arXiv:1809.09382v1 [nucl-th] 25 Sep 2018 neetosadpoosadinrsnurn,ti fifth this neutrons, ignores to and analogue protons of acts and sort electromagnetism electrons a on is where force except This electromagnetism, nucleus. width the atomic times an protopho- several short-range of only a extremely distances over an carry acts is would that which force particle 29], “protopho- a [28, a fifth-force Such be bic could boson”. particle X new bic consis- this properties colleagues that and for Feng concluded mass results, looking experimental with After other with boson [27]. tent new MeV possible 17 a about proposed state ground etlsuiso easo nectdsaeof state excited an of experi- decays the of Recently, studies [17–26]. mental carriers fifth-force as rate. ticles faster causing increasingly is an energy at stan- dark expand the why to explain Also universe the not matter. can of dark model model explain standard dard the to interest of physics growing inability particle has the mat- forces of all new because of for mainly blocks searching building While the ter. are that particles atomic evidence existing found gravity. not However, non-Newtonian have such still 11–16]. of people 3, now, [2, to boson up a by exchanging poten- gravitational through effectively normal tial the characterized non- to term Yukawa models, be a simplest adding can the gravity In and Newtonian experimental [2–10]. 3]. important results hypothesis of [2, theoretical number gravity theory large non-Newtonian M a spawned or of theory enunciation string The using studied further h atri h nvre h o-etna grav- Non-Newtonian Fischbach The by suggested universe. was the all ity between in interactions matter The the the govern particles. generally subatomic forces between in four prevail interactions forces nuclear galaxy tiny weak of the and scale strong enormous the the and at gravita- clusters universe The the gravity. the of sculpts above exception tion life the practically daily with in for scale, encounters nuclear responsible one phenomena one electrically the the all between is type occurs a particles, that force, charged interaction electromagnetic physical The of and forces. strong nuclear the weak and electromagnetism gravity, i.e., forces, cetsshv itrclypooe aiu xtcpar- exotic various proposed historically have Scientists sub- fundamental the describes model standard The ti nw oalta hr r orfundamental four are there that all to known is It opoetepoohbcfit oc ndnemte ynucleu by matter dense ener in kinetic worldwide. force high equipments fifth experimental very protophobic current at evide the ratio force probe pions effects fifth to charged protophobic negligible negatively form the has that and While force matter dense fifth collisions. in nucleus protophobic force new the this fifth of protophobic expected, existence the the probe refute or to confirm To attention. much h rpsdpoohbcfit oc in force fifth protophobic proposed The .INTRODUCTION I. nttt fMdr hsc,CieeAaeyo cecs L Sciences, of Academy Chinese Physics, Modern of Institute osbepoeo h e fhforce fifth new the of probe possible A tal et 1 n hnwas then and [1] 8 et its to Be a-hnYong Gao-Chan hs e.Lett. Rev. Phys. ieeouino h igepril hs pc distri- space phase particle describes single model function the bution transport of of BUU evolution evolution The time dynamical describing reaction. in which nuclear successful [31], quite model is transport (BUU) Uehling-Uhlenbeck i fhfrei es atrfre nnucleus-nucleus in formed protopho- matter the dense collisions. probe in ki- to force high used fifth very be bic at could thus ratio energies pion netic charged high The very at energies. ratio final kinetic pions of charged value negatively fi- the and the the affects positively While evidently on force collisions. effects fifth protophobic nucleus-nucleus negligible in protopho- has observables the generally nal that force show fifth results nucleus- proto- bic in The the this formed probe that collisions. of matter to nucleus dense existence method in a the force fifth propose refute phobic I or here confirm force, much To dark causes and [30]. attention much debate attracts nature of force have that universe the observed. of be mass to the entry- yet of an bulk the be up particles also making mysterious might matter, dark force understanding into fifth way The Theory. Unified ignores and neutrons and protons. electrons between works force rg oeta energy potential erage rbto ucindet t rnpr n enfield, mean and item dis- collision transport side space right-hand its phase the to and particle due of function evolution tribution time the denotes nry hc seult iei energy kinetic to equal is which energy, h rbblt ffidn atcea time at particle a finding of probability the mentum i n nlsi w oycollisions. body two elas- inelastic by and function tic distribution space phase of modification h hs pc itiuinfunction distribution space phase The h td sbsdo h eilsia Boltzmann- semiclassical the on based is study The oaastepsil icvr fteukonfifth unknown the of discovery possible the Nowadays Grand the achieve scientists help might force fifth The tyaet h au ffia positively final of value the affects ntly 117 nteuulosralsi nucleus- in observables usual the on ~p is h inltu ol eused be could thus signal The gies. neato,hr rps method a propose I here interaction, ∂f di ulu-ulu olsos As collisions. nucleus-nucleus in ed ∂t -ulu olsos xeiet at experiments collisions’ s-nucleus tposition at I H METHODOLOGY THE II. 783(2016) 071803 , + nhu700,China 730000, anzhou f ∇ ( ~ p ~r, ~p, t E ∇ · U ~ r ,wihreads which ), h ethn ieo q (1) Eq. of side left-hand The . hl h enfil potential mean-field the While . ~r f ∇ − eetyattracts recently ~r E ∇ · E I ~ p c f spril’ total particle’s is f E consfrthe for accounts ( = kin ~r, ~p, t I c lsisav- its plus t . denotes ) ihmo- with (1) 2

U of the particle is given self-consistently by its phase

space distribution function. 1.50 8

w/o fifth force Au+Au, E = 0.4 GeV

beam

In the model, I use a 17 MeV mass gauge boson [27, with fifth force

b= 2 fm 1.45 6

(y/y ) < 0.2

28], mediating a Yukawa force with range 12 fm. The beam c.m.

−3 strength of the coupling for u quark is −3.5 × 10 e, 1.40 4 for d quark is 7 × 10−3e, where e = 0.303 is the usual 1.35 2

(a) (b) free - / electromagnetic coupling constant [28]. The fifth force

1.50 8 + between two particles is then given by [32] (n/p) (c) (d)

w/o f if th f orce

1.45 with f if th f orce, 6

ij −3 1 1 −r/λ nn only ×

F =1.44 10 c c ( + )e , (2) 1.40 4 fifth i j r2 λr

1.35 2 where λ = 12 fm, ci and cj are the strength parame- ters of coupling of two interacting particles and r is their 0.15 0.20 0.25 0.30 0.35 0.0 0.1 0.2

c.m. distance. The strength parameters of different particles E (GeV) used in Eq. (2) are shown in Table I. kin

FIG. 1: (Color online) Effects of the fifth force on the kinetic TABLE I: The Strength parameters used in Eq. (2). energy distributions of the neutron to proton ratio of free − particle i strength parameter ci/e nucleons and the π /π+ ratio in the Au + Au reaction at 0.4 p 0 GeV/nucleon (in center-of-mass system). The cases in panels −2 n 1.05 × 10 (a) and (b) consider the fifth force interactions among all − −2 π 1.05 × 10 kinds of particles in heavy-ion collisions, including neutron, π π0 0 meson and ∆ resonances with different charge states. While − π+ −1.05 × 10 2 the cases in panels (c) and (d) only consider the fifth force − − ∆ 2.1 × 10 2 interactions between neutrons. ∆0 1.05 × 10−2 ∆+ 0 ++ −2 ∆ −1.05 × 10

8

From Eq. (2), it is seen that the strength of this fifth Au+Au, E = 0.4 GeV, b= 2 fm

beam

7 force decreases as increase of the distance of two particles (y/y ) < 0.2, t= 20 fm/c and there is a sharp increase at short distance. The force beam c.m .

with fifth force is repulsive between two neutrons. From Eq. (2) and 6

w/o fifth force Table I, it is shown that there is no fifth force between ++ / -

0 + two protons. And proton, π , ∆ have no fifth force 5 interactions with other particles. More interestingly, π− + and π have opposite fifth force interactions with other 4 particles.

3

III. RESULTS AND DISCUSSIONS

0.0 0.3 0.6 0.9

c.m .

E (GeV)

The fifth force should have less effects on general ob- k in servables in heavy-ion collisions but may have effects on light meson production. Because π− and π+ have oppo- site fifth force interactions with other particles, to probe FIG. 2: (Color online) Effects of the full fifth force interac- the fifth force in dense matter that formed in heavy-ion tions on the kinetic energy distribution of ∆−/∆++ ratio at − + collisions, it is naturally to think of the observable π /π maximal compression stage of the central Au + Au collision ratio. Since the strength of the fifth force is large at at 0.4 GeV/nucleon incident beam energy. short distance, it is naturally to think that the emitting particles from the compression region in heavy-ion colli- sions may carry more information about the fifth force. collisions thus more free neutrons are produced. There- Fig. 1 shows kinetic energy distribution of the emitting fore the fifth force increases the value of free n/p ratio as neutron to proton ratio of free nucleons (n/p)free and shown in panel (a) in Fig. 1. The right upper panel of the π−/π+ ratio in central Au + Au reaction at 0.4 Fig. 1 shows the corresponding π−/π+ ratio in the same GeV/nucleon. From panel (a), it is seen that due to the Au + Au reaction at 0.4 GeV/nucleon. Due to more free repulsive fifth force interactions of neutrons, more neu- neutron emissions, the dense matter formed in heavy-ion trons are emitted from dense matter formed in heavy-ion collisions is neutron-deficient. Less neutron-neutron col- 3 lisions cause less π− productions [33], thus the value of − + the π /π ratio decreases with the fifth force interac- tions as shown in the right upper panel (b) of Fig. 1. As 5 comparison, the cases in panels (c) and (d) of Fig. 1 only

Au+Au, E = 0.4 GeV consider the fifth force interactions among neutrons, i.e., beam

b= 2 fm turning off the fifth force interactions of ∆ resonances 4 and π mesons. It is seen that the effects of the fifth force (y/y ) < 0.2

beam c.m . become smaller comparing to that with full fifth force + / - consideration (the full fifth force considers all the fifth 3 force interactions among all kinds of particles) as shown in panels (a) and (b). One thus can deduce that the be-

with fifth force havior of resonance dynamics affects the free neutron to 2 proton ratio and the π−/π+ ratio at relative low kinetic energies. w/o fifth force

In heavy-ion collisions, relative numbers of very ener- 0.2 0.4 0.6 0.8 1.0 getic neutrons (protons) emitted from dense matter are

c.m . mainly affected by the relative numbers of produced en- E (GeV) k in ergetic ∆− (∆++) in dense matter (∆− mainly decays into π− and neutron and ∆++ mainly decays into π+ − and proton). From Table I, it is seen that ∆ ’s suffer FIG. 3: (Color online) Effects of the fifth force on the − stronger repulsive fifth force interactions than neutrons energetic π /π+ ratio in the Au + Au reaction at 0.4 ++ while ∆ ’s suffer attractive fifth force interactions from GeV/nucleon. The fifth force interactions are from all kinds − matter, thus there should be more energetic ∆ produc- of particles. tions than ∆++. Fig. 2 shows the effects of the fifth force on the ∆−/∆++ ratio in dense matter as a function of kinetic energy. It is clearly shown that the value of the collisions at intermediate energies is definitely from many ∆−/∆++ ratio increases with the fifth force at very high − rounds’ nucleon-nucleon or baryon-baryon scatterings, kinetic energy, i.e., there are relative more ∆ produc- thus both the colliding energy and the effects of the fifth ++ tions than ∆ at very high kinetic energy. This is the force are accumulated. From Fig. 3, it shows that the reason why the effects of the fifth force interactions are so maximal effect of the fifth force on the final π−/π+ ratio small with only neutron-neutron fifth force interactions is larger than 25%. The uncertainty band denotes the as shown in panels (c) and (d) of Fig. 1. From the upper statistical error. The number 25% is deduced from the panels (a) and (b) of Fig. 1, it is seen that maximal effects lower band of the value of π−/π+ ratio with the fifth − + of the fifth force on the free n/p ratio and the π /π ratio force (3.5) and the upper band of the value of the π−/π+ in central Au + Au reaction at 0.4 GeV/nucleon are only ratio without the fifth force (2.8). It is worth mentioning about 1.5% and 3.5%, respectively. However, since Fig. that such effect could be measured at current experimen- − ++ 2 shows that the energetic ∆ /∆ ratio is more sensi- tal equipments worldwide, such as heavy-ion collisions at − + tive to the fifth force, the energetic π /π ratio should RIBF-RIKEN in Japan [34], at CSR in China [35, 36] or be also more sensitive to the fifth force. This is because at GSI in Germany [37]. In the study, each event pro- − + the ratio of π and π yields are mainly decided by the duces about 3 × 10−5 pions with kinetic energy of about − ++ relative numbers of the ∆ and ∆ resonances [33]. 0.8 GeV in Au + Au reaction at 400 MeV/nucleon. One To show more clearly the effects of the fifth force on the thus needs to accumulate at least millions of events. − π /π+ ratio, in Fig. 3, we plotted kinetic energy distri- Because the strength of the fifth force is very small, the − bution of the π /π+ ratio at very high kinetic energies. collision of two heavy nuclei is best used to probe the fifth In the simulations, we accumulated 5,000,000 events for force. The heavy collision system can get stronger effects − each case. Energetic π and π+ mesons emitted from the of the fifth force due to the superposition of fifth forces dense matter are mainly from energetic nucleon-nucleon among more particles. And also the heavy collision sys- − collisions (neutron-neutron collision mainly forms ∆ tem can form larger dense matter size thus more rounds − resonance and it decays into neutron and π meson and of nucleon-nucleon or baryon-baryon scatterings accumu- proton-proton collision mainly forms ∆++ resonance and late more effects of the fifth force in nucleus-nucleus col- it decays into proton and π+ meson) in the compression lision. With the increase of the incident beam energy of stage in heavy-ion collisions. Because the fifth force re- nucleus-nucleus collisions, the effect of the fifth force on − pels ∆ ’s and attracts ∆++’s (as shown in Table I and the energetic positively and negatively charged pions ra- − Fig. 2), more ∆ ’s are produced than ∆++’s at very high tio is negligible as the thermal or random energies also kinetic energies. It is thus not surprising that one sees become larger. − + the fifth force increases the value of the π /π ratio at While the nuclear symmetry energy may affect the very high kinetic energies as shown in Fig. 3. value of π−/π+ ratio in heavy-ion collisions at inter- The production of very energetic pions in heavy-ion mediate energies [33]. For very energetic π−/π+ ratio, 4 however, the symmetry energy should show no effects. evident enhancement of energetic π−/π+ ratio at pion This is because after many rounds’s nucleon-nucleon or kinetic energies around 900 MeV are mostly due to the baryon-baryon scatterings, the isospin effects on the fi- fifth force because the very energetic pions almost have nal energetic nucleon-nucleon or baryon-baryon collisions no much time to suffer the mean-field potential from sur- are averaged and thus planished. The produced ener- rounding nuclear matter. getic pions are therefore not affected by the exact form of the symmetry energy. Similarly, the isospin-dependence of the in-medium baryon-baryon cross section in the IV. CONCLUSIONS above energetic collisions is also planished due to many rounds’s different isospin-dependent baryon-baryon scat- terings. Since there are no effects of the symmetry en- Once the recently proposed new fifth force is confirmed ergy on the produced energetic nucleons and resonances, by experiments, it would have profound influence in fun- the threshold effect on energetic pion production can be damental physics. To confirm or refute this kind of new also neglected [38]. Of course, there are also some others interaction, I propose to probe the fifth force in dense theoretical uncertainties which affect the value of ener- matter that formed in nucleus-nucleus collisions. It is getic charged pion ratio, such as initialization of nucleon shown that the protophobic fifth force affects the value distribution in coordinates and momentum space in pro- of final positively and negatively charged pions ratio at jectile and target nuclei, dynamical self-energies due to very high kinetic energies in nucleus-nucleus collisions at nucleon-nucleon short-range correlations and core polar- intermediate energies. The heavy collision system can ization [39], off-shell pion transport [40], etc. However, effectively enlarge the effects of such dark force on the all the above effects on energetic charged pion ratio are ratio of yields of positively and negatively charged pi- negligible [39, 40]. ons. This signal could be used to probe the protophobic There are still other mechanisms which cause the en- fifth force by nucleus-nucleus collisions’ experiments at hancement of energetic π−/π+ ratio with the fifth force current experimental equipments worldwide. in heavy-ion collisions at intermediate energies. For ex- Acknowledgements ample, based on the ∆-hole model and the relativistic Vlasov-Uehling-Uhlenbeck transport model with the rel- The author acknowledges Jonathan L. Feng for helpful ativistic nonlinear NLρ interaction, the pion p-wave po- communications. The work was carried out at National tential may enhance the value of the energetic π−/π+ Supercomputer Center in Tianjin, and the calculations ratio at pion kinetic energies above 200 MeV [38], al- were performed on TianHe-1A. The work is supported by though the pion potential may have no effects at pion the National Natural Science Foundation of China under kinetic energies around 200 MeV [41]. Nevertheless, the Grants No. 11775275, and No. 11435014.

[1] E. Fischbach, D. Sudarsky, A. Szafer, C. Talmadge, S. H. Rev. Lett. 98, 021101 (2007). Aronson, Reanalysis of the Eoumltv¨os experiment, Phys. [9] E.G. Adelberger et al., Particle-Physics Implications of Rev. Lett. 56, 3 (1986). a Recent Test of the Gravitational Inverse-Square Law, [2] A.O. Sushkov, W.J. Kim, D.A.R. Dalvit, S.K. Lamore- Phys. Rev. Lett. 98, 131104 (2007). aux, New Experimental Limits on Non-Newtonian Forces [10] C.D. Hoyle et al., Submillimeter Test of the Gravitational in the Micrometer Range, Phys. Rev. Lett. 107, 171101 Inverse-Square Law: A Search for “Large” Extra Dimen- (2011). sions, Phys. Rev. Lett. 86, 1418 (2001). [3] S.Q. Yang et al., Test of the Gravitational Inverse Square [11] Y. Fujii, Dilaton and possible non-newtonian gravity, Na- Law at Millimeter Ranges, Phys. Rev. Lett. 108, 081101 ture 234, 5 (1971). (2012). [12] E. Fischbach, C.L. Talmadge, [Phenomenological De- [4] E. Fischbach, C. Talmadge, Six years of the fifth force, scription of Non-Newtonian Gravity] The Search for Nature 356, 207 (1992). Non-Newtonian Gravity, [11-65] (Springer-Verlag, New [5] D.H. Eckhardt, C. Jekeli, A.R. Lazarewicz, A.J. Ro- York, Inc. 1999). maides, R.W. Sands, Tower Gravity Experiment: Evi- [13] E.G. Adelberger, B.R. Heckel, A.E. Nelson, Tests of dence for Non-Newtonian Gravity, Phys. Rev. Lett. 60, the Gravitational Inverse-Square Law, Annu. Rev. Nucl. 2567 (1988). Part. Sci. 53, 77 (2003). [6] M.V. Moody, H.J. Paik, Gausss law test of gravity at [14] J.P. Uzan, The fundamental constants and their vari- short range, Phys. Rev. Lett. 70, 1195 (1993). ation: observational and theoretical status, Rev. Mod. [7] J. Chiaverini, S.J. Smullin, A.A. Geraci, D.M. Weld, Phys., 75, 403 (2003). A.Kapitulnik, New Experimental Constraints on Non- [15] D.H. Wen, B.A. Li, L.W. Chen, Supersoft Symmetry En- Newtonian Forces below 100 µm, Phys. Rev. Lett. 90, ergy Encountering Non-Newtonian Gravity in Neutron 151101 (2003). Stars, Phys. Rev. Lett. 103, 211102 (2009). [8] D.J. Kapner et al., Tests of the Gravitational Inverse- [16] J. Xu, B.A. Li, L.W. Chen, H. Zheng, Non-Newtonian Square Law below the Dark-Energy Length Scale, Phys. gravity in finite nuclei, Journal of Physics G. 40, 035107 5

(2013). Anomaly in Beryllium Nuclear Decays, Phys. Rev. D 95, [17] P. Fayet, U(1)A symmetry in two-doublet models, U 035017 (2017). bosons or light scalars, and ψ and γ decays, Phys. Lett. [30] E. Cartlidge, Has a Hungarian physics lab found a fifth B 675, 267 (2009). force of nature?, Nature, doi:10.1038/nature.2016.19957. [18] C. Boehm, D. Hooper, J. Silk, M. Casse, J. Paul, MeV [31] G.F. Bertsch, S. Das Gupta, A guide to microscopic mod- Dark Matter: Has It Been Detected? Phys. Rev. Lett. els for intermediate energy heavy-ion collisions, Phys. 92, 101301 (2004). Rep., 160, 189 (1988). [19] M.I. Krivoruchenko, F. Simkovic,˘ A. Faessler, Con- [32] J.L. Feng, private communication. straints for weakly interacting light bosons from existence [33] B.A. Li, G.C. Yong, W. Zuo, Near-threshold pion pro- of massive neutron stars, Phys. Rev. D 79, 125023 (2009). duction with radioactive beams, Phys. Rev. C 71, 014608 [20] C. Boehm, P. Fayet, Scalar dark matter candidates, Nucl. (2005). Phys. B 683, 219 (2004). [34] R. Shane et al., SπRIT: Atime-projection chamber [21] P. Jean et al., Early SPI/INTEGRAL measurements of for symmetry-energy studies, Nuclear Instruments and galactic 511 keV line emission from positron annihilation, Method A 784, 513 (2015). A&A 407, L55 (2003). [35] Z.G. Xiao et al., Probing nuclear symmetry energy at [22] S.H. Zhou, U boson at the BES III detector, Phys. Rev. high densities using pion, kaon, eta and photon pro- D 75, 115004 (2007). ductions in heavy-ion collisions, Eur. Phys. J. A 50, 37 [23] S.N. Gninenko, Stringent limits on the π0 → γX, X → (2014). − e+e decay from neutrino experiments and constraints [36] L.M. L¨uet al., Conceptual design of the HIRFL-CSR on new light gauge bosons, Phys. Rev. D 85, 055027 external-target experiment, Sci. China-Phys. Mech. As- (2012). tron. 60, 012021 (2017). [24] S.N. Gninenko, Constraints on sub-GeV hidden sector [37] P. Russotto et al., Results of the ASY-EOS experiment gauge bosons from a search for heavy neutrino decays, at GSI: The symmetry energy at suprasaturation density, Phys. Lett. B 713, 244 (2012). Phys. Rev. C 94, 034608 (2016). [25] M. Reece, L.T. Wang, Searching for the light dark [38] Z. Zhang, C.M. Ko, Medium effects on pion production gauge boson in GeV-scale experiments, JHEP 0907, 051 in heavy ion collisions, Phys. Rev. C 95, 064604 (2017). (2009). [39] F. Zhang, G.C. Yong, Effects of the high-momentum [26] G.C. Yong, B.A. Li, Effects of nuclear symmetry energy tail of nucleon momentum distribution on the isospin- on η meson production and its rare decay to the dark sensitive observables, Eur. Phys. J. A 52, 350 (2016). U-boson in heavy-ion reactions, Phys. Lett. B 723, 388 [40] W. Cassing, S. Juchem, Semiclassical transport of (2013). hadrons with dynamical spectral functions in A+A col- [27] A.J. Krasznahorkay et al., Observation of Anomalous In- lisions at SIS/AGS energies, Nucl. Phys. A 672, 417 ternal Pair Creation in 8Be: A Possible Indication of (2000). a Light, Neutral Boson, Phys. Rev. Lett. 116, 042501 [41] W.M. Guo, G.C. Yong, H. Liu and W. Zuo, Effects (2016). of pion potential and nuclear symmetry energy on the − + [28] J.L. Feng et al., Protophobic Fifth Force Interpretation of π /π ratio in heavy-ion collisions at beam energies the Observed Anomaly in 8Be Nuclear Transitions, Phys. around the pion production threshold, Phys. Rev. C 91, Rev. Lett. 117, 071803 (2016). 054616 (2015). [29] J.L. Feng et al., Particle Physics Models for the 17 MeV