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JHEP12(2007)033 of U = 2, d U d hep122007033.pdf smology of December 10, 2007 November 14, 2007 November 28, 2007 September 14, 2007 nparticle operators Revised: re comparable for Received: Accepted: Published: ngly on the dimension nts. In general, astrophysical orial unparticle couplings are sions and compare the limits with http://jhep.sissa.it/archive/papers/jhep122007033 /j = 1, 5th force experiments yield exceedingly strong [email protected] , U d Published by Institute of Physics Publishing for SISSA Conformal and W Symmetry, Classical Theories of Gravity, Co We investigate the effects of all flavor blind CP-conserving u [email protected] Institut Theoretische für Physik, Zürich, Universität Winterthurerstrasse 190, CH-8057 Switzerland Zürich, E-mail: SISSA 2007 c the unparticle operator. While for ° Theories beyond the SM. bounds, the bounds from stellar and supernova cooling are mo with stellar cooling beinggenerally most stronger than restrictive. those Bounds on scalar on ones. vect Keywords: Ayres Freitas and Daniel Wyler Abstract: on 5th force experiments, stellareach cooling, supernova other explo and withbounds those obtainable are from considerably collider stronger, experime however they depend stro Astro unparticle physics JHEP12(2007)033 1 2 8 4 12 ons 10 rpose we only need the r Bij [2]. The picture is nd Standard Model fields ion of the standard model couple weakly to Standard ailed and complete treatment raints can highly restrict the ection [3, 4]; in this paper we At low energies the unparticle ons of the SM as they are not . [5] a list of operators composed ariant sector with a non-trivial . A different, but in effect similar iance is not broken in the infrared, , which is in general non-integer. U d – 1 – In this paper we want to assess possible effects of this extens We consider only couplings between SM singlet unparticles a fixed point was proposed byModel Georgi [1]. (SM) particles, These new are fields,described quite which in different terms from of other particles butdeviation extensi rather from by the ”unparticles” standardvalid model up has to a been certain proposedsector scale, above by is which Van characterized the de by picture a changes. scaling dimension 1. Introduction Recently the possible existence of a non-trivial scale-inv 2. Constraints from 5th force experiments 3. Constraints from stellar cooling 4. Constraints from SN 1987A 5. Comparison to reach of colliderA. experiments and Phase-space conclusi integrals Contents 1. Introduction in astrophysics. There arecombine by the now different several manifestations and studiesof give in the also this various a unparticle dir more operators. det Ifas the it conformal invar is assumedinteractions throughout between unparticle this and paper, SM astrophysical fields. const through CP-conserving and flavor blind interactions.of In SM ref fields with dimensions 4 or less has been given. For our pu JHEP12(2007)033 le . U does (1.2) (1.1) (1.4) (1.3) (2.1) O ssible µ -boson U S2 U c Z O f ∂ O µ 5 µ U γ µ f ∂ O f ∂ 5 µ ¯ µ γ fγ µ ¯ U f ∂ fγ oupling constants d ¯ Z µ fγ P2 U c U d ¯ Z M U S2 fγ d U O U P2 c M U C + Λ O d U S2 The term with ρσ U C + Λ F + ρσ ass, chosen as the e: dard tests of new particles U F epending on their chemical U µν e space integration is more with some useful results. + . . . arametrized by the potential ty [6, 7]. The most stringent , f O es between SM particles that hat a new (fifth) force would F 5 µν U + rived on different length scales ation. F f O r/λ r µ , which were originally pioneered 5 − µνρσ ¯ ¯ e DfO fD/f fD/γ ǫ µνρσ j U U ǫ ¯ ¯ U DfO fD/f fD/γ d d B Z Z d ˜ ieQA γ Z S1 i P1 U ˜ γ c ˜ c U U γ d U B M M c d U d U S1 ˜ γ M P1 + ) C 4 C C Λ Λ µ 1 4Λ + + ∂ − + + H µ U − 1 U = µ U ( U 2 O f O µ – 2 – O 5 f O D γ µν 5 µν µ γ F F ξGm µ ¯ fγ µν ± µν ¯ 1 F fγ F − 1 U U A U d − )= is the covariant derivative, which introduces four-partic Z d c Z d U A γγ r U γγ C ( c d U M C M D 5 4Λ 4 Λ V + − + µ U = ≡− µ U γγ f O f O has the same structure as the effective coupling of axions. Po U µ µ L ˜ γ ¯ fγ ¯ ˜ γ fγ c 1 1 − − U V V U d c Z C d U M Λ , so that the only unknown quantities are the dimensionless c Z ≡ = M For photons we have In the following sections, we will consider the various stan For the experimental analyses, the fifth force is typically p ff U . In the above equations, L X couplings involving the photon through not contribute to physical processes due to current conserv Here the coefficients in eq. (1.2) have been scaled to a common m c mass couplings to fermions and gauge bosons. With fermions we hav The term with couplings to gluons are not considered. and forces and reachinvolved than our for conclusions. usual particles, Since we have unparticle added phas an appendix 2. Constraints from 5th force experiments New massless or light degrees oflead freedom to can an mediate effective new modification forc constraints of come the from composition-dependent Newtonian and experiments Pekár law Fekete of [8].by gravi Eötvös, They make use ofin the general fact act t differentlycomposition on [9]. different Limits bodies for suchranging of an from equal interaction sub-meter have mass, to been d astronomical de scales of order AU. JHEP12(2007)033 ≈ en ) 1 (2.8) (2.6) (2.2) (2.4) (2.3) (2.7) H 1 ( m , ≈ 2 U e interaction is d i,j /p − ν cleons and atoms 2 m p ) µ iǫ p = 2. an be applied to the − U + are the baryon numbers d 2 ces therein), results are readily translated to the sion limit at length scale p µν i,j ange can be derived from g . , − ons with , B ( . − -energy limit one obtains 2 2) 2) 2) ) / / − / ) ) 12] for gives 1 1 U U ) 1 1. do not contribute to long-range s on eq. (2.1) come mainly from U U d U 2 U d d − − 2) (2.5) S1 πd d − d ≥ λ c P2 U U Γ( Γ( − U ) A Γ( d d C is exponentially suppressed, while for 1 d , c U 5 Γ( Γ( d H 2 sin( P1 Γ( V 1 i U U j ( , U d d , c 2 d 2 2 Γ(2 = λ m 2 , up to a prefactor, and yields Z S2 i − − i c U 2 2 V Z Z d ≫ M m c π . , Gm M A M ∝ (0) ) r 1 – 3 – j 2 S1 ν U U U − 2 A 2 V lim c d d B c c U ξ O 2 i 2) d ) U 1 2 / − 2 U U d B x 2 / 2 d d r Z . For − 2 2 ( / 1 2 2 ) e / λ 1 ) ) ) µ U 1 U M + 1 1)Γ(2 π π π π π π ≈ ≈ π 2 V U Newton’s constant of gravity. (2 − (3 c (2 4 (2 Cα d r T O | U G lim has been normalized to the gravitational interaction betwe = , 0 Γ( d = = = = h U ξ U U U U Γ( α α ipx V α α U 2 d / x e 2 5 4 ) π d π 16 (2 Z atoms, with 1 = ≡ H U 1 µν F d (1) accounts for the quark and electron density inside the nu contributes in the same way as A are the masses of the electrons and nucleons between which th ∆ O A c = i,j the experiments are less sensitive [7]. Therefore the exclu m C . λ Z Taking the experimental values (see ref. [7, 13] and referen The experimental limits on an interaction of type eq. (2.1) c The potential for interaction due to vector unparticle exch Also M is ≪ 1 90 This result agrees well with the power-law analysis in ref. [ two hydrogen shown for different scalingaxial-vector dimensions and in scalar figure cases. 1. They can be unparticle force eq. (2.4) bymeasurements at observing a that length the scale constraint λ The scalar and pseudo-scalar interactions where of the two test objects. its propagator [10, 11] where the coupling constant of the test objects. For a conservative limit, we take r non-relativistic forces. However, the contribution from By taking the Fourier transform of this propagator in the low where exchanged. The major contribution here comes from the nucle JHEP12(2007)033 . X (3.1) + e + e 12 12 10 10 → e + ams are shown in 10 e 10 10 is the axion-electron , X aee 88 10 ton process is + g 10 10 In the hot and dense envi- e complicated. Due to the he following, we will focus pe fifth-force experiments at tal-branch stars, the Comp- rticles can be derived from 2+ rnovae [17], horizontal-branch r helium ignition flash [16, 21 – n emission, with the strongest produced efficiently and would ng stars, which would lead to a 66 10 10 + He e . The dashed lines indicate the overall 2 U ¸ → d e 10 10 ω 2+ m [m] [m] · λλ 22 44 2 e [1], the final state integration requires some 10 10 + He 2 aee m 2 – 4 – e 3 αg − , U d 0 0 X = ) 10 10 c a 2 U + σ p + )( 3 3 / / 2 U -2 is the incoming photon energy. p 10 10 ( + H ω θ = 1 = 4 = 5 = 2 e range. ) and bremsstrahlung involving Hydrogen and Helium nuclei as U U U U 0 U λ d d d d p → -4 -4 -2 ( X 10 10 θ + + U , for various scaling dimensions d e , where λ e A + H -6 -6 m → e 10 10 0 0 e -3 -3 -9 -9 -6 -6 -21 -21 -15 -15 -18 -18 -24 -24 -12 -12 10 10 10 10

3 3 ≪ 10 10 10 10 10 10

10 10 10 10 10 10 10 10 10 10 +

V V ω

γ c c Limits on vector unparticle interactions from Eötvös-ty is the axion or unparticle. The corresponding Feynman diagr X The total cross-section for axion emission through the Comp For unparticle emission, the calculations are somewhat mor Mainly two processes contribute to energy loss from horizon figure 2 (a) and (b). well as electrons, different length scales Here Figure 1: limit derived from the whole coupling. phase space factor 3. Constraints from stellar cooling Constraints from stellar cooling on fermion couplings. ronment of stars, lightcontribute weakly interacting to particles the can coolingwhite be of dwarfs the [14 – 16], star. thestars with ignition Constraints a condition on helium-burning for core such [1823]. type – pa 20], I These and supe red processes giants have nea bounds been coming studied from extensively helium-burning foron stars axio the and evaluation red of unparticle giants.reduction emission of In from the t helium-burni lifetime of the horizontal-branch stars. ton process in the limit JHEP12(2007)033 e U U (3.4) (3.2) (3.3) (3.6) (3.9) (3.5) (3.7) (3.8) one finds e γ m ≪ , ke processes, (b) , ω U d U 2 d 2 ¸ γ ¸ γ Z γ e Z ω πM ω πM 2 cally, with a distribution 2 , · , , , 2 · hen 2 2 ) ) . In the limit − − − ) ω U U ( U U U d d d d c U 2 d 2 2 . ¸ σ , U e,N U e,N U e,N ) ¸ ¸ 1 2 rticle-photon couplings. ) Z γ 2 Z Z )Γ(2 2 U n )Γ(2 ω − U U ρ d e (H) ω ω ω U d 2 d πM πM πM d m + 2 − 2 2 + ω/T π · U · · H γ γ γ e ωωn U d ) ) ) X d d U 2 U U )(3 + 2 – 5 – d ∞ d )= d )(3 + 2 U ) 0 1+ U d Z U (2 + 2 d d (2 + 2 )Γ(2 T,ω ≈ U )Γ(2 )Γ(2 = 1 ( U d e U U γ d 2 U d d n , n (1 + 2 ee e e e e c (1 + 2 ) 2(2 + e,N e,N e,N T 2 P2 ( 2 P1 (1 + 2 2 e (1 + 2 (1 + 2 αc 2 e Q 2 αc m 2 Z m 2 2 e 2 e 2 A 2 S1 2 V π π M m m αc 4 16 αc αc = = = = = e e e U U U A V , , S1 P1 P2 , , , c c U U c U

c c U U σ σ σ σ σ Feynman diagrams for unparticle emission in (a) Compton-li (a) (b) (c) γ γ γ e e e bremsstrahlung-like processes and (c) processes with unpa care. The important integrals are collected in the appendix Figure 2: In the hot environment of a star photons are generated thermi for the Compton process production of unparticles The thermally averaged unparticle energy emission rate is t with the electron density JHEP12(2007)033 , , 1 2 2 − − the , U U , d d . 2 U 2 2 (3.17) (3.12) (3.15) (3.14) (3.16) (3.13) (3.11) (3.10) d − ¸ ¸ U 2 d U 2 2 i i ¸ Z Z 2 d β β 2 2 i ¸ Z e e ¸ β 2 i Z e πM πM 2 i m m β Z e . β πM · · m e r 1 carry a third πM · m πM m 4+ 1) 1) · > · T 2) − − / U ) 2) d 2) U r U / 1; below we will find / d d + 7 1 ) Maxwellian distribution + 7 → U ) − U ´ d U )Γ(4 )Γ(4 U 2 U 2 i , U d U U d πd 5 i T )Γ(4 + π d however only for the leading d d πd )Γ( ) relativistic limit one finds r 2 β 2 2 ) ucleus collisions is, in the limit mβ U z U )Γ( l. U d n U − − − ty of the particles and therefore )Γ(2 e πd d ³ πd U n (4 + ) csc(2 is the proton number of the nucleus. ) csc(2 2 d ζ )Γ(4 2 e U 2 aee exp )Γ(2 )Γ(2 Z U )Γ(4 2 U d g / U U d low, screening and degeneracy (Pauli blocking) d 2 U 2 i 1 ) csc(2 2 d d d Cn ) csc(2 α − U 2 + 6 2 − U πβ d π )(3 + 2 + 6 4 d − Z U 5 [24, 21] U 2 )= U d / )Γ( e d e − d 3 T U )Γ( )(3 + 2 )(3 + 2 ( – 6 – ´ m d U U c U U πm 2 d 8(2 + 3 d d Q 24(1 ≪ πT m (1 + 2 − 2 i 135 2 (15 + 14 the mass fraction of hydrogen. The averaging gives β ³ (15 + 14 2 1) )(1 + 4 2 H π U )(1 + 4 )= π − . The suppression can be understood from the fact that i d X − )= U 1)(1 + 2 1)(1 + 2 ⇒ β i − β U d ( d − − eZ a (2 U U T, β ( Q d d z e r 4(1 + 2 n (1 + 2 in the non-relativistic limit, as one may see when expanding (2 (2 n e z z i z z n n Cω n β n n e i e e e β n n n n e i i )= i i β β m β β ω 2 Z ( 2 A 2 P2 2 P1 2 S1 2 V e e e e c U c c c c c M 2 2 2 2 2 σ m m m m α α α α α coupling is suppressed. Because however unparticles for 2 2 2 2 2 are the electron and nucleus densities and Z Z Z Z Z is the total density and R e,z ρ = = = = = n − A V The bremsstrahlung emission rates have to be averaged over a L , S1 , P2 P1 Since the density of horizontal-branch stars is relatively , , , eZ U eZ U 1 eZ U eZ U eZ U Q Q Q Q Q One can see that the rate for the axial vector vanishes for effects are negligible [14, 23]. where this behaviour also forpower other in the processes. velocity This results holds where correct expression in powers of The emission rate for axionfor bremsstrahlung small from incident electron-n electron velocities the electron-nucleon scattering is independent of the chirali For unparticle emission through bremsstrahlung in the non- polarization degree of freedom, the suppression is not tota JHEP12(2007)033 3 2 2 2 2 ´ ´ ´ ´ ) or i Z Z Z Z (3.19) (3.18) β T T T T g/cm M M M M ( ollisions 4 ³ ³ ³ ³ Q 10 2 V 2 A 2 V 2 A 2 S1 2 S1 2 P2 2 P1 2 P1 2 P2 2 2 2 2 2 2 2 2 2 2 aee aee aee aee aee aee aee aee aee aee in c c c c c c c c c c g g g g g g g g g g × 2 e . 8 2 6 . n 32 64 25 71 36 14 36 77 09 ...... 4 r/ 0 is suppressed by 0 0 0 4 18 0 0 0 0 3 2 ) e z ≈ → 3 3 3 3 3 3 m / / / / / / . ) 3 3 3 3 z ocesses play a role in ρ 2 2 2 2 2 2 i / / / / 2 n − − − 4 4 − − − 4 4 T/m β ) and the bremsstrahlung e ( ¶ c ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ooling can be evaluated n (2 Z Z Z Z Z Z Z Z Z Z H Q 2 eZ U T T T T T T T T T T ρ ¶ M M M M M M M M M M Z m . Q ates in a stellar plasma of tem- r ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ U ) µ i mmarized in table 1. plicity, we give here the com- d 2 n constraints which have been densities 2 A 2 A 2 V 2 V 2 S1 2 S1 2 P2 2 P2 2 P1 2 P1 2 2 2 2 2 2 2 2 2 2 aee aee aee aee aee aee aee aee aee aee H c c c c c c 3+ c c c c g g g g g g g g g g i with mass ment T, β X µ ery similar results as bremsstrahl- ( 6 7 2 . . e 76 67 72 66 04 58 29 68 Γ ...... n 2 5/3 0 1 0 9 38 2 0 2 2 10 i 1+ / 1 β 3 3 3 3 3 3 − d ≈ / / / / / / 3 3 3 3 π 4 4 4 4 4 4 ) / / / / ∞ 2 − − − 2 2 − − − 2 2 0 He ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ Z C n Z Z Z Z Z Z Z Z Z Z T T T T T T T T T T = M M M M M M M M M M – 7 – )= ³ ³ ³ ³ ³ ³ ³ ³ ³ ³ + 4 T H ( 2 A 2 A 2 V 2 V 2 S1 2 S1 2 P1 2 P1 2 P2 2 P2 2 2 2 2 2 2 2 2 2 2 aee aee aee aee aee aee aee aee aee aee c c c c n c c c c c c g g g g g g g g g g ( eZ U e 9 4 5 3 4 Q . . . . . n 99 21 22 69 22 . . . . . 36 1 18 73 13 2 17 9 4/3 1 4 . ≈ 2 z / 2 2 2 1 2 2 n − − − ) 2 − − z ´ ´ ´ ⇒ ´ ´ Z Z Z Z Z Z T T T ), respectively [21]. Bremsstrahlung in nucleus-nucleus c /m T T M M M z M M e T X ³ ³ ³ ( ³ ³ m e r ( i Q 2 V 2 A 2 S1 2 P1 2 P1 2 2 2 2 2 aee aee aee aee aee n 2 V c c 2 S1 2 P2 2 P2 c c c 2 2 2 2 aee aee aee aee g g g g g c c c c ∼ g g g g in C β 4 4 0 e . . . 25 34 2 e i, . . n 126 5 31 126 118 0 185 31 1 3 10 )= /β 2 i z ), as well as different values of the scaling dimension β i, √ 12 12 12 12 23 23 ( 12 12 12 12 β . Separately shown are the rates from the Compton process ( eZ Comparison of unparticle emission rates to axion emission r 10 10 10 10 eZ U 10 10 T → Q 10 10 10 10 Q × × × × × × z × × × × n S1 P1 P2 P2 V A V A S1 P1 The impact of weakly interacting particle emission on star c One needs to observe that both bremsstrahlung and Compton pr c a c a e , , , , , , , , , , c a c a eZ a eZ a eZ a eZ a eZ a eZ a eZ U eZ U eZ U eZ U eZ U eZ U c U c U c U c U Q n Q Q Q U Q Q Q Q Q Q 2 Q Q Q Q Q Q Q Q Q Q d so that process ( perature Z Table 1: Bremsstrahlung in electron-electron collisions leadsung to in v electron-nucleus collisions, except for the replace Furthermore, summing over the relevant nuclei, is negligible since the radiation of unparticles from nucle powers of with a numerical codeparison for of stellar the evolution unparticle [22,analyzed emission 23]. earlier rate For [18 to – sim 20]. the axion The emissio relation between thered two is giant su environments. At typical horizontal-branch star JHEP12(2007)033 2 , U U , κ U a + + d 2 (4.1) e n + ¸ (3.20) (3.21) (3.22) e + Z → e e → ω πM → e + 2 γ n + , while proton ¸· + γ U ) [21], we find the e 2 + 14 /κ − n 2 ω , 10 + ernova SN 1987A puts a are generalized hyperge- ; 4 screening of the Coulomb n × is very mild; changing ¶¸ If unparticles only couple U p to 20%. Therefore these q ss accounts for 90% of the 2 5 d 2 F . κ ng of axions to photons. By own very weakly interacting → ω process), see figure 2 (c). In p 5 κ ) 4 . This has been analyzed for 2 2+ om the supernova core. The ough the process n he unparticle couplings. Using l code for stellar evolution. ]. . , < ross-section for 1 n supernova cores is smaller than U /κ e + s with electrons, 4] is extended to derive limits for − 2 1+ d s m n ω µ + ) 3 1 2 ; 4 U − aγγ , log d U g U d ¶ d 2 ; 2 )Γ(2 ω U κ and 1+ U 4 erg cm d , d 2 [19, 20] translates into the limits in table 2. U 33 ) d – 8 – 1+ 1+ , 13 U 10 + d − U GeV µ 2 (1 + d 1 2 8 × 10 − ; − 3 Γ(2 U 1 2 1+ × d 10 · , 6 keV, the bremsstrahlung process contributes roughly < ∼ . 2 U × d X 2 < ∼ 1+ 5 2 aγγ Q 8 . 1+ , , U aee α g K = 8 d 2 g (1 = 7 8 3 1+ = 2 F , 10 p κ a 4 σ (1 U ≈ 2 d F T 3 the cross-sections for this kind of process are − ˜ γ · e p ˜ γ 6 keV, . σ m 2 γγ ˜ γ ˜ 2 γ 2 ≪ α c κ /c 2 ω K = 8 π is the radius, inverse which Debye-Hückel accounts for the 2 γγ 8 2 c κ 10 = = In previous studies, limits have been derived for the coupli The observation of the length of the neutrino burst of the sup ≈ p γγ σ potential of the electron in a free stellar plasma [25, 26, 21 via t-channel photon exchangethe (usually limit called the Primakoff ometric functions. comparing the unparticle production cross-section to the c and temperatures Several processes candominant contribute effect to comes unparticle frombremsstrahlung emission is neutron less fr bremsstrahlung, important since thethe proton neutron density density. i In principle bremsstrahlung processe where Constraints from stellar coolingto photons, on they photon would mainly couplings. contribute to star cooling thr 10% of the totalrate axion [23]. emission rate, Then the while bound the Compton proce strong constraint on the allowed(un)particles energy [18], loss rate due to unkn by an order of magnituderesults changes should the be limits in reliable table even 3 without only a by detailed u numerica 4. Constraints from SN 1987A Unparticle emission would alsovector influence unparticles in supernova ref. cooling [3,other 4]. types Here in the unparticle analysis couplings, in as ref. in [ eq. (1.2). limits in table 3. Note that the dependence of the results on these limits can beT translated to corresponding limits for t JHEP12(2007)033 , 2 . , − 2 U , U nn , − d d U U 2 d U (4.3) (4.2) (4.5) (4.4) d ¸ 2 d 2 e able 2 Z ¸ ¸ ¸ Z Z 2 i Z 2 T i β πM e collinear β T n 2 n πM πM · πM 2 m 2 m 2 2) · · · / ) ) 2) U 2) U is the typical / 2) d +5 / d / 2 U denotes the neutron collision process and +5 d ) +5 3 +5 U cm 4 U − d U nn U d nteractions. In addition d 3 d )Γ(2 27 e spin dependence of the cles, the matrix elements )Γ(3+2 )Γ(3 + 2 U nds in the non-relativistic s − U unction [28, 29]. Following 3 4 gcm d )Γ(2 d + 6 U )Γ(2 U ning effects in the dense core 10 )Γ(2 nd when the bremsstrahlung 14 U d ns. Here one can distinguish d 3 U d U do-scalar couplings) [28]. itive to the nucleon spin (vector lar unparticle emission couples d ”hard” U × 10 d 4 and 0.0014, depending on s and Standard Model fermions. + 60 d + 6 2 × +60 )(5+2 25 3 2 U 3 U U d U )(5+2 d + 31 )(5+2 d ∼ d )(5+2 ≈ U 2 U d U n d U d nn 0 n d + 31 + 228 σ 30 MeV is much smaller than the neutron )(3+2 +228 2 U collision process and soft radiation from one U U ≈ U d )(3+2 d d + 55 )(3+2 d )(3+2 U T U nn – 9 – U d U d d d 8 + 55 − U 1)(1+2 d − 39 + 1073 8 1)(1+2 1)(1+2 U (39 + 1073 1)(1+2 − − 2(21 d − − U (2 U d U d d nn 0 (2 2(21 (2 (2 σ 2 n nn 0 nn 0 nn 0 n σ σ σ 2 n 2 2 n 2 n / n ) has the same dimensional dependence as in ref. [4], but we ar n 5 n 2 n 2 T / 2 3 / i ( 7 / , can be important due to collinear enhancement. However, th 7 m i 1 n 3 β 2 V β T U , 2 n m π T n 2 V 2 / nn U 2 + 2 1, the bounds in table 2 are obtained. / 3 m 2 S1 e π m 3 Q π √ Cc 2 π ≥ + 2 V 2 S1 Cc e 32 32 √ C is the incident neutron velocity and C c C c → i )= )= e β = = T T ( ( + V For vector and scalar unparticle-quark interactions, one fi Since the supernova core temperature Convolution with the Maxwellian thermal distribution give For the emission of axial-vector and pseduo-scalar unparti e , S1 V , , S1 transition, which is given by the dynamical spin structure f , nn U nn U nn U nn U Q Q Q ”soft” unparticle radiation frombetween the one case when of the bremsstrahlung the couplingand is external scalar insens neutro couplings ofemission the couples unparticles to to the the nucleon spin quarks) (axial-vector [27] and a pseu limit mass, the neutron bremsstrahlung process factorizes into a phase space region is suppressedplasma, due see to e.g. strong ref. Coulomb [21]. scree density. where factorize in a similar way into the on-shell Thus we arrive atwe somewhat obtain bounds weaker for bounds scalarAssuming for interaction between the unparticle unparticle i and Our result for Q scattering cross section at the given energy [27]. of the external legs.to Since the the spins axial-vector ofnn and the pseduo-sca nucleons, one needs to take into account th to identify an additional numerical prefactor between 0.00 JHEP12(2007)033 U is d X (4.8) (4.7) (4.6) c = 1, the , , 2 U ) 2 d ) T 2 T for the nucleon 1 as before, the ons , / 2 tending towards / σ /T 2 ≥ U σ ) /T σ d σ T Γ C 2 Γ + (Γ / luminosity. Of course, + (Γ 2 σ perimental error of the /T ngs derived from astro- 2 x σ ooling are much stronger r vector particle (see also x ts for limits from current x ailable from the literature Γ generally weaker, which is rally considerably stronger x − timate of the possible reach perators. For tion based on experimental ever for n structure function, which + (Γ e − itive to a certain coupling, as ooling provides the strongest e 2 e assessment of the sensitivity umed that it can perform the ess particle, so that our limits +2 x . These couplings could also be or/axial couplings. For small +2 U x d U − 2 d e 2 U d xx 2 xx d rge systematic uncertainties could arise in the d ∞ xx 0 ∞ 450 MeV for the typical temperature and d . Since these errors are difficult to quantify they 0 Z Z ≈ U ∞ d 0 U σ 2 d Z 2 ¸ – 10 – 2 ¸ Z − Z U T d πM 2 T 2 πM 100 MeV. Taking this value and ¸ 2 · Z ≈ · ) ) T σ U πM n U d n 2 d n for the case that only one of the unparticle couplings n · 2 2 2 ) )Γ(2 T n T )Γ(2 U U n d U d 2 P1 2 2 P2 d T Γ(2 C c 2 A C c 2 3 3 π C c 4(1 + 2 (1 + 2 4 )= )= )= T T T is the spin fluctuation rate. Using a one-pion exchange model ( ( ( A σ , P2 P1 , , nn U nn U nn U Q For the unparticle-photon couplings, our bounds from star c It can be seen that the constraints for astrophysics are gene Q Q In the derivation of the stellar energy loss constraints, la 2 density inside the supernovanucleon core scattering [29]. data A [28]can more finds be robust parametrized a evalua by smaller using value Γ for the spi scattering kernel, one obtains the estimte Γ bounds in table 2 are derived. 5. Comparison to reach of colliderIn experiments tables and 2 conclusi andphysical constraints. 3 we The summarizeastrophysical bounds our observations, correspond limits to on the unparticle 90% coupli CL ex where Γ calculation of nuclear interactionshave not and been stellar taken evolution into account here. ref. [28, 30], we obtain unparticle scaling behavior correspondsalso to apply a to regular anyref. massl model [31]). which includes a new massless scalar o than the limits from supernova cooling, taken from ref. [32] non-zero at a time.(LEP, Tevatron) and For future comparison colliders (LHC,of we ILC). also a To get future show an earlier internationalsame es linear resul kind collider of (ILC), measurementsonly as we a LEP, have proper but ass analysis with canof a go ILC. 1000 beyond The this times order-of-magnitud blanks higher for in the the given table interaction. indicatedenoted Some by that of a the no bar processes results in are are the not av table. sens than those from colliders. The strongest bounds are for vect two all constraints becomebound. similarly For important; scalar here andmainly star pseudoscalar due c couplings to the the bounds higher are dimensionality of the interaction o limits on a 5th force are by far the dominant constraints; how JHEP12(2007)033 [33, 34] [35] [35] [36] [37] [38] [34] 5 7 4 7 4 − − − − − 1 1 10 10 10 2 2 1 — · · · 10 10 > > 0.03 · · 1 1 5 0.008 . . . 3 9 1 4 2 related work 8 3 3 5 10 8 10 − − − − − − 1 1 − 10 10 10 10 10 — · · · · 10 5/3 5/3 · > > 10 0.04 0.01 · 5 3 3 5 · 3 . . . . P2 2 . 1 c 1 1 3 3 7 A c , 2 3 5 6 10 13 16 12 − − − − P1 1 1 − − − c 10 10 10 10 — · · · 4/3 4/3 · > > 10 10 10 0.15 imits were obtained, since 4 1 3 · · · 0.045 5 . . . d flavor-diagonal unparticle perators that only couple to . 8 2 2 1 1 1 experiments at low energies, 5 eriments at short but not at m astrophysical constraints and 3 8 d. 15 8 24 11 − − ors [51] can lead to new effects, − ] or the Higgs boson [49, 37, 50]. − 1 1 − − e available from the literature, while 10 10 1 1 1 rature, as indicated by the references 10 — · · 10 0.1 · > > 10 10 tive to operators that involve flavor sical bounds have been derived in this · 2 5 · · 3 . . 4 . 4 2 3 5 6 7 4 3 5 5 3 − − − − − − 1 1 13 10 10 10 2 2 . · · · 10 10 10 > > 0.01 0.45 0 · · · 5 2 7 . . . 2 3 2 3 3 1 3 8 5 10 9 6 4 − − − – 11 – at LEP and ILC, but this has not been analyzed − − − − 1 1 10 10 10 10 · · · 10 10 10 U 0.1 5/3 5/3 · > > 0.04 0.05 0.02 · · · 3 6 6 0.025 8 . . . 1 1 3 . + 1 1 6 1 V γ S1 c 6 8 3 c 13 15 12 7 − − − − − − − 1 1 → 10 10 10 10 10 10 · · · 10 0.4 0.2 4/3 4/3 · · · > > − 0.25 0.01 0.25 · 4 5 4 0.045 2 4 5 . . . e 7 . . . 2 3 1 1 1 2 + e 9 4 22 8 9 24 15 − − − − − 1 1 − − 10 10 1 1 1 10 · · 10 10 · > > 10 10 · · 3 6 0.005 · · 5 . . 8 1 . 7 5 1 1 6 Shortly before finishing this manuscript, we became aware of Comparison of limits for unparticle-fermion couplings fro This analysis is restricted to the leading CP-conserving an U U SN 1987A LEP ILC Coupling d 5th force (”Eötvös”) Energy loss from stars ILC LHC Electroweak precision Quarkonia Positronium d 5th force (”Eötvös”) Energy loss from stars SN 1987A LEP Tevatron Coupling from present and futurework, collider while experiments. the The collider astrophy boundsin have the been right taken column. fromthe Blank the bars spaces lite denote are that left no where bound no on results the ar coupling can be determine constrained by the process Table 2: on 5th force experimentsthese authors [52] included where only similar,astronomical results distances. though from weaker Newtonian-law l exp Note added. interactions. The astrophysical constraints arechanging not sensi neutral currents, whichsuch can as heavy-flavor be mixing tested and decaysthird-generation in [39 fermions – precision 45], [46, as 47], well heavyFurthermore, as gauge direct to bosons o CP-violation [48 inwhich the cannot unparticle be operat tested in astrophysics. so far. JHEP12(2007)033 , 1 2 − (A.2) (A.3) (A.1) n + |M| U ) d p 2 ) [32] ′ − p θ k 6 , 4 − − 2 cos ′ − p 10 2 ω · 10 is the angle between · + |M| 7 (2 . ′ 8 ′ 2 1 p k − θ ( U + 1) d 9 (4) ) 5 n δ − 2 astrophysical constraints. The ′ ) − 4 p ) + n 10 g integrals appear: areful reading and interesting ds. We thank Pedro Schwaller · 10 π 5/3 gration then yields · − U 3 unparticle emission processes in + e the supernova bounds have been . ) one finds in the non-relativistic ′ 4 (2 d ′ ˜ γ p 5 U 2 ˜ γ ) in the non-relativistic limit, with k θ ′ d ( − he right column. k , c U U ( d 11 cos γγ 6 Γ(2 ) U ′ c − + 1)Γ( 2 − ′ )+ ′ k U 10 ωp q )+ 10 · 4/3 d ( ′ · ) ( 7 (2 p 4 . Z 2 ( Γ( ′ 2 1 ′ e k p ( = )+ ′ ′ – 12 – θ → p ) 2 p it is useful to choose a reference frame where the 14 ′ 0 d ( ) 7 − ′ e − k + 1 e k − U p ( ( d θ 1 n θ m ) 10 γ 10 ′ → · 1 · 4 ′ p + 5 cos ) ) 9 . k ≪ q )+ ω π 5 4 U − ( 2 p d d ′ ω 0 ( (2 Z p 2 e Z θ ′ = p Z = )+ 0 θ e p cos k n ( 1 + m e ω U 2 d (2 d cos 3 2 ′ ) ) n 1 p ′ π 0 3 + p Z d θ U (2 d ′ ω p ,... 2 e Z ′ 3 (cos U p m , d ′ ω is the squared and spin-averaged matrix element, and 2 d 2 U p e A d , ′ θ 2 π p m Comparison of limits for unparticle-photon couplings from θ A U 4 32 d Energy loss from stars SN 1987A Coupling = 1 |M| cos = = d cos n ω For bremsstrahlung For the Compton process 1 2 c U 0 0 σ Z Z Table 3: with where the incident photon and the outgoing electron. The followin A. Phase-space integrals In the following the relevant phase-space integrals for the bounds from star coolingtaken have from been the literature, derived as in indicated this by work, the whil reference in t Acknowledgments This work was supported by thefor Schweizerischer discussions. Nationalfon We areand grateful helpful comments. to the journal referee for c this article are summarized. the initial photon energy electron in the initial state is at rest. The phase space inte JHEP12(2007)033 , 0 nn , /p | 2 (A.8) (A.6) (A.9) (A.5) (A.4) (A.7) | t, the → ~p | (A.11) (A.10) 0 2 ,..., . = , 3 nn ,..., 2 |M , i 2 3 / +2 2 β |M| , 2 , U ) 2 d |M| , 4 q i |M| ), the situation ′ 2 β = 1 ) 2 |M| − k − q − ( = 1 2 U p 2) U d − U − / d ectively, U − ) ¶ p 2 d 2 ′ 2 | ′ ) )+ | 2 ′ q ′ + 3 ′ | 2 k − , n k | ~ q 0 k ~ ′ ′ 4 2 U ( + | , n | | ) ( q k ′ ~ − ′ d ′ n ) 2 ′ n k ~ − | p k ~ |M +1 U | + k + − | | | 2 ) n ′ ( d ′ )+ + U ) q ′ 2 p + θ )Γ(2 πd d k 2 2 ′ ~ f ) p ) | U 2 / U k ( 2 ′ β 0 ) he strongly interacting f − d + d ( d 2 ¶ f 4 i k n β ′ ) p ) 2 β ( − 2 β f k 2 it there are some differences due to f (4) θ csc(2 − β ( − 2 − i ) − → δ 2 β i 2 4 − ′ ′ 2 2 β i β / 2 ) ) 2 i (4) ( ( ) q k 2 3 1. +3 − − β q e β − 2 e δ π 4 ( ( n U ( U π 4 2 U i m + 2 n n d d ≥ ) n m d (2 0 β ) ′ m + πd π ( 1 4 m 2 Z p ¶ + 1) Γ( U ′ 0 ( e ( 2 ′ )+ U Z d k µ (2 Z | 2 U k ′ p d m z ′ × π d (4) ( k ~ ) ( k × × 4 δ 1 2 π n µ m dΩ ′ 4 – 13 – 4 , ) ) csc(2 2 ) 2(2 k − | , dΩ )Γ(2 n f f ) ( n 3 π 2 ′ 2 Z 2 U 1 2 ) θ β β ) ) q = ′ ) ′ d 1+ (2 2 1+ Z − π 2 3 f p ′ ~p 0 2 ′ + − n f f ′ π d β k , q (2 p i i 4 Γ( β β − ( − 3 π 2 dΩ β β + − θ ) 4 π d 1)Γ( ~p ′ − + Z 2 dΩ f 2 ( i U i k i R Z 2 2 Γ( e β − β d | 4 ′ log β β ′ ( 2 f Z 1 p d − U m 2 e − k f ~ β 3 2 | 2 f 2 d R β 2 i d m 2 Γ( f β = log i ′ 3 3 1 ′ β β 1 4 Γ( ) β q 1 f R p ( U U 3 d π 3 2 e 4 e β d d µ = i d 2 = 2 d d β (2 n ) 2 m m is the mass of the nucleus. After including the matrix elemen ) i | 0 R 2 f 2 ′ f β )] ′ z ′ Z β k 0 β ~ = = Z p ~p i Z m 3 | | i 2 4 ′ − β − e d β ) ) n − 2 z k i ~ ′ ′ 2 i 2 z i | m R ~p β ~p ~p U m β m β d ( d 3 U m 1 1 ( ( 4 U U ) · d e − − π d d , and 2 A f n f π f ′ 4 ′ 0 β A ~p ~p are the velocity of the incoming and outgoing electron, resp m β β A A k ~ ( ( [ − 64 4 f f ′ /p ′ ′ 2 i | p k p = = i,f β β β ′ π π π ( = = β d d ~p 4 e 4 4 | i i 2 dΩ dΩ dΩ nn nn 0 U m β β For bremsstrahlung off a neutron pair, eZ U σ σ 0 0 0 = σ Z Z Z Z Z Z f limit typical integrals are β where Most of the above integrals are valid only for is very similar to electron-nucleus-bremsstrahlung,fact albe that this process can only be evaluated by factorizing t scattering. In the center-of-mass frame JHEP12(2007)033 . ] 4 (A.13) (A.12) ) ′ ]. ~p Phys. ]. · (1986) 402. , , ′ Phys. Rev. Phys. Rev. k ~ , (2006) 56 , k-energy length + ( 4 B 166 ) (2007) 275 hysics ~p B 636 · are the velocities of (1992) 207. ′ k ~ hep-ph/0611223 i,f [( β B 650 scattering, which in the 356 ′ hep-ph/0703260 (1969) 358. k π 4 Phys. Lett. 10 nn dΩ , elations between Γ-functions ore, Phys. Lett. iables of the external particles, Z Nature , ]. , 1 6 ,..., Phys. Lett. 3 − ] and (A.9) one needs the integrals , , ] (2007) 131104 [ n Yad. Fiz. 2 4 , , ) (2007) 221601 [ m ′ 98 ~p 98 = 1 → + 2 ~p ]. Unparticle constraints from SN1987A / Collider signals of unparticle physics e ]. ]. hep-ph/0611184 + ( m – 14 – 4 , n ) Contributions to the law of proportionality of inertia ′ n ~p 2 | ′ − ~p | Phys. Rev. Lett. ~p n Six years of the fifth force (1955) 1501. 2 , (1922) 11. [( | Phys. Rev. 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