Neutral–pion reactions induced by chiral anomaly in strong magnetic fields
We investigate decay and production of neutral pions in strong magnetic fields. In the presence of strong magnetic fields, transition between a neutral pion and a virtual photon becomes possible through the triangle diagram relevant for the chiral anomaly. We find that the decay mode of a neutral pion into two photons cannot persist in the dominant mode in strong magnetic fields, and that decay into a dilepton instead dominates over the other modes. We also investigate the effects of magnetic fields on prompt virtual photons created in ultrarelativistic heavy-ion collisions. There is no anisotropy in the spectrum at the stage of creation of prompt virtual photons, but after traversing the strong magnetic field that is induced perpendicularly to the reaction plane, virtual photons turn into neutral pions, leading to an anisotropic spectrum of dileptons as a feasible signature in the measurement.
Much attention has been paid to strong magnetic fields in nature and laboratories. Especially, magnitudes of magnetic fields are thought to reach T in strongly magnetized neutron stars as known as magnetars TD (), and T at the impact of ultrarelativistic heavy–ion collisions in RHIC and LHC KMW (); estimates (); Itakura_PIF (). A magnitude of the latter field provides a scale as large as pion mass , implying that effects of the strong magnetic field on light hadrons could become as important as strong interaction. In this Letter, we address the effects of such extremely strong magnetic fields on neutral pion reactions through the chiral anomaly anomaly (); anom_rev ().
First, we show that the leading decay mode and lifetime of neutral pion change as magnitude of an external magnetic field approaches the neutral–pion mass squared, , and/or a propagating pion carries large energy. In the strong field limit, a neutral pion dominantly decays into a dilepton without being accompanied by any real photon in the final state (see Fig. 1(c)).
The inverse process provides a neutral–pion production mechanism (see Fig. 1()) known as the Primakoff effect, i.e., a conversion of a real photon into a neutral pion in atomic Coulomb fields Pri (). Remarkably, the Primakoff effect has been a standard method in measurement of neutral–pion lifetime Anom_review (), and also has been applied to search for a hypothetical pseudo–scalar particle “axion” axion (). We investigate a neutral–pion production mechanism applied to the prompt virtual photon from hard parton scatterings in ultrarelativistic heavy–ion collisions. Since this scattering process takes place in a time scale much shorter than that of a rapidly decaying strong magnetic field KMW (); estimates (), prompt photons enjoy much chance to interact with the magnetic fields. We show that an oriented production rate with respect to the external magnetic field gives rise to an anisotropic spectrum of dileptons originating from the prompt virtual photons. The second Fourier coefficient of the azimuthal angle dependence, conventionally called , will be analytically related to the Fourier component of the spacetime profile of the magnetic field. We will find that energy transfer from a time–dependent magnetic field enlarges the final–state phase space, leading to a non–vanishing dilepton in a wide kinematical window.
It has been shown that conservation of the axial vector current is anomalously violated anomaly ():
where the fine structure constant is given by with unit electric charge “”. This “anomaly relation” indicates a coupling between neutral–pion field composing and two photon fields, and is consistently taken into account as an effective vertex called the Wess–Zumino–Witten (WZW) term WZW (),
A coupling constant is specified by the number of color degrees and pion decay constant MeV. In the presence of an external field, we divide the photon field into dynamical and external fields as , and correspondingly the field strength tensor as . Substituting these into the WZW term (2), we obtain not only the conventional vertex with two dynamical photons,
but also a three–point vertex including an external field,
The Adler–Bardeen theorem tells that the anomaly relation (1) and thus the WZW term (3) persist in the all–order perturbation theory AB (), which implies that only the triangle diagram without higher–order radiative corrections is relevant. In the absence of an external field, the WZW term (3) describes the decay of into two photons (Fig. 1 (a)), providing a decay width to be, eV in excellent agreement with measurement: eV. The next–to–leading decay mode follows from a QED correction to one of the two photons, i.e., the Dalitz decay (Fig. 1 (b)), without decay into photons more than two. The Adler–Bardeen theorem also confirms that a coupling of to an external field is determined by the vertex (4) NQED ().
Focusing on the vertex (4), we find a decay mode possible only in the presence of strong external fields. We shall consider the case of magnetic fields below. As depicted in Fig. 1 (c), a neutral pion couples to an electromagnetic current through the vertex (4), corresponding to a dilepton in the final state. While the amplitude of this process, , is suppressed compared to decay mode (a) by an order in QED coupling constant, , with an inverse pion mass square from the virtual–photon propagator, a large value of a strong external magnetic field compensates the suppression when the magnitude becomes strong beyond the “critical field” defined by . Therefore, dilepton mode (c) overwhelms the Dalitz decay (b) and two–photon mode (a) as the magnetic field becomes strong.
where the “lepton tensor” provides a squared amplitude of the process, , which is related to an imaginary part of the photon vacuum polarization tensor, . For the photon propagator and the imaginary part of the vacuum polarization tensor, we incorporate the lowest–order contribution in the order of the QED coupling and the magnetic field: namely, the free photon propagator (in arbitrary gauge) and one–loop vacuum polarization tensor in the ordinary vacuum. Without momentum transfer from a constant magnetic field, a pion with can decay into but not , so that this lepton mass corresponds to electron mass below.
Inserting the free photon propagator and the one–loop vacuum polarization tensor into the decay rate (5), one finds that the gauge–dependent terms in the propagators drop in contracting with the transverse projection operator in , and thus the decay rate is obviously gauge invariant. With , the decay rate is obtained as
where a squared momentum is denoted as with and being a spatial component of the momentum parallel to the external field and an angle measured from the direction of the field, respectively. This angle dependence gives rise to an elliptically–anisotropic production rate, and the decay rate grows quadratically with respect to by the factor as mentioned above.
Figure 2 shows the lifetime of in a strong magnetic field, obtained as an inverse of the total decay rate, . While dashed lines show the lifetime against decay modes (a) and (b) in the ordinary vacuum, solid lines indicate lifetime including decay mode (c) as well as the other two, which decreases by a few orders when the magnetic field becomes strong. The decay width in modes (a) and (b) are referred to the measured values eV and eV with an appropriate kinematical factor providing decay widths in arbitrary Lorentz frame, . As this kinematical factor represents the relativistic time delay, a fast moving pion in a weak field has larger lifetime than that of a rest pion. However, Fig. 2 shows that a fast moving pion decays faster than a rest pion in strong magnetic fields because of an enhancement of the decay rate (6) by a factor, .
To show the dominant decay mode in the presence of strong magnetic fields, we define a branching ratio by for “”–mode among (a), (b) and (c). Magnetic–field–strength dependence of is shown in Fig. 3. Decay mode (c) overwhelms the other two modes not only in the strong field limit, but also in relatively weak field if pions carry large energy. Dashed vertical line shows a field strength of the order of which are thought to accompany the magnetars.
Very recently, pion spectrum in cosmic rays was constructed from two photons SNR (), of which origin is attributed to energetic collisions of highly accelerated protons in the region of supernova remnants. In such radical events in stars, strong magnetic fields extending in macroscopic scales may act energetic neutral pions to modify the lifetime and branching ratio, and in turn successive processes of energetic nuclear reactions.
One would expect that the strong magnetic fields created in ultrarelativistic heavy–ion collisions also provide any signature of the interaction through the anomaly relation. Indeed, it is proposed that photon production is possible in strong magnetic fields by conversions from a dilatation current through conformal anomaly BKS () and an axial vector current through chiral anomaly FM (). However, it should be elaborated whether light–meson components in those currents could interact with the rapidly decaying magnetic field, because formation of light mesons would take a longer time than the lifetime of the magnetic field. This also holds in the present case. Neutral pions are expected to be formed in a time scale of the order of its mean lifetime that is much larger than the lifetime of the magnetic field, even if we take into account the short lifetime due to the magnetic field shown in Fig. 2. Therefore, we do not expect that significant amount of neutral pions decay just after the collisions. However, it opens another possibility if we consider the inverse process, namely, a neutral pion production from a prompt virtual photon. In the rest of this Letter, we propose that of the dilepton originating from the prompt virtual photon emerges as a reflection of the oriented neutral–pion production in the strong magnetic field though the WZW term (4) (see Fig. 1 ()).
First, recall that a cross section of the dilepton production from a prompt virtual photon in a nucleon+nucleon collision is expressed as (see Appendix B in Ref. dilepton ())
where and is an invariant mass of a dilepton. The cross section of virtual photon production is given by perturbative QCD calculation pQCD (). Note that the final–state momenta of leptons are integrated out in Eq. (7), and thus this expression provides an integrated cross section at a given virtual photon momentum, . By Glauber–modeling of high–energy collisions, total yield in a heavy–ion collision event is obtained by scaling the cross section in a proton+proton collision with the collision geometry encoded in the overlap function between nuclei A and B at impact parameter Gla (), resulting in a simple scaling formula, .
where the final–state pion phase space is integrated out with respect to , so that Eq. (8) provides an integrated cross section of the pion production from virtual photons carrying a given momentum . In integral (9), Fourier component of an external magnetic field is denoted as . Note that dependences of the cross section on an angle and the magnetic field strength follow from a Lorentz contraction, , and that an angle dependence measured from the reaction plane is thus found to be with and .
Some of the virtual photons created at the hard collision are converted into neutral pions in the presence of strong magnetic field. Since the conversion rate depends on the angle, it effectively gives rise to an anisotropy in the spectrum of dileptons. By using the cross sections (7) and (8), a reduced amount of dilepton yield in the presence of the magnetic field may be given by where isotropic and elliptically–anisotropic components are expressed as
In the above, we find negative for dilepton production, because the neutral pion production mechanism works more efficiently in perpendicular to the magnetic field as inferred from –dependence in Eq. (8).
We capture the magnetic field created in peripheral heavy–ion collisions as a time–dependent, but spatially homogeneous, field oriented in perpendicular to the reaction plane (). Motivated by the Liénard–Wichert potential, the non–vanishing component is simply modeled as . This mimics profiles obtained in numerical simulations estimates () if we take a spatial scale , magnitude of the magnetic field and beam rapidity to be fm, and for nucleus–nucleus collisions at TeV, respectively. It is necessary for investigating impact–parameter dependence to take into account a charge distribution in nuclei by more sophisticated analyses, and thus we do not go into this point in the present Letter. Fourier component of the magnetic field, , is then analytically obtained as with the help of modified Bessel function of the first kind, . Inserting this into Eq. (9), the integral is carried out as
where square of delta functions in spatial components are understood as a multiplication of the delta function and three–dimensional volume, , and shorthand notations are introduced as, and . To obtain the cross section (8), reaction rate has to be averaged over a ‘time scale ’ of the reaction in the magnetic field, which is here taken to be of the order of lifetime of the rapidly decaying magnetic field, fm/.
Figures 4 and 5 show dilepton at mid–rapidity with respect to the invariant mass and transverse momentum , respectively. Solid (dashed) line shows a result for () pair. In Fig. 4, the anisotropy becomes large in low invariant mass region, since, as seen in Eq. (8), pion production is enhanced in this region with less suppression by the quartic factor from the virtual photon propagator. An energy transfer from the time–dependent magnetic field stretches the kinematical window from a single point, . We find in Fig. 5 that the anisotropy becomes large in high transverse momentum region. The reason is twofold. First, anisotropy of the longitudinal–momentum square in Eq. (8) becomes stronger, providing a large in Eq. (11). The other depends on profiles of time–dependent magnetic field. Noticing that a large transverse momentum provides a small difference in Eq. (12), high momentum region reflects a value around the origin, . The Fourier component in the present model takes the largest value at the origin, and thus provides a large anisotropy. This would hold more generally for a wide variety of the profiles.
In summary, we investigated both decay and production of neutral pions in strong magnetic fields. These reactions become possible as conversions between neutral pions and virtual photons in external fields. The dominant decay into a dilepton provides a large decay width and thus short lifetime of , when the magnetic field strength is large and/or the pion energy is large. The decay width is large for pions propagating in perpendicular to the magnetic field, since the conversion is the most efficient in this direction. This angle dependence of the conversions, when applied to production from the prompt virtual photons in ultrarelativistic heavy–ion collisions, gives rise to an origin of an anisotropic dilepton spectrum totally different from hydrodynamic flow. The magnitude of the anisotropy could be as large as, or even larger than, that of the thermal dilepton emitted in the quark–gluon plasma phase dilepton_v2 (). The negative of dileptons would be more suitable for measurement rather than the positive of neutral pions, due to a huge background in the pion spectrum and an energy loss in the hot matter. We will further pursue a possibility if the effect of the strong magnetic field is indeed observed in ultrarelativistic heavy–ion collisions. This would require more detailed investigations including estimate of a signal significance, i.e., the isotropic part (10) combined with a virtual–photon yield from perturbative QCD.
This work was supported in part by the Korean Research Foundation under Grant Nos. KRF-2011-0020333 and KRF-2011-0030621, and also in part by “The Center for the Promotion of Integrated Sciences (CPIS)” of Sokendai.
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