ 5 JQ5 22 tb ts PP After taking into account the three SM neutrino species, the differential decay width is SM 22 2 2 d* D GF *23/2 VV C10 f 2752 tb ts dEKBS m 444222222 where O mmqmqmqmm222 BK B KO BK Where we have made use of the form factors (') () ' Kp sJ PP bBp p p f q f Kp(') s5 bBp () 0 JJP P Note also that we shall use the propagator for the scalar unparticle field as 2 4.iA 2dU dxeiP x 0()(0)0 TOxO dU P ³ UU 2sin dUS In the case of B ĺ K U the following scalar operators can contribute 1 sJJ CCw5 bO dU PPSP U /U In which case the matrix element is C 22 2 S S ªº MfmmfqO BK U dU ¬¼ /U In which case the decay rate is SU 2 d* Ad 2 2222 dU U 2 2 2 CEmmmEmSKKBKKB 8 dU dEKBUS m / 2 22 2 2 ªºfm m fm 22 m mE ¬¼BK B K BK Similarly for the vector unparticle operators 1 P 1 sJJ CC 5 bO dU P VA U /U We get the matrix element C V V ªº' P P MfppfqO 1 P U dU ¬ ¼ /U And the differential decay rate VU 2 d* Ad 22 2222 dU U 2 2 22CfVKKBKKB Emmm Em 8 dU dEKBUS m / 2 2 ªºmm «»222 BK mm mE 22 «»BK BKmm2 mE ¬¼BK BK To obtain the total decay width we integrate from 2 2 mK < EK < (mB + mK )/2mB Note also that the energy distribution of final hadrons is very different from the SM compared to unparticles, such that, though the present limits are one order of magnitude above the SM expectation (SuperB will fix this) we can still place constraints now. 0.001 SM Scalar operator Vector operator Here we plot the decay rate 0.0001 against E for the different K K )/dE − ν ν decay modes. 1e-05 K → (B Γ Note the striking difference in d the high energy regime for the 1e-06 vector operators 1e-07 0.5 1 1.5 2 2.5 EK (GeV) 0.001 0.001 SM SM Λ = 1000 GeV Λ = 1000 GeV Λ = 2000 GeV Λ = 2000 GeV Λ = 5000 GeV Λ = 5000 GeV 0.0001 0.0001 K + missing ) K + missing ) → Expt. Limit → Expt. Limit 1e-05 1e-05 Br(B Br(B 1e-06 1e-06 1.4 1.5 1.6 1.7 1.8 1.9 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 dU dU Here we look at the branching ratio against dU for different values of /U As can be seen, it is very sensitive to both dU and /U And that the vector operator is more strongly constrained than the scalar 0.0001 0.1 SM dU = 1.5 SM dU = 1.8 dU = 1.5 d = 1.8 dU = 2.1 0.01 U dU = 2.1 0.001 K + missing ) Expt. Limit K + missing ) → 1e-05 → 0.0001 Br(B Br(B Expt. Limit 1e-05 1e-06 0.0001 0.001 0.01 1e-07 1e-06 1e-05 0.0001 CS CV Finally we plot the branching ratio against CS and CV for various values of dU These plots show how the scalar process constrains CS whilst the vector process constrains CV To conclude, these operators also contribute to other processes, such as meson anti-meson mixing etc, however, our constraints here are much stronger Example 5: Lepton Anomalous Magentic Moments K. Cheung, W-Y. Keung and T-C. Yuan, arXiv:0704.2588[hep-ph] If we replace one photon exchange in QED by the P unparticle associated with the vector operator OU , one can derive the unparticle contribution to the lepton anomaly 'aƐ = (gƐ –2)/2 1 2 2 dU 321 cZVd§·m * dd * 'a U A UU A 4222¨¸ S ©¹/*UUd where mƐ is the charged lepton mass and that here we assume cV is lepton flavour blind. 2 2 As dU ĺ 1 we get 'aƐ ĺ cV /8S , and if we set cV to e we reproduce the well known QED result Muon Anomaly versus dU Λ 1 1 Here we plot the muon 1. 10 8 aΜexp aΜSM anomalous magnetic 1. 10 10 moment contribution from 1 10 the unparticle versus d Μ U a 12 1. 10 10 2 for various cV’s 1. 10 14 The horizontal line is the 10 3 experimental value less 1. 10 16 the SM contribution 1.2 1.4 1.6 1.8 2 dU It is amusing to see that current experimental data of the muon anomaly can give bounds to the effective coupling cV and the scale dimension dU already Example 6: Mono-photon events in e+e- collisions K. Cheung, W-Y. Keung and T-C. Yuan, arXiv:0704.2588[hep-ph] The energy spectrum of the mono-photon from the process e- e+ ĺ J U can also be used to probe the unparticle Its cross-section is given by 2 2 dU 1 2 EA §·P dMV J dU U dEd: 21632¨¸ 2 s //UU©¹ with the matrix element squared S 22 2J 2 222ut2 sP MeQc 2 U eV ut 2 where PU is related to the photon energy by the 2 1/2 recoil mass relation PU = s – 2 s EJ 10-1 Here we have plotted the mono-photon ) distribution for various -1 choices of d (GeV U γ 10-2 /dE σ dU=1.2 d σ d =1.5 1/ U The sensitivity of the dU =1 d =3 scale dimension can U dU=2 10-3 be easily discerned 10 20 30 40 50 60 70 80 90 100 Eγ (GeV) Note that mono-photon events have been searched quite extensively at LEP experiments in other contexts and a more detailed study by K. Cheung et al. is expected soon Summary • Unparticle physics, due to conformal invariance, might appear at the TeV scale • An effective field theory can be used to explore the unparticle effects • It can lead to interesting phenomenological consequences, due to the scale dimension being able to take non-integer values, which can be checked at low energy experiments • Such as particular missing energy distributions • The unparticle propagator in the time-like region has interesting properties that force us to re-examine preconceived ideas about interference. • In the LFV decay e+ + e- ĺ P P, we demonstrated the sensitivity of these processes to the scaling dimension and other parameters. Such that a study of other LFV processes will place strong constraints on this model. • In our study of unparticle physics in the LFV P ĺ e- e+ e- we determined the decay width is sensitive to the virtual effects of these unparticles • In our study of B ĺ K + missing energy, we were able to constrain the scalar and vector operators from experimental data (future results will further constrain this) • Current experimental bounds can be used to constrain some of our parameters from (g – 2)P • Furthermore, these operators will appear elsewhere, placing additional constraints on them.