Sensitivity of microwave transition in H2O2 to variation of the electron-to-proton mass ratio
aa r X i v : . [ a s t r o - ph . C O ] O c t Sensitivity of microwave transition in H O to variation of the electron-to-proton massratio M. G. Kozlov
Petersburg Nuclear Physics Institute, Gatchina 188300, Russia (Dated: March 27, 2018)Recent observation of several microwave transitions in H O from the interstellar medium[Bergman et al. Astron. Astrophys., , L8 (2011)] raised interest to this molecule as yet an-other sensitive probe of the tentative variation of the electron-to-proton mass ratio µ . We estimatesensitivity coefficients of the microwave transitions in H O to µ -variation. The largest coefficientfor 14.8 GHz transition is equal to 37, which is comparable to highest sensitivities in methanol andan order of magnitude higher than sensitivity of the tunneling transition in ammonia. PACS numbers: 06.20.Jr, 06.30.Ft, 33.20.Bx, 95.85.Bh
Molecules with an internal motion of large ampli-tude recently attracted much attention as high sensi-tive probes of the tentative variation of the electron-to-proton mass ratio µ = m e /m p . In 2004 van Veld-hoven et al. [1] pointed out that inversion transition inammonia is very sensitive to µ -variation. Later, sev-eral groups used this transition to place stringent lim-its on µ -variation on a large timescale of the orderof 10 Gyr [2–5]. At the same time ammonia spectrafrom our Galaxy were used to study possible depen-dence of µ on the local mater density [6, 7]. More re-cently the sensitivity to µ -variation was studied for othermolecules with tunneling motion, including hydronium(H O + ) [8, 9], methanol (CH OH) [10–12], and methy-lamine (CH NH ) [13]. All these molecules are observedin the interstellar medium (ISM) and potentially can beused as probes of µ -variation. Sensitivities of the tunnel-ing transitions are two orders of magnitude higher com-pared to optical transitions in molecular hydrogen, whichis traditionally used to study µ -variation at high redshifts(see [14] and references therein). Even higher sensitivi-ties can be found for certain mixed tunneling-rotationaltransitions. Recently methanol and methylamine weredetected at redshift z = 0 .
89 [15]. First limit on µ -variation using methanol is reported in [11].Peroxide molecule (H O ) is one of the simplestmolecules with large amplitude tunneling mode. It isvery well studied both theoretically and experimentally.However, to the best of our knowledge, there is no anal-ysis of the sensitivity of the microwave transitions in thismolecule to µ -variation. This is probably explained bythe fact that H O was not observed in ISM. Situationhas changed after the first observation of four microwavelines of H O from the molecular cloud core ρ Oph Ain our Galaxy by Bergman et al. [16]. In this communi-cation we estimate sensitivities of the microwave mixedtunneling-rotational transitions to µ -variation.In equilibrium geometry H O is not flat; the angle 2 γ between two HOO planes is close to 113 ° . Two flat config-urations correspond to local maxima of potential energy;the potential barrier for trans configuration (2 γ = π ) issignificantly lower, than for cis configuration ( γ = 0): U π ≈
400 cm − and U ≈ − . To a first ap- proximation one can neglect the tunneling through thehigher barrier. In this model peroxide is described bythe slightly asymmetric oblate top with inversion tunnel-ing mode, similar to ammonia and hydronium. A moreaccurate theory accounts for tunneling through both bar-riers [17]. In this case torsion motion can be describedas hindered rotation. For sufficiently high angular quan-tum numbers J and K A this internal motion stronglyinteracts with overall rotation. According to [18] this in-teraction becomes important for K A ≥
7. Such levels lievery high and can not be observed from ISM. Below 70Kthere are only levels with quantum numbers J ≤ K A ≤
2. For such levels the simpler model is sufficientlyaccurate and we will use it here.When tunneling through both barriers is taken into ac-count the ground torsion state splits in four componentsdesignated by the quantum number τ = 1 – 4. Because trans barrier is much lower than the cis one, the split-tings between the pairs τ = 1 , τ = 3 , τ = 1 and τ = 3 states[17, 18].Low energy effective Hamiltonian has the form (atomicunits are used throughout the paper): H eff = E τ + H rot , (1)where E τ is tunneling energy and H rot is Hamiltonian ofthe rigid asymmetric top, whose constants A , B , and C weakly depend on the quantum number τ (see Table I). TABLE I: Parameters of the effective Hamiltonian (1) in GHzfrom Ref. [18]. τ E τ A τ B τ C τ . .
873 26 .
193 25 . .
885 301 .
585 26 .
142 25 . Dimensionless sensitivity coefficient Q µ of the transi-tion ω to µ -variation is defined so that:∆ ωω = Q µ ∆ µµ . (2)In order to find these coefficients for transitions describedby Hamiltonian (1) we need to know µ -dependence ofthe parameters from Table I. Rotational parameters A , B , and C scale linearly with µ . Thus, purely rotationaltransition has sensitivity Q µ, rot = 1. Sensitivity of thetunneling energy E τ to µ -variation can be estimated withthe help of semiclassical Wentzel-Kramers-Brillouin ap-proximation. Following [19], we can write: E τ = 2 E π e − S , (3)where S is the action over classically forbidden region and E is the zero point energy for the inversion mode. Ex-pression (3) gives the following sensitivity to µ -variation: Q µ,τ = S + 12 + S E U max − E ) , (4)where U max is the barrier hight. Numerical factor in thesecond fraction depends on the barrier shape. For thetriangular and square barriers it is 2 times smaller and1.5 times larger respectively [2]. The factor in Eq. (4)corresponds to the parabolic barrier [9].The potential for the tunneling coordinate γ was foundby Koput et al. [20] in a form of Fourier expansion with U max = U π = 387 cm − . Numerical solution of the onedimensional Schr¨odinger equation for this potential gives E = 169 cm − E τ = 13 . − and Q µ,τ = 2 .
44. Usingthis value for E and tunneling energy from Table I wecan find S from Eq. (3) and Q µ,τ from Eq. (4): S = 2 . , Q µ,τ = 2 . . (5)If we scale potential from [20] to fit experimental tunnel-ing frequency from Table I, the numerical solution gives Q µ,τ = 2 .
56. All three values for Q µ,τ agree to 5%. Thesemiclassical value corresponds to the experimental tun-neling frequency and is less sensitive to the details of thepotential shape. Therefore, we use it in our calculationsand assign it 5% uncertainty.We see, that tunneling energy is 2.5 times more sensi-tive to µ -variation than rotational energy. Sensitivity ofthe mixed tunneling-rotational transitions ω = E τ ± ω rot , is a weighted average of the tunneling and rotational con-tributions [21]: Q µ = E τ ω Q µ,τ ± ω rot ω Q µ, rot . (6)This sensitivity is further enhanced for the frequencies | ω | ≪ E τ .Results of the numerical calculations with Hamiltonian(1) are given in Table II. In this table we list six transi-tions in the range 200 – 700 GHz, which were studied inRef. [16] (four of them were observed and other two werenot) and transitions from the JPL database [22] with fre-quencies below 100 GHz.Note that pure rotational transitions with ∆ τ = 0 forperoxide are not observed. Because of that all sensi-tivities in Table II significantly deviate from unity. As TABLE II: Numerical calculation of the Q -factors for low fre-quency mixed transitions in peroxide using Hamiltonian (1)and Eq. (5). Experimental frequencies are taken from JPLCatalogue [22]. E up is upper state energy in Kelvin. J K A ,K C ( τ ) E up ω (MHz) Q µ upper lower (K) theory exper.Transitions below 100 GHz0 , (3) – 1 , (1) 17 14818.8 14829 . . . , (1) – 1 , (3) 21 37537.0 37518 . − . . , (3) – 1 , (1) 19 67234.5 67245 . . , (3) – 2 , (1) 24 68365.3 68385 . . , (3) – 3 , (1) 31 70057.4 70090 . . , (3) – 4 , (1) 41 72306.0 72356 . . , (3) – 5 , (1) 53 75104.6 75177 . . , (3) – 6 , (1) 68 78444.7 78545 . . , (1) – 2 , (3) 28 90399.8 90365 . − . , (3) – 2 , (1) 31 219163.2 219166 . . , (1) – 5 , (3) 66 252063.6 251914 . − . , (3) – 3 , (1) 41 268963.7 268961 . . , (3) – 4 , (1) 53 318237.7 318222 . . , (3) – 6 , (1) 67 318635.6 318712 . . , (3) – 0 , (1) 32 670611.9 670595 . . expected, lower frequency transitions have higher sensi-tivities. Transitions with the frequency | ω | < E τ fallin two categories. For transitions τ = 3 → τ = 1 thetunneling energy is larger than rotational energy and | ω | = E τ − ω rot . For such transitions Q -factors are posi-tive. For transitions τ = 1 → τ = 3, | ω | = ω rot − E τ and Q -factors are negative.Let us discuss the accuracy of our estimates. Calcu-lated frequencies agree with experiment to 0.1%, or bet-ter in spite of the simplicity of Hamiltonian (1). Thatmeans that centrifugal corrections to the rotational en-ergy are indeed unimportant for the transitions with lowrotational quantum numbers considered here. Even inthis case the scaling of the rotational energy with µ isnot exactly linear as rotational constants of the effectiveHamiltonian are averaged over the vibrational wave func-tions of the states τ = 1 ,
3. Respective corrections to therotational sensitivity coefficients are of the order of 1% –2% [11]. Note that τ -dependence of the rotational con-stants can be considered as centrifugal corrections to thetunneling energy [9]. According to Refs. [9, 11] the ac-curacy of the semiclassical expressions for tunneling (3,4) is comparable, about 3%. Additional uncertainty isassociated with the zero point energy E , which is notdirectly observable and we take it from calculation withpotential [20]. This potential was used in a number oflater papers, including [23, 24]. We estimate the accu-racy of the tunneling sensitivity coefficient (5) to be 5%,which agrees with numerical estimates made above. Un-certainties in rotational and tunneling sensitivities resultin the final uncertainties for mixed transitions given inTable II. These uncertainties are typically about 10%.Such accuracy is sufficient for the analysis of the astro-physical spectra.To summarize, we have estimated sensitivity coeffi-cients for the low frequency mixed tunneling-rotationaltransitions in H O . Transitions below 100 GHz appearto be highly sensitive to µ -variation with sensitivitycoefficients of both signs. Maximal relative sensitivityis found for transitions 14.8 and 37.5 GHz, where∆ Q µ = +36 . − ( − . ≈
50. This is comparableto the highest sensitivities in methanol [10, 11] andan order of magnitude larger than for ammonia [1, 2].The lines recently observed by Bergman et al. [16]have frequencies above 200 GHz and significantly lowersensitivities. Three of the observed transitions havesensitivities, which are close to each other. However,the sensitivity of the fourth observed transition issignificantly different. Transitions 219 GHz and 252GHz have relative sensitivity ∆ Q µ = 4 .
5. This is close to the sensitivity of the ammonia method, where∆ Q µ = 3 .
5, but with the advantage that both linesbelong to one species. This eliminates very importantsource of systematic errors caused by the differencein spatial and velocity distributions of species [7].We conclude, that microwave transitions in H O potentially can be used for µ -variation search in as-trophysics. Note, that high sensitivity transitionscorrespond to low rotational quantum numbers J . Thismakes peroxide a potential candidate for laboratorytests on µ -variation using molecular beam technique [25].I am grateful to S. A. Levshakov, A. V. Lapinov, P.Jansen, and H. L. Bethlem for helpful discussions. Thiswork was supported by RFBR grants 11-02-00943 and11-02-12284. [1] J. van Veldhoven, J. K¨upper, H. L. Bethlem, B. Sartakov,A. J. A. van Roij, and G. Meijer, Eur. Phys. J. D , 337(2004).[2] V. V. Flambaum and M. G. Kozlov, Phys. Rev. Lett. ,240801 (2007), arXiv: 0704.2301.[3] M. T. Murphy, V. V. Flambaum, S. Muller, andC. Henkel, Science , 1611 (2008), arXiv:0806.3081.[4] C. Henkel, K. M. Menten, M. T. Murphy, N. Jethava,V. V. Flambaum, J. A. Braatz, S. Muller, J. Ott, andR. Q. Mao, Astron. Astrophys. , 725 (2009), arXiv:0904.3081.[5] N. Kanekar, Astrophys. J. Lett. , L12 (2011),arXiv:1101.4029.[6] S. A. Levshakov, P. Molaro, and M. G. Kozlov (2008),arXiv: 0808.0583.[7] S. A. Levshakov, A. V. Lapinov, C. Henkel, P. Molaro,D. Reimers, M. G. Kozlov, and I. I. Agafonova, As-tron. Astrophys. , A32 (2010), arXiv:1008.1160.[8] M. G. Kozlov and S. A. Levshakov, Astrophys. J. ,65 (2011), arXiv:1009.3672.[9] M. G. Kozlov, S. G. Porsev, and D. Reimers, Phys. Rev.A , 052123 (2011), arXiv:1103.4739.[10] P. Jansen, L.-H. Xu, I. Kleiner, W. Ubachs, and H. L.Bethlem, Phys. Rev. Lett. , 100801 (2011).[11] S. A. Levshakov, M. G. Kozlov, and D. Reimers, Astro-phys. J. , 26 (2011), arXiv: 1106.1569.[12] P. Jansen, I. Kleiner, L.-H. Xu, W. Ubachs, and H. L.Bethlem, ArXiv e-prints (2011), 1109.5076 (submittedto Phys. Rev. A, 2011).[13] V. V. Ilyushin and H. L. Bethlem, private communication (2011).[14] F. van Weerdenburg, M. T. Murphy, A. L. Malec,L. Kaper, and W. Ubachs, Phys. Rev. Lett. , 180802(2011).[15] S. Muller, A. Beelen, M. Gu´elin, S. Aalto, J. H.Black, F. Combes, S. Curran, P. Theule, and S. Long-more, ArXiv: (2011), 1104.3361 (accepted to Astron-Astrophys. 2011).[16] P. Bergman, B. Parise, R. Liseau, B. Larsson, H. Olofs-son, K. M. Menten, and R. G¨usten, Astron. Astrophys. , L8 (2011).[17] J. T. Hougen, Canadian Journal of Physics , 1392(1984).[18] F. Masset, L. Lechuga-Fossat, J.-M. Flaud, C. Chamy-Peyret, et al., J. Phys. France , 1901 (1988).[19] L. D. Landau and E. M. Lifshitz, Quantum mechanics (Pergamon, Oxford, 1977), 3rd ed.[20] J. Koput, S. Carter, and N. C. Handy, J. Phys. Chem. A , 6325 (1998).[21] M. G. Kozlov, A. V. Lapinov, and S. A. Levshakov, J.Phys. B , 074003 (2010), arXiv: 0908.2983.[22] P. Chen, E. A. Cohen, T. J. Crawford, B. J. Drouin, J. C.Pearson, and H. M. Pickett, JPL Molecular SpectroscopyCatalog , URL http://spec.jpl.nasa.gov/ .[23] S. Y. Lin and H. Guo, J. Chem. Phys. , 5867 (2003).[24] S. Carter, A. R. Sharma, and J. M. Bowman, J. Chem.Phys.135