Unitary Limit of Two-Nucleon Interactions in Strong Magnetic Fields
William Detmold, Kostas Orginos, Assumpta Parreno, Martin J. Savage, Brian C. Tiburzi, Silas R. Beane, Emmanuel Chang
IINT-PUB-15-044, NT@UW-15-10, NSF-KITP-15-119, MIT-CTP-4704
Unitary Limit of Two-Nucleon Interactions in Strong Magnetic Fields
William Detmold, Kostas Orginos,
2, 3
Assumpta Parre˜no, Martin J.Savage, Brian C. Tiburzi,
6, 7, 8
Silas R. Beane, and Emmanuel Chang (NPLQCD Collaboration) Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795, USA Jefferson Laboratory, 12000 Jefferson Avenue, Newport News, VA 23606, USA Dept. d’Estructura i Constituents de la Mat`eria. Institut de Ci`encies del Cosmos (ICC),Universitat de Barcelona, Mart´ı Franqu`es 1, E08028-Spain Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, USA Department of Physics, The City College of New York, New York, NY 10031, USA Graduate School and University Center, The City University of New York, New York, NY 10016, USA RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA Department of Physics, University of Washington, Box 351560, Seattle, WA 98195, USA (Dated: October 16, 2018)Two-nucleon systems are shown to exhibit large scattering lengths in strong magnetic fields atunphysical quark masses, and the trends toward the physical values indicate that such featuresmay exist in nature. Lattice QCD calculations of the energies of one and two nucleons systemsare performed at pion masses of m π ∼
450 and 806 MeV in uniform, time-independent magneticfields of strength | B | ∼ —10 Gauss to determine the response of these hadronic systemsto large magnetic fields. Fields of this strength may exist inside magnetars and in peripheralrelativistic heavy ion collisions, and the unitary behavior at large scattering lengths may haveimportant consequences for these systems.
PACS numbers: 11.15.Ha, 12.38.Gc, 13.40.Gp
In most physical situations, external electromagnetic(EM) fields have only small effects on hadronic and nu-clear systems, whose structure and dynamics are dom-inated by the internal strong interactions arising fromQuantum Chromodynamics (QCD) and internal EM in-teractions. However, there are specific situations involv-ing extremely large EM fields, created either naturallyin astrophysical environments or in particle colliders, forwhich the effects of external fields are important. In mag-netars , high magnetic field rotating neutron stars [1], sur-face magnetic fields are observed up to O (10 ) Gauss(for reviews, see e.g. Ref. [2, 3]), and it is conjecturedthat interior magnetic fields reach up to O (10 ) Gauss[4]. In heavy ion collisions, the currents produced by rel-ativistic nuclei lead to large magnetic fields within theprojectiles, particularly during (ultra-)peripheral colli-sions [5]. It is estimated that fields of O (10 ) Gaussare experienced by the nuclei during the femtosecondsof the nuclear crossings [5]. Neither of these environ-ments are easy to probe in a controlled way, and thedetailed behavior of nuclei in such fields is an open ques-tion. As a step toward exploring nuclei in these extrememagnetic environments, we present the results of calcu-lations of the effects of uniform, time-independent mag-netic fields on two-nucleon (as well as two-hyperon) sys- tems performed with the underlying quark and gluon de-grees of freedom. We find that such fields can poten-tially unbind the deuteron and significantly modify thenucleon-nucleon (NN) interactions in the S channel. Atthe unphysical quark masses where the calculations areperformed, the scattering lengths in both the S – D and S channels diverge at particular values of the fieldstrength. Near these values, the low energy dynamicsof these systems will become unitary. The trends seentowards the physical values of the quark masses suggestthat this feature may exist in nature in some of these sys-tems. The prospect of such resonant behavior in nuclearsystems is exciting and it will be important to incorpo-rate this effect into models of magnetars and heavy ioncollisions in which the relevant field strengths are probed.Before presenting the results of our calculations, it isinteresting to consider phenomenological expectations forthe behavior of such systems. For small, constant mag-netic fields, the responses of the two-nucleon systems be- Significant effort has been devoted to understanding the natureof the QCD vacuum in strong magnetic fields (see Ref. [9] fora review), but effects specific to hadronic systems are not wellstudied. a r X i v : . [ h e p - l a t ] M a r yond their charges are governed by their magnetic mo-ments if the system has spin, and otherwise by their mag-netic polarizabilities. The deuteron has a magnetic mo-ment such that in a magnetic field in the z directionthe j z = +1 component is positively shifted in energywith respect to the breakup threshold and so an ap-proach toward unbinding in a magnetic field is plausible.However, higher order responses to the magnetic fieldmay be important, and at intermediate field strengths, | e B | ∼ m π , significant deviations from linearity should beanticipated. In the opposite limit of extremely large mag-netic fields, where | e B | (cid:29) Λ , the asymptotic freedomof QCD implies [10] that the eigenstates evolve towardsweakly-interacting up and down quarks in Landau lev-els. Hence, as the magnetic field tends to infinity, theground states of dilute systems tend to threshold. Whenthe density of the system is also large and comparable tothe scale of Landau orbits, more exotic phases may occur(see Ref. [11] for a review).In this work, the numerical technique of Lattice QCD(LQCD) is applied to study two-nucleon systems in uni-form, time-independent background magnetic fields, fol-lowing methods used in previous studies of the magneticmoments [12] and polarizabilities [13] of nucleons andlight nuclei up to atomic number A = 4. To understandthe phenomenological effects of the strong fields in nu-clear environments, a first task is to ascertain the ef-fects on the two-nucleon interactions. Two particle scat-tering phase shifts can be accessed in LQCD from thevolume dependence of two-nucleon energies (the L¨uschermethod [14, 15]), but here a simpler approach is under-taken in which only the bound states of the two-nucleonsector are addressed. The primary goal of these calcu-lations is to investigate how the binding energies of thetwo-nucleon states respond to applied magnetic fields.LQCD calculations were performed using two ensem-bles of gauge-field configurations generated with a clover-improved fermion action [16] and the L¨uscher-Weiszgauge action [17]. The first ensemble had N f = 3 de-generate light-quark flavors with masses tuned to thephysical strange quark mass, producing a pion of mass m π ∼
806 MeV, and used a volume of L × T = 32 × N f = 2 + 1 quark flavorswith the same strange quark mass and degenerate up anddown quarks with masses corresponding to a pion mass of While the deuteron magnetic moment is positive, it is less thanthe sum of the neutron and proton magnetic moments. In apotential model the difference is due to the d -state admixtureinto the predominantly s -wave deuteron wave function, while inNN effective field theories (EFTs) this is encapsulated in short-distance two-nucleon interactions with the magnetic field. At unphysically large values of the light quark masses, both thedeuteron and dineutron are bound, as are various two baryonhypernuclei [18]. m π ∼
450 MeV and a volume of L × T = 32 ×
96. Bothensembles had a gauge coupling of β = 6 .
1, correspondingto a lattice spacing of a ∼ .
11 fm. The ensembles con-sisted of ∼ ,
000 gauge-field configurations at the SU(3)point and ∼
650 configurations at the lighter pion mass,each taken at intervals of 10 hybrid Monte-Carlo trajec-tories. We have extensively studied these ensembles inprevious works, and have found that the finite-volumeeffects to both the single nucleon and two-nucleon boundstate energies are small [18, 19].As in Refs. [12, 13, 20], background EM ( U Q (1)) gaugefields were implemented through the gauge-links, U ( Q ) µ ( x ) = e i πQq ˜ nL x δ µ, × e − i πQq ˜ nL x δ µ, δ x ,L − , (1)that give rise to uniform magnetic fields along the x -direction. These were multiplied onto each QCD gaugefield in each ensemble (separately for each quark flavor ofcharge Q q ). The combined QCD+EM gauge fields wereused to calculate up-, down-, and strange-quark propa-gators, which were then contracted to form the requi-site nuclear correlation functions using the techniquesof Ref. [21]. To ensure periodicity, ˜ n ∈ Z , and thevalues ˜ n = 0 , , − , , , − ,
12 were used on the SU(3)symmetric ensemble, while ˜ n = 0 , , − , m π ∼
450 MeV ensemble. The corresponding fieldstrengths are quantized as | e B | = 6 π | ˜ n | / ( aL ) , givinga field of O (10 ) Gauss for ˜ n = 1. On each con-figuration, quark propagators were generated from 48uniformly distributed Gaussian-smeared sources for eachmagnetic field. For further details of the production atthe SU(3)-symmetric point, see Refs. [12, 18, 19] and inparticular, Ref. [13]. Analogous methods were used forthe light mass ensemble.This work focuses on the dineutron, the diproton, andthe maximal | j z | = j = 1 spin state of the deuteron,all of which remain isolated, sub-threshold states in thepresence of a magnetic field. The I z = j z = 0 neutron-proton systems with ( j = 1; I = 0) and ( j = 0; I = 1)mix in a magnetic field and have been considered previ-ously in Ref. [20] to determine the cross section for theradiative capture process np → dγ . States with the quan-tum numbers of h = n, p, nn, pp, d | j z | =1 are accessedfrom correlation functions C h ( t ; B ) = (cid:104) | χ h ( t ) χ h (0) | (cid:105) B computed in the presence of the background magneticfield B from source and sink interpolating operators withthe requisite quantum numbers, as discussed in detail inRef. [13]. Representative correlation functions for theheavier mass ensemble can be found in Ref. [13] for eachhadron/nucleus and background magnetic field. Ratiosof these correlation functions to those without the mag-netic field, R h ( t ; B ) ≡ C h ( t ; B ) /C h ( t ; ), are also shownin Ref. [13], and are used to extract the magnetic mo-ments and polarizabilities of the respective systems. Forthe m π ∼
450 MeV ensemble, the ratios behave in aqualitatively similar manner and the signals are of com-parable quality. As the central focus of this study is onthe difference between the effect of the field on the two-nucleon systems and on the nucleons in isolation, thefurther ratios δR A ( t ; B ) = R A ( t ; B ) (cid:30) (cid:89) h ∈A R h ( t ; B ) , (2)are of primary importance. In this expression, A refersto the composite system and the product is over its con-stituent nucleon correlator ratios (e.g., for A = d j z =+1 the contributions are from p ↑ and n ↑ ). The late time ex-ponential decay of this ratio is dictated by the bindingenergy of the system in the presence of the field [13], δR A ( t ; B ) t →∞ −→ Z A ( B ) e − (cid:32) δE A ( B ) − (cid:80) h ∈A δE h ( B ) (cid:33) t . (3)Fig. 1, shows these ratios for the m π ∼
450 MeV en-semble along with the results of single exponential fits totime ranges in which the individual correlation functionsentering the ratios are consistent with single exponen-tial behavior. As discussed in Ref [13], multiple differentinterpolating operators are investigated for each state inthis study and the resulting differences are used to gauge,in part, the systematic uncertainty. In the figures below,we focus on a particular set of interpolating operators forclarity but have verified that other choices of interpola-tors provide consistent results. The analogous results forthe heavier mass ensemble are presented in Ref. [13]. �������� δ � �� � � ��������� δ � � � � � � �� �� �������� � / � δ � �� � � � � �� �� �� � / � � � �� �� �� � / �� = � � =- � � = � FIG. 1: Correlator ratios defined in Eq. (2) for the nn ,the j z = +1 deuteron and pp systems for field strengths˜ n = 1 , − ,
4, for the m π ∼
450 MeV ensemble. The bands cor-respond to the exponential fit and its statistical uncertaintiesassociated with the shown fit interval. Systematic uncertain-ties from the choice of fit range are separately assessed.
The energy shifts∆ A ( (cid:101) n ) ≡ δE A ( B ) − (cid:88) h ∈A δE h ( B ) (4)in the dineutron and deuteron ( j z = +1) channels areshown in Figs. 2 and 3, respectively. As the strength ofthe applied magnetic field is increased, the ground stateenergies of the systems are shifted closer to threshold,and at a given field strength it appears that the states m (cid:1) =
806 MeV - - (cid:1) nn ( n (cid:2) ) [ M e V ] m (cid:1) =
450 MeV - - (cid:1) nn ( n (cid:2) ) [ M e V ] |n| ~ FIG. 2: Response of the binding of the dineutron system toapplied magnetic fields. The upper panel shows the result at m π = 806 MeV, while the lower panel is for m π = 450 MeV.The shaded regions correspond to the envelopes of successfulfits to the energy shifts using linear and quadratic polynomialsin ˜ n to data points in the corresponding range indicated bythe shaded region. The horizontal bands indicate the bindingthreshold. m π =
806 MeV - - Δ d , j z = + ( n ) [ M e V ] m π =
450 MeV - - | n | Δ d , j z = + ( n ) [ M e V ] FIG. 3: Response of the binding of the j z = +1 state ofthe deuteron to applied magnetic fields. The shaded regionscorrespond to the envelopes of successful fits to the energyshifts using polynomials in ˜ n of up to 4 th (2 nd ) order forthe m π = 806 (450) MeV ensemble. The horizontal bandsindicate the binding threshold. unbind. For the deuteron, this behavior is not clearlyresolved at the lighter mass because of the uncertainties.The approach to threshold and subsequent turnover isseen at both quark masses in the dineutron system, andthe point of minimum binding decreases as the quarkmass is lowered, ˜ n (max) nn ∼ m π ∼
806 MeV and˜ n (max) nn ∼ m π ∼
450 MeV. The dineutron is unboundin nature and the present results suggest that magneticeffects would push the system further into the contin-uum. On the other hand, it is possible that the deuteron m π =
806 MeV - - - - Δ pp ( n ) [ M e V ] m π =
450 MeV - - - - | n | Δ pp ( n ) [ M e V ] FIG. 4: Response of the binding of the diproton to ap-plied magnetic fields. The shaded regions correspond to theenvelopes of successful fits to the energy shifts using polyno-mials in ˜ n of up to 4 th (2 nd ) order for the m π = 806 (450)MeV ensemble. The horizontal bands indicate the bindingthreshold. m π =
806 MeV - - | n | Δ ΛΛ ( n ) [ M e V ] FIG. 5: The energy splitting in the H dibaryon channel at m π = 806 MeV. The horizontal band indicates the bindingthreshold. could be unbound by the presence of magnetic fields ofstrength comparable to those expected in magnetars andheavy ion collisions, potentially modifying the dynamicsof those systems. A particularly interesting aspect of thebehavior in both of these channels is the approach to theunitary regime in which the binding energies decreaseto zero and consequently the scattering lengths diverge. In atomic physics, such behavior is routinely used to in-vestigate the universal physics that emerges in systemsinteracting near unitarity [22], but they have not beenobserved in nuclear physics.The energy shifts of the diproton are shown in Fig. 4.For this system, the extracted energies are not as cleanlydetermined as for the dineutron, but a trend towardstrengthening attraction is seen at both quark masses It is expected that the range of the interaction (set by hadronicscales) is only weakly affected by the magnetic field, so the vol-ume effects in the two-nucleon systems are not expected to beunmanageable even as the scattering length diverges. as the field strength increases. This is interesting in lightof a recent suggestion [23] that the diproton can over-come the Coulomb repulsion and form a bound state insufficiently large magnetic fields. A naive extrapolationof the slope of the shift linearly in m π indicates that fora field of | e B | ∼ Gauss, corresponding to ˜ n ∼ . H -dibaryon at heavier quark masses [24, 25]. Thischannel exhibits a slight reduction of the binding energyfor intermediate field strengths, comparable in size tothat of the dineutron system, but does not exhibit res-onant behavior in the range of field strengths that areprobed as the binding energy is significantly larger. Discussion:
Having found significant changes in thebinding of two-nucleon systems immersed in strong mag-netic fields at two values of unphysical quark masses, itis conceivable that similar modifications occur in nature.To solidify this discussion, calculations would need tobe performed at or near the physical quark masses andthe continuum and infinite volume limits would requirecareful investigation. While the responses of these sys-tems can as yet only be estimated at the physical quarkmasses, the calculated trends provide an interesting start-ing point to consider possible consequences. To this end,it is conjectured that two-nucleon systems will exhibitunitary behavior, with the deuteron unbinding in a largemagnetic field and the diproton system becoming bound.On the other hand, the dineutron will be pushed furtherinto the continuum as the field strength increases. In-terestingly, it may be possible to find values of the fieldstrength and quark masses where all NN states are atthreshold simultaneously, realizing the low energy confor-mal symmetry postulated by Braaten and Hammer [26].Given the observed behavior of bound states, it is nat-ural to expect that the NN scattering phases shifts andmixing angles will also be modified at a similar level insuch fields. These modifications would be interesting toprobe in future LQCD calculations utilizing the L¨uschermethod [14, 15] to analyze the spectra of NN systems.In ultra-peripheral heavy ion collisions, one can spec-ulate that the reduced binding between pairs of nucle-ons, along with the reduction in the nucleon mass, willincrease the size of each nucleus as they interact with Based on studies of binding energies on these and other relatedensembles, we are confident that the current calculations do notsuffer from large volume or scaling artifacts. the field of the other nucleus. Ignoring other potentialeffects, purely geometrical considerations will result inlarger than expected interaction cross-sections that willincrease with the collision energy for a given impact pa-rameter and potentially larger fluctuations in collisioncross sections. However, considering the transient natureof such a collision, and the difference between the internaltime-scales associated with a rearrangement of the nu-cleons comprising each nucleus and that of the collision,more detailed analyses must be performed before even aqualitative understanding can be established. The effectsof large magnetic fields in magnetars through the mag-netic moments of nucleons and electrons have been con-sidered through a number of model approaches [4, 27–31],and in some cases lead to significant modifications. Themore complicated effects from magnetic shifts in bindingand hadronic interactions likely also induce significantmodifications that deserve further investigation.We would like to thank Zohreh Davoudi, DaekyoungKang and Krishna Rajagopal for several interesting dis-cussions. This research was supported in part by the Na-tional Science Foundation under Grant No. NSF PHY11-25915 and WD and MJS acknowledge the Kavli Institutefor Theoretical Physics for hospitality during completionof this work. Calculations were performed using compu-tational resources provided by the Extreme Science andEngineering Discovery Environment (XSEDE), which issupported by National Science Foundation grant numberOCI-1053575, NERSC (supported by U.S. Departmentof Energy Grant Number DE-AC02-05CH11231), and bythe USQCD collaboration. This research used resourcesof the Oak Ridge Leadership Computing Facility at theOak Ridge National Laboratory, which is supported bythe Office of Science of the U.S. Department of Energyunder Contract No. DE-AC05-00OR22725. The PRACEResearch Infrastructure resources Curie based in Franceat the Tr`es Grand Centre de Calcul and MareNostrum-IIIbased in Spain at the Barcelona Supercomputing Centerwere also used. Parts of the calculations used the Chromasoftware suite [32]. SRB was partially supported byNSF continuing grant PHY1206498 and by U.S. Depart-ment of Energy through Grant Number DE-SC001347.WD was partially supported by the U.S. Departmentof Energy Early Career Research Award DE-SC0010495.KO was partially supported by the U.S. Department ofEnergy through Grant Number DE- FG02-04ER41302and through contract number DE-AC05-06OR23177 un-der which JSA operates the Thomas Jefferson NationalAccelerator Facility. 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