Bounds on new light particles from high-energy and very small momentum transfer np elastic scattering data
BBounds on new light particles fromhigh-energy and very small momentumtransfer np elastic scattering data Yuri Kamyshkov ∗ , Jeffrey Tithof † , University of Tennessee, TN 37996-1200, USA
Mikhail Vysotsky ‡ ITEP, Moscow, 117218, Russia
Abstract
We found that spin-one new light particle exchanges are stronglybounded by high-energy and small momentum transfer np elastic scat-tering data; the analogous bound for a scalar particle is considerablyweaker, while for a pseudoscalar particle no bounds can be set. Thesebounds are compared with the bounds extracted from low-energy n − P b scattering experiments and from the bounds of π and K + meson decays. The Standard Model of three fundamental forces describes interactions ofelementary particles very well. While the electromagnetic force has a longinteraction range, the short radius of the “weak force” ( ∼ ∼ ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - ph ] N ov f ∼ ∼ /r law,usually by an additional Yukawa term that can be parameterized with twoparameters α , the relative strength of new interaction, and λ , the character-istic radius of the interaction. Applied in the analysis of experimental dataat macroscopic distances down to ∼ a micrometer, this ansatz describes thepossible deviations from classical gravity; (b) as a quantum field theory de-scription of the interactions (excluding gravity) in a covariant form, whichcan be expressed in the lowest perturbation order through the coupling con-stant g and the mass of the exchanged particle mediating the interaction µ .Covariant forms that can be consistently considered in this description arescalar (S), pseudoscalar (P), vector (V), and axial vector (A). We can arguethat higher spins of the intermediate particle should not be considered sincethey lead to non-renormalizable theory. The particles that mediate the newforce could be absent from the spectrum of known particles [1] due to theirsmall mass and coupling constant or due to some other reason that is helpingthem avoid detection. In any case, if these particles are not observed, directexperimental limits on their existence in terms of g and µ are required.In Section 2 of the present paper, we reanalyze the experimental small-angle np -elastic scattering data at high energy [2] in terms of bounds on theexistence of new forces expressed as S, P, V, or A covariant interactions. InSection 3, we examine bounds that can be obtained from lower energy data. np scattering The data for small-angle np -elastic scattering at high-energy were obtainedin the NA-6 experiment [2] performed at CERN SPS a quarter century ago.Incident neutron energy in the experiment was 100–400 GeV, while the square2f the 4-momentum transfer | t | was varied in the range 6 · − to 5 · − GeV .The data of this experiment are consistent with extrapolation of the hadronicamplitude from higher | t | values, while at | t | < − GeV the differentialcross-section rises due to Schwinger scattering, which is the interaction ofthe neutron’s magnetic moment with the Coulomb field of the proton orelectron. The purpose of NA-6 [2] was to measure hadronic interactions athigh s in the region of momentum transfer ( ∼ | t | < − GeV ) that wasusually inaccessible in the scattering of charged hadrons due to Coulombinteractions. This is the region where the effect of a new force mediated bya light particle may be present.Figure 1 (similar to Fig. 16 from [2]) demonstrates that np elastic scat-tering data in this experiment are well described by the following formula: dσdt = A exp[ bt ] − (cid:18) αk n m n (cid:19) πt , (1)where A = (79 . ± . mb/ GeV and b = (11 . ± .
08) GeV − weredetermined from the fit to the data (data are taken from Table 7 in [2]), m n isthe neutron mass, and k n = − .
91 is the neutron magnetic moment in nuclearmagnetons. The factor of 2 in the Schwinger term, as will be discussed later,accounts for the scattering of the neutron’s magnetic moment on the protonplus an incoherent contribution of scattering on electrons (gaseous hydrogenwas used in [2] as a target). Smaller effects due to neutron polarizabilityare not included in the description of the data. This description (1) worksrather satisfactorily with χ =41.5 for 31 degrees of freedom. We will referto this description as the “zero model” since no new force contributions areincluded here.Although Quantum Chromodynamics does not provide a detailed theo-retical description of the hadronic elastic scattering at small | t | , i.e. at largeimpact parameters, hadronic scattering has been studied experimentally ingreat detail in the past and was phenomenologically well understood, e.g. inthe framework of Regge models. The description of elastic data by a singleexponent was a general universal feature of hadronic scattering observed atlow | t | in the region where it was not obstructed by Coulomb scattering (forexample, see [3] and also the comparison with other experiments in [2]). Thisjustifies, at a phenomenological level, our choice of the hadronic scatteringdescription with a single exponent. However, in an attempt to improve thedescription of the data [2], we have tried several alternative modifications of3 |t| [(GeV/c) -6 -5 -4 -3 -2 -1
10 1 ] / d t[ ( m b / ( G e V / c ) σ d Pure HadronicDiffractionPure Hadronic DiffractionSchwinger ScatteringContribution
NA6 np Scattering ExperimentI pp Experiments
Coulomb Interference I Figure 1:
Elastic differential neutron-proton cross-sections measured in experi-ment [2]. For comparison, the | t | region measurable in pp scattering is shown withthe effect of the Coulomb interaction indicated by the dashed line. χ per degrees of freedom was slightly increased demon-strating that more complicated modifications of the “zero model” are notstatistically justifiable.We describe the contribution of a new interaction in the following way:Let us suppose that a new light particle with mass µ exists which interactswith the neutron and proton with couplings g n and g p correspondingly. As-suming scalar, pseudoscalar, vector, and axial vector couplings of this particlewith nucleons, we obtain the following addition to expression (1): dσ i dt ( g, µ ) | new = | A i | · F F πs ( s − m ) , (2)where s = ( p n + p p ) is the invariant energy square and m is the nucleonmass. We parameterize the hadronic form factor’s contribution as: F F = 1(1 − t/ Λ ) , (3)which comes from a 1 /q decrease of the nucleon form factor, and we set Λequal to the mass of the lightest meson resonance with appropriate quan-tum numbers ( η (cid:48) in the case of pseudoscalar). Finally, we use the followingamplitude squares for different couplings: | A S | = g S ( t − µ ) (4 m − t ) , (4) | A P | = g P t ( t − µ ) , (5) | A V | = 4 g V ( t − µ ) [ s − m s + 4 m + st + 12 t ] , (6) | A A | = 4 g A ( t − µ ) [ s + 4 m s + 4 m + st + 12 t + 4 m t µ + 8 m tµ ] , (7)where coupling constants g i ≡ g ip g in .It is quite natural to suppose that a new light particle’s couplings withnucleons originates from its couplings with quarks. In this case, (6) and57) are modified. For the vector exchange, the induced magnetic moment’sinteraction term should be added to the scattering amplitude. Since itsnumerator contains momentum transfer divided by m N , which in consideredkinematics gives a factor much smaller than 1, we can safely neglect it anduse (6) in what follows. The case of the axial vector exchange is more delicateand discussed in detail in the Appendix.We can now turn to the discussion of other features of the np elasticscattering amplitude. Though the strong interaction amplitude cannot bedetermined theoretically from the first principles, our confidence that 1 /t dependence is absent in strong interactions for | t | < m π opens the road tobounding the light particle exchange if its mass is smaller than that of the π -meson. Experimental data at | t | < m π matter for our bounds, which makesthe precise value of Λ in the expression for F F not important, since in therelevant domain of | t | the form factor is close to 1. For the same reason, noform factor is introduced for the Schwinger term in (1).For each fixed set of parameters g i and µ describing the possible contri-bution of a “new force”, we are fitting the experimental distribution with acombined function (1)+(2), where parameters A and b describing the stan-dard hadronic contribution are free. Then the maps of A , b , and the minimumvalues of χ are composed as functions of g i and µ . Analyzing the χ map,we determined the level of χ [4] above which parameters of the “new force”become incompatible with experimental data at a confidence level (C.L.)greater than 90%. At this level we also examined and ensured that parame-ters A and b remain within the 90% C.L. close to those in the “zero model”.In this way, we can ensure that the “new force” contribution does not sub-stitute for the standard hadronic plus electromagnetic contributions in thedescription of the data.Figure 2 shows, for comparison, fits to the data for the “zero model”and for several excluded models for the new vector particle exchange withparameters slightly beyond the excluded limits for µ V and g V .Two comments need to be made on formulas (4)–(7): (a) the amplitudeswith the exchange in the t -channel of a point like particle with spin α de-pend on s as s α . That results in an amplitude behavior of s for scalar andpseudoscalar and of s for vector and axial vector particles. This propertyof high-energy scattering amplitudes would allow us to determine the valueof the spin of the “new physics” mediator; (b) the pseudoscalar exchangevanishes at t = 0. These comments explain why we will get the strongestbounds on g A and g V , a weaker bound on g S , and no bound on g P .6 |t| [(GeV/c) -6 -5 -4 -3 -2 -1
10 1 ] / d t[ ( m b / ( G e V / c ) σ d Figure 2:
Several fits to the experimental data of [2]: long-dash line – single expo-nent without the Schwinger contribution; solid line – the “zero model” descriptionof (1); dotted line – the “zero model” plus the new vector particle contributionwith µ = 1 MeV and g = 0 . µ = 10 MeV and g = 0 . µ = 40 MeV and g = 0 . | Re/Im | < . t = 0 , unlike the square of the Schwinger amplitude,which contributes significantly at small | t | because of 1 /t behavior.The mass of a light particle µ was bounded in our fits to be below 100MeV, and the range of the coupling constants varied depending on the par-ticular model. Compilation of the bounds obtained from the χ limit for P,S, V, and A models in coordinates g versus µ is presented in Figure 3.In the next step of analysis, we checked that for each model, fitted pa-rameters A and b corresponding to the boundary of the excluded domainof g i and µ must not deviate from their “zero model” values by more than1 . σ , where σ is the corresponding error of parameters A and b from the“zero model.” We found that these conditions are satisfied if µ < M eV forthe S, V, and A-models, and cannot be satisfied for any value of µ for theP-model. Thus, our bounds on the coupling strength g shown in Figure 3should only be referred to in these validated domains (indicated in Figure 3by brackets). No consistent limit can be set for the P-model. In addition, weshould notice that the very high value of g P obtained from the χ analysisfor the P-model makes our perturbative approach of formula (5) not valid.We should conclude therefore that the experimental data [2] do not provideany limit for the pseudoscalar exchange.The factor of 2 in the Schwinger term of the “zero model” in (1) is comingfrom both n − p and n − e scattering and is an estimate of equal contributionfrom both. However, n − e scattering occurs in a different kinematical rangeand the event selection criteria in [2] could suppress the detection of electrons.Consequently, we varied the factor in the Schwinger term of the “zero model”(1) from 1 (no n − e contribution) to 3 (double n − e contribution) and foundin analysis that this variation was not very significant, changing our limitingvalue for g by ±
8% for a fixed value of µ .One can notice that since the average s ≈
540 GeV in experiment [2], The Schwinger part of the interference term contains q µ /q , q ≡ t , which multiplies q µ ≡ ( p − p ) µ cancelling the 1 /q enhancement, or ( p + p ) µ giving zero. [GeV]µMass, -4 -3 -2 C oup li ng , g -4 -3 -2 -1
10 110 PseudoscalarScalarVector Axial Vector ]]X
Figure 3:
Compilation of the upper bounds obtained in the current analysis interms of g and µ at a 90% C.L. No limit can be set for the pseudoscalar exchange.Brackets indicate the interval of mass µ where the analysis was validated (see text). A V ) and axial vector ( A A ) amplitudes, as follows from Eq. (6) and(7), are practically the same (see Figure 3), except when µ (cid:46) M eV . Inthis case, the last two terms of Eq. (7) arising from the q µ q ν /µ part of thepropagator of the axial vector particle start to dominate the amplitude.Our bounds on the parameters g V and g A (Figure 3) are rather strong;say, for µ = 10 MeV, g V,A < · − at 90% C.L., which corresponds to g V,AN < . , (8)four times smaller than the QED coupling constant √ πα (cid:39) .
3. For thescalar exchange, taking µ = 10 MeV, we get a much weaker bound, g S < . We will now compare our results from the previous section with other searchesfor new interactions [5]–[14] in which new light particles participate. In theliterature, the effect of new forces is usually parameterized as a deviationfrom the Newtonian gravitational potential: V ( r ) = − G N m m r [1 + α G exp( − r/λ )] , (9)which is an adequate approximation for the description of the effect of a newparticle exchange between nonrelativistic constituents. The following rela-tionship exists between the coupling constant α G and characteristic length λ and our parameters g i and µ in cases of vector and scalar exchanges: α G = g V,S πG N m p m n = 1 . · g V,S , lg α = lg g V,S + 37 . , (10) λ (cm) = 1 µ (MeV)5 . · , lg λ = − lg µ − . . (11)The pseudoscalar exchange would not modify the potential in a nonrel-ativistic approximation, while axial coupling leads to an interaction amongspins of constituents.(A) References [5]–[7] show e.g. that values of α G larger than 1 areexcluded for λ larger than 0.1 mm, while for smaller λ the upper boundson α G rapidly grow, reaching 10 at the micron scale.10n papers [8]–[14], analogous bounds for shorter distances are presented.We see e.g. that for λ = 10 − m, α G should be less than 10 , or g V,S lessthan 10 − . The corresponding value of µ is 2 MeV. In Figure 4, our limitsfor the V and S particle exchanges in terms of α G and λ parameters arecompared with the limits obtained in papers [5]–[14].(B) Additionally, the data on low-energy (1 keV < E n <
10 keV) neutronscattering on
Pb [15] were applied in paper [16] to obtain bounds on thepossible contributions of a light scalar particle exchange to neutron-nucleuspotential. The upper bound on the coupling constant of a 10 MeV boson toa nucleon obtained in [16] corresponds to g V,S < · − . This rather restric-tive bound was obtained from the analysis of the shape of the differentialcross-section of low-energy n − Pb scattering, where the additional term,originating from the light scalar boson exchange, leads to a modification ofangular dependence not observed in the experimental data. Let us stressthat the same bound is valid for g V .According to [15, 16], experimental data in the keV energy range are welldescribed by the following expression: dσd Ω = σ π [1 + ωE cos θ ] , (12)where (cid:112) σ / π ≈
10 fm, and ω = (1 . ± . − keV − . These numericalvalues are very reasonable from the nuclear scattering point of view and,from the demand that these values are not spoiled by a Yukawa potentialcontribution originating from a light boson exchange, bounds on g and µ wereobtained in [16]. The point is that the Yukawa amplitude, interfering withthe strong interaction amplitude, will show up in the following contributionto ω for E → | ∆ ω | = 16 m n (cid:112) σ / π λ n π Aµ , (13)and from the demand that ∆ ω < ω , the above mentioned bound was ex-tracted. For an update of results obtained in [16], see [17].(C) It is quite natural to assume that the coupling of a new light bosonwith nucleons originates from its coupling with u - and d -quarks. In this case,bounds from pion and kaon decays [18] are applicable. Let us start withvector coupling. According to CVC, couplings to nucleons are equal to thesum of the couplings to quarks: 2 f uV + f dV for a proton and f uV + 2 f dV for11 -14 -12 -10 -8 -6 -4 ) G l og ( SeattleLamoreauxStanfordDecca et al.Mostepanenko et al. Mohideen et al.Nesvizhevsky et al. NesvizhevskyBordag et al. & ProtasovPokotilovski This Work
ScalarVector
Figure 4:
Experimental limits on α G and λ from [5]–[14] parameterizing devia-tions from Newton’s law. Our limits transformed into coordinates α G and λ arealso shown for comparison.
12 neutron ( f i are analogous to our g iN ). The π → V V decay contributes to π → invisible decays and, using the experimental bound Br ( π → νν ) < . · − in [18], the following bound was obtained: (cid:113) | f uV − f dV | ≤ · − , (14)which is automatically satisfied for an isoscalar coupling, f uV = f dV . How-ever, the bound on the π → γV decay, which contributes into π → γ + invisible mode, allows bounding of isoscalar couplings as well [18]:2 f uV + f dV < . · − . (15)Here, the experimental bound Br ( π → γνν ) < · − was used. Thesenumbers should be compared with our bound on g VN (8).Since π → SS and π → Sγ decays violate the corresponding P - and C -parities, we do not obtain bounds on f S from these decays. C -parityconservation forbids the π → γA decay as well, while from the bound on π → invisible decays, we get the coupling constant bound (14) for the axialvector boson.More stringent upper bounds on the coupling constants follow from verystrong experimental limits on the branching ratio Br ( K + → π + + νν ) < · − . The longitudinal component of the axial vector boson contributesto the decay amplitude proportionally as (2 m q /µ ) f qA [18], and even if theaxial vector boson couples only with light quarks, we obtain: f u,dA < ∼ − µ (MeV) . (16)The factor of 2 m q /µ is absent when the axial vector interaction is substi-tuted by the scalar interaction, and thus we obtain: f u,dS < ∼ − . (17)Fortunately, CVC forbids K → πV decays for µ = 0, so that is why thebound on the vector coupling for light µ is not very strong: f u,dV (cid:18) µm K (cid:19) < ∼ − . (18)13 Conclusions
Our bounds obtained from high-energy and very small momentum transfer np elastic scattering data [2] provide exclusions of new forces at distancesabove 5 fermi, which corresponds to exchanged particle masses lighter than 40MeV. These bounds are extracted in a covariant approach, as an alternativeto the bounds on couplings at larger distances, extracted from the absenceof deviations from the Newtonian gravitational law.Both low-energy n − Pb and high-energy np scattering data lead tosimilar upper bounds on the coupling constants for ≈
10 MeV vector bosons,though upper bounds from n – Pb scattering on the coupling constant g VN are ∼
30 times lower and close to the bounds from π → invisible and π → γ +invisible decays on the vector coupling constants with quarks.Strong upper bounds on the “new physics” contribution into the K + → π + + invisible decay allows us to get very strong bounds for scalar and axialvector bosons: g A,SN < ∼ − for a 10 MeV boson mass.We are grateful to A. B. Kaidalov, B. Z. Kopeliovich, L. B. Okun, and Yu.N. Pokotilovsky for useful discussions and comments. M. V. was partiallysupported by Rosatom and grants RFBR 07-02-00021, RFBR 08-02-00494and NSh-4568.2008.2. Y.K. would like to acknowledge research support fromthe United States Department of Energy HEP grant DE-FG02-91ER40627. For vanishing light quark masses, their isotriplet axial current is conserved.The same should hold for the nucleon currents and is achieved by accountingfor pion exchanges:˜ A A = g A ¯ nγ β γ n (cid:18) g αβ − k α k β k − m π (cid:19) ( g αµ − k α k µ µ ) k − µ (cid:18) g µν − k µ k ν k − m π (cid:19) ¯ pγ ν γ p == g A k − µ (cid:20) g αβ − k α k β µ ( m π − µ m π + k µ )( k − m π ) (cid:21) ¯ nγ α γ n ¯ pγ β γ p . (19)For a massless pion, the 1 /µ singularity cancels out and the expressionin square brackets contains k α k β /k , which, acting upon fermionic axial cur-rents, becomes (2 m N ) /k . The numerator of the expression for differential14ross-section is regular at k ≡ t = 0 since the square of the pseudoscalarexchange amplitude contains t in the numerator (see (5)), while the inter-ference of the axial vector and pseudoscalar exchanges is proportional to t (the denominator equals ( t − µ ) independently of the spin of the exchangedboson).In real life, light quarks, as well as pions, have nonzero masses, and toobtain an amplitude square for the axial vector boson exchange we shouldsubstitute µ by ˜ µ in the square brackets of (7), where1˜ µ = 1 µ m π − µ m π + tµ ( t − m π ) , and for t, µ (cid:28) m π we get ˜ µ = µ .The numerator of the expression for the differential cross-section is sin-gular for µ →
0. However, in renormalizable theory, the mass of the axialvector boson equals its gauge coupling constant ( g A in our case) times thevacuum average of the corresponding higgs field.As an example, one can have in mind the expansion of the Standard Modelwith two higgs doublets with opposite hypercharges, where Peccei–Quinn U (1)-symmetry is spontaneously broken producing an axion. In order tosuppress axion couplings to quarks and leptons, the additional singlet neutralhiggs field N is usually added, which makes the axion invisible. Gauging ofPeccei–Quinn U (1) leads to the axial coupling of the corresponding vectorboson to matter. Such a light axial vector boson is discussed in particularin [18], where it is light due to the smallness of the gauge coupling constant,while the vacuum average < N > (cid:29)
100 GeV, making it superweaklycoupled to matter ( g A /µ ∼ / < N > ). References [1] C. Amsler et al., The Review of Particle Physics, Physics Letters B667,1 (2008).[2] A. Arefiev et al., Nucl. Phys.
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