Probing Extreme Electromagnetic Fields with the Breit-Wheeler Process

Daniel Brandenburg, for the STAR Collaboration (Shandong University & BNL/CFNS)

March 3, 2020 36th Winter Workshop on Nuclear Dynamics Puerto Vallarta, Mexico Outline of this Talk 1. Intro: What is the Breit-Wheeler Process? 2. Results from STAR Collaboration 3. Vacuum Birefringence in Extreme Magnetic Fields 4. The Magnetic Field in Heavy Ion Collisions 1. Measuring the “Initial” (� = 0) Magnetic Field 2. Evidence for Long-lived Magnetic Field or Medium Effects? 5. Conclusions

3/3/20 Daniel Brandenburg 2 Fundamental Interactions : Light & Matter � �⋆

� Photo Electric Effect Bremsstrahlung Compton Scattering 1887 Hertz, Ann Phys 1895 Röntgen, Ann Phys 1906 Thomson, Conduction of (Leipzig) 31, 983 (Leipzig) 300, 1 Electricity through Gases

Bethe-Heitler Pair Single Photon Dirac Annihilation Breit-Wheeler Production Annihilation 1934, Klemperer, pair production 1932, Anderson, 1933, Blackett & Proc Camb Phil Predicted 1934 Science 76,238 Occhialini, Proc R Soc 30, 347 Soc Lond A 139, 699

February 5, 2020 Daniel Brandenburg Based on slide by O. Pike 3 The Breit-Wheeler Process : �� → ��

�

� o Breit-Wheeler process is by definition the lowest-order, tree level process o Two diagrams contribute at the lowest-order o t-channel process, specifically note: � = � + �

February 5, 2020 Daniel Brandenburg 4 Ultra-Peripheral Heavy-Ion Collisions Ultra-relativistic charged nuclei produce highly Lorentz- contracted electromagnetic field

Weizäcker-Williams Equivalent Photon Approximation (EPA): � ≈ � → In a specific phase space, transverse EM fields can be quantized as a flux of real photons Weizsacker̈ , C. F. v. Zeitschrift fur̈ Physik 88 (1934): 612

� ∝ �⃗ = �×� ≈ � ≈ � �

�� ≈ 1 → High photon density Ultra-strong electric and magnetic fields: � ≈ � → Expected magnetic field strength � ≈ �� − �� T Skokov, V., et. al. Int. J. Mod. Phys. A 24 (2009): 5925–32

� � Test QED under extreme conditions

3/3/20 Daniel Brandenburg 5 �� → �� Process in UPCs

´103 Zero Degree Calorimeter 60 Au+Au UPC Data 1 neutron 50 2 neutrons 40 3 neutrons 4 neutrons 30 Total Fit 20 dN/d(ADC Sum West ZDC)

0 200 400 600 800 1000 1200 ADC Sum West ZDC Breit-Wheeler �� → �� Mutual Coulomb excitation and pair production process nuclear dissociation • Provides efficient trigger condition →Provides high statistics sample (>6,000 �� pairs from data collected in 2010) → Allows for multidifferential analysis

February 5, 2020 Daniel Brandenburg 6 Total �� → �� cross-section in the STAR Acceptance A Pure QED 2 → 2 scattering :

)) 1 arXiv : 1910.12400 2 STAR ��⁄�� ∝ � ≈ � Au+Au UPC gg® e+e- (QED) No vector meson production gg® e+e- (gEPA) → Forbidden for real photons with -1 10 gg® e+e- (STARLight) helicity ±1 (i.e. 0 is forbidden) Scale Uncertainty : ± 13% (mb/(GeV/c ) -

e + � �� → � � in the STAR Acceptance: -2 e 10 Data : 0.261±0.004 (stat.) ± 0.013 e ® P > 0.2 GeV/c & |he| < 1.0

dM (sys.) ± 0.034 (scale) mb

g ee g |y | < 1.0 & P < 0.1 GeV/c STARLight gEPA QED (

s -3 0.22 mb 0.26 mb 0.29 mb d 10 0.5 1 1.5 2 2.5 M (GeV/c2) Measurement of total cross ee section agrees with theory STARLight: S. R. Klein, et. al. Comput. Phys. Commun. 212 (2017) 258 gEPA & QED : W. Zha, J.D.B., Z. Tang, Z. Xu arXiv:1812.02820 [nucl-th] calculations at ±�� level February 5, 2020 Daniel Brandenburg 7 Toy MC setup ØIn e+e- pair rest frame, the . is defined as the angle between positron momentum and the beam line • The . distribution for the --->e+e- has e+e- pair mass dependence �� → ��• The ⁄�cos��. distribution for the hadronic � two-body decay is flat C 1 �� → �� : Individual �/� preferentially arXiv : 1910.12400 STAR 2 aligned along beam axis [1]: Mee = 2 GeV/c 0.4 Au+Au2 UPC

0.8 (mb) Mee = 1.2 GeV/c - gg2® e+e (XnXn)´0.88 ) - Mee = 0.4 GeV/c 4� 4� e 2 + -

+ Isotropic e e 1 − sin � cos � + Mee = 0.01 GeV/c 4� � � ')| 0.3 e

� � = 2 + 4 1 − 0.6 q

� ® 4�

→ 1 − 1 − cos � NOTE: for virtual photons � g

g 0.2 isotropic (flat) distribution

0.4( d|cos( s o Highly virtual photon interactions should d have an isotropic distribution 0.1 0.2 2 0.4 < Mee < 0.76 GeV/c o Measure �, the angle between the � and the beam axis in the pair rest frame. 0 0 0 0.2 0.4 0.6 0.8 0 0.5 1 1.5 2 2.5 3 12/6/17 |cos(q')| 3 q for gg->e+e- process

[1] S. Brodsky, T. Kinoshita and H. Terazawa, Phys. Rev. D4, 1532 (1971) STARLight: S. R. Klein, et. al. Comput. Phys. Commun. 212 (2017) 258

February 5, 2020 Daniel Brandenburg 8 �� → �� ⁄�� C �� → �� : Individual �/� preferentially arXiv : 1910.12400 STAR aligned along beam axis [1]: 0.4 Au+Au UPC (mb) gg® e+e- (XnXn)´0.88 ) - 4� 4� e + -

+ Isotropic e e 1 − sin � cos � + 4� � � ')| 0.3 e

� � = 2 + 4 1 − q

� ® 4� 1 − 1 − cos � � g

g 0.2 ( d|cos( s o Highly virtual photon interactions should d 0.1 have an isotropic distribution 2 0.4 < Mee < 0.76 GeV/c o Measure �, the angle between the � and the 0 beam axis in the pair rest frame. 0 0.2 0.4 0.6 0.8 |cos(q')| ⇒Data are fully consistent with �(�) distribution expected for �� → �� [1] S. Brodsky, T. Kinoshita and H. Terazawa, Phys. Rev. D4, 1532 (1971) ⇒Measurably distinct from isotropic STARLight: S. R. Klein, et. al. Comput. Phys. Commun. 212 (2017) 258 distribution

February 5, 2020 Daniel Brandenburg 9 �� → � � ⁄�� B arXiv : 1910.12400 o High precision data – test theory 5 STAR predictions Au+Au UPC o STARLight predicts significantly 4 gg® e+e- (STARLight) lower ⟨�⟩ than seen in data

3 arXiv : 1910.12400 (mb/(GeV/c)) ) - e + 2 e

dP g 1

g 2 0.4 < Mee < 0.76 GeV/c ( s

d 0 0 0.02 0.04 0.06 0.08 0.1 P (GeV/c) QED and STARLight are scaled to match measured �(�� → ��) STARLight: S. R. Klein, et. al. Comput. Phys. Commun. 212 (2017) 258 QED : W. Zha, J.D.B., Z. Tang, Z. Xu arXiv:1812.02820 [nucl-th] February 5, 2020 Daniel Brandenburg 10 �� → � � ⁄�� B arXiv : 1910.12400 o Data are well described by 5 STAR leading order QED Au+Au UPC calculation (�� → � � ) with + - 4 gg® e e (QED) quasi-real photons + - gg® e e (STARLight) o STARLight predicts significantly 3 arXiv : 1910.12400 (mb/(GeV/c)) lower ⟨�⟩ than seen in data ) - STARLight calculations do not have e o + 2 centrality-dependent � e

distribution ®

dP g 1

g 2 0.4 < Mee < 0.76 GeV/c (

s o Experimentally investigate d 0 0 0.02 0.04 0.06 0.08 0.1 impact parameter dependence : P (GeV/c) Compare UPC to peripheral → QED and STARLight are scaled to match measured �(�� → � � ) collisions (come back to later) STARLight: S. R. Klein, et. al. Comput. Phys. Commun. 212 (2017) 258 QED : W. Zha, J.D.B., Z. Tang, Z. Xu arXiv:1812.02820 [nucl-th] 11/05/19 Daniel Brandenburg 11 Classical Electromagnetism • Maxwell’s equations are linear ØSuperposition principle holds 1 � �ℒ ℒ = − � � = � = �� 2� � 1 �ℒ � = � � = − � ��

→ Unique speed of light in vacuum: � = = 299792458 m/s

February 5, 2020 Daniel Brandenburg 12 Quantum Electrodynamics Three important discoveries that alter the classical picture: o Einstein’s energy-mass equivalence: � = �� o Uncertainty principle: Δ�Δ� ≥ ℏ/2 Einstein o Existence of positron : Dirac predicts negative electron energy states (1928), Anderson discovered positron in 1932

Anderson

February 5, 2020 Daniel Brandenburg 13 Quantum Electrodynamics Three important discoveries that alter the classical picture: o Einstein’s energy-mass equivalence: � = �� o Uncertainty principle: Δ�Δ� ≥ ℏ/2 Einstein o Existence of positron : Dirac predicts negative electron energy states (1928), Anderson discovered positron in 1932 → Vacuum fluctuations o 1936: Euler & Heisenberg present modified Lagrangian

ℒ = − � + − � + 7 ⋅ � + ⋯ Anderson oNon-linear → Superposition principle is broken!

NB: in 1951 Shwinger derived the Lagrangian within QED formalism

February 5, 2020 Daniel Brandenburg 14 Vacuum Magnetic Birefringence � = BUT � ≠ � and � ≠ � ∥ ∥ Light behaves as if it is traveling through a medium with an index of refraction � ≠ 1

Guido Zavattini ICNFP2019

February 5, 2020 Daniel Brandenburg 15 wikipedia Optical Birefringence Birefringent material: Different index of refraction for light polarized parallel (�∥) vs. perpendicular (�) to material’s ordinary axis → splitting of wave function when �� = �∥ − � ≠ �

Birefringent Material Linearly polarized (vertical)

Ordinary ray

Extra-ordinary ray

Linearly polarized (horizontal)

February 5, 2020 Daniel Brandenburg 16 Vacuum Birefringence Vacuum birefringence : Predicted in 1936 by Heisenberg & Euler. Index of refraction for � interaction with �-field depends on relative polarization angle i.e. �� = �∥ − � ≠ �

Empty space + R. P. Mignani, et al., Mon. Not. Roy. Astron. Soc. 465 (2017), 492 Ultra-strong Magnetic Field Linearly polarized (vertical)

Linearly polarized (horizontal)

February 5, 2020 Daniel Brandenburg 17 Birefringence of the QED Vacuum Vacuum birefringence : Index of refraction Feynman Diagram for Vacuum Birefringence for � interaction with �-field depends on relative polarization angle i.e. �� = �∥ − � ≠ � � from �

Lorentz contraction of EM fields → Quasi-real photons should be linearly polarized (� ⊥ � ⊥ �)

Recently realized that a consequence of Probe � Observed � �∥ − � ≠ � in �� → � � collisions is a �� modulation [3] between the pair momentum and the daughter momentum. ��(�) = transmission process �� → �� (�) = absorption process �� → �� (diagram cut) Can we observe vacuum birefringence [1]S. Bragin, et. al., Phys. Rev. Lett. 119 (2017), 250403 [2]R. P. Mignani, et al., Mon. Not. Roy. Astron. Soc. 465 (2017), 492 in ultra-peripheral collisions? [3] C. Li, J. Zhou, Y.-j. Zhou, Phys. Lett. B 795, 576 (2019)

February 5, 2020 Daniel Brandenburg 18 [1] C. Li, J. Zhou, Y.-j. Zhou, Phys. Lett. B 795, 576 (2019) Birefringence of the QED Vacuum QED calculation: arxiv : 1911.00237 1000 Recently realized, Δ� = �∥ − � ≠ 0 0.45 < M < 0.76 GeV/c2 leads to ��(��) modulations in / 20) 900 ee STAR p polarized �� → � � [1] Au+Au UPC Au+Au 60-80% ´ 0.5 � = 200 GeV Total Cross-section Measured STARLight800 gEPA QED Fit: C´( 1 + A cos 2Df + A cos 4Df ) ± 1 s 2Df 4Df Δ� = Δ� �0.261+ � 0.004, � (stat.)− � 700 + ±

( e e)mb 0.013 (syst.) 0.034 (scale) 0.22counts / ( 0.26 0.29 ! ≈±Δ� � + � ±, � 600 Ultra-Peripheral Peripheral HHICs Differential Quantities Total Cross-section Measured STARLight gEPA QED 500 Quantity Measured QED 2/ndf Measured QED 2/ndf 0.261 0.004 (stat.) 400 + ± ( e e)mb 0.013 (syst.)A4 (%)0.034−� (scale)(%) 0.2216 16.8.8 ±2.52.5 2216 0.26.5 18.8 0.29 / 16 27 6 39 10.2 / 17 ! ± | |± ± ± 300

Ultra-PeripheralA2 (%) 2.0Peripheral Peripheral2.4 0 (60 HHICs− 18.880%) / 16 6 6 0 10.2 / 17 Differential Quantities | | ± ± 200 MeasuredP QED2 (MeV/2Quantityc/)ndf Measured 38.1 0.9 37.6 QED —2/ndf 50.9 2.5 48.5 — h ?i ± ±100 + - q Polarized gg ® e e (QED) arXiv : 1910.12400 A (%) 16.8 2.5 22 18.8−� / 16(%) 2727 6± 6 3934.5 10.2 / 17 | 4| ± Table 1: The top row reports± the total measured cross-section within STAR acceptance for + 0 p p e e in (XnXn) events compared with three theory calculations. The lower rows report Df = f - f A (%) 2.0 2.4! 0 18.8 / 16 6 6 0 10.2 / 17 2 ee e | 2| ± measurements of and ± P 2 from UPCs and peripheral HHICs with the corresponding h ?i P 2 (MeV/c) 38.1 0.9theory 37.6 calculations — where 50.9q applicable.2.5 The 48.5 uncertainties — reported here are the statistical and h ?i ± February 5, 2020± Daniel Brandenburg2 19 q systematic uncertainties added in quadrature. The theory calculations for the P are from h ?i Table 1: The top row reports the totalRef. ( measured24). The QED cross-section calculations within for the STAR modulations acceptance arefor provided by Ref.q (13). + e e in (XnXn) events compared with three theory calculations. The lower rows report ! measurements of and P 2 from UPCs and peripheral HHICs with the corresponding h ?i theory calculations whereq applicable. The uncertainties reported here are the statistical and 2 systematic uncertainties added in quadrature. The theory calculationsSource of for the P are from 2 Distribution h ?i Fit Result /ndf Ref. (24). The QED calculations for the modulations areContamination provided by Ref.q (13).

0 + M ⇢ e e 0.36 1.2 (% of total ) 106 / 98 ee ! ± + ! e e 0.17 0.35 (% of total ) 106 / 98 ! ± + Source of e e +20.57 0.24 (% of total ) 104 / 98 Distribution Contamination Fit Result! /ndf± + 0 + cos ✓0 Isotropic e e +0.9 1.7 (% of total ) 7.7 / 12 M ⇢ e e | 0.36| 1.2 (% of total ) 106 / 98± ee ! ± + P (60 80%) Broadening 14 4 (stat.) 4 (syst.) (MeV/c) 3.4 / 6 ! e e ? 0.17 0.35 (% of total ) 106± / 98 ± ! ± + Table 2: The result from fits to various possible sources of contamination. For each source, the e e +0.57 0.24 (% of total ) 104 / 98 ! given distribution± was fit to the combination of the Breit-Wheeler shape and the listed contam- 2 +ination shape. The /ndf of each fit is also shown. cos ✓0 Isotropic e e +0.9 1.7 (% of total ) 7.7 / 12 | | ± P (60 80%) Broadening 14 4 (stat.) 4 (syst.) (MeV/c) 3.4 / 6 ? ± ± 24 Table 2: The result from fits to various possible sources of contamination. For each source, the given distribution was fit to the combination of the Breit-Wheeler shape and the listed contamination shape. The 2/ndf of each fit is also shown.

24 [1] C. Li, J. Zhou, Y.-j. Zhou, Phys. Lett. B 795, 576 (2019) Birefringence of the QED Vacuum QED calculation: arxiv : 1911.00237 1000 Recently realized, Δ� = �∥ − � ≠ 0 0.45 < M < 0.76 GeV/c2 leads to ��(��) modulations in / 20) 900 ee STAR p polarized �� → � � [1] Au+Au UPC Au+Au 60-80% ´ 0.5 � = 200 GeV Total Cross-section Measured STARLight800 gEPA QED Fit: C´( 1 + A cos 2Df + A cos 4Df ) ± 1 s 2Df 4Df Δ� = Δ� �0.261+ � 0.004, � (stat.)− � 700 + ±

Ultra-PeripheralA2 (%) 2.0Peripheral Peripheral2.4 0 (60 HHICs− 18.880%) / 16 6 6 0 10.2 / 17 Differential Quantities | | ± ± 200 MeasuredP QED2 (MeV/2Quantityc/)ndf Measured 38.1 0.9 37.6 QED —2/ndf 50.9 2.5 48.5 — h ?i ± ±100 + - q Polarized gg ® e e (QED) arXiv : 1910.12400 A (%) 16.8 2.5 22 18.8−� / 16(%) 2727 6± 6 3934.5 10.2 / 17 | 4| ± Table 1: The top row reports± the total measured cross-section within STAR acceptance for + 0 p p e e in (XnXn) events compared with three theory calculations. The lower rows report Df = f - f A (%) 2.0 2.4! 0 18.8 / 16 6 6 0 10.2 / 17 2 ee e | 2| ± measurements of and ± P 2 from UPCs and peripheral HHICs with the corresponding → First Earthh ?i-based observation (�. �� level) of vacuum birefringence P 2 (MeV/c) 38.1 0.9theory 37.6 calculations — where 50.9q applicable.2.5 The 48.5 uncertainties — reported here are the statistical and h ?i ± February 5, 2020± Daniel Brandenburg2 20 q systematic uncertainties added in quadrature. The theory calculations for the P are from h ?i Table 1: The top row reports the totalRef. ( measured24). The QED cross-section calculations within for the STAR modulations acceptance arefor provided by Ref.q (13). + e e in (XnXn) events compared with three theory calculations. The lower rows report ! measurements of and P 2 from UPCs and peripheral HHICs with the corresponding h ?i theory calculations whereq applicable. The uncertainties reported here are the statistical and 2 systematic uncertainties added in quadrature. The theory calculationsSource of for the P are from 2 Distribution h ?i Fit Result /ndf Ref. (24). The QED calculations for the modulations areContamination provided by Ref.q (13).

24 Connection to the Initial Magnetic Field Two direct connections to the arXiv : 1910.12400 �-field 1. Total �� → �� cross section 2. Strength of the vacuum birefringence phenomena

This field density is used in the QED calculations for Breit-Wheeler (�� → �� ) process and vacuum birefringence that achieve good agreement with all data.

Peak value for single ion: � ≈ 0.7 ×10 Tesla ≈ 10,000× stronger than Magnetars February 5, 2020 Daniel Brandenburg 21 Connection to the Initial Magnetic Field o How sensitive are these measurements to the peak field? o How sensitive to the geometry of the fields? QED: two-photon overlap probability

Au Au

oMost �� interactions in region where field from one ion is maximum �×� ∝ � × � ≈ �, ×�� (for large impact parameters)

February 5, 2020 Daniel Brandenburg 22 Mapping the Initial Magnetic Field Strength • At large impact parameters �×� ∝ � × � ≈ �, ×�� Zha, W., Brandenburg, J. D., Tang, Z. & Xu, Z. Phys. Lett. B800, 135089 (2020). Numerical QED calculation using arbitrary Four-Potential as input

Full QED calculation shows � ∝ |�|^2 For large � Note: � and � are fixed, the only free parameter is the Form-Factor (i.e. configuration of charges) Uncertainty from numerical integration Assumptions: o Spherically symmetric o Woods-Saxon charge distribution

February 5, 2020 Daniel Brandenburg 23 4

The numerical results for the computed azimuthal asymmetries forthedifferent collisions species and centralities are presented in Figs.2 and 3. Here the azimuthal asymmetries, i.e. the average value of cos 4φ are defined as, dσ cos 4φ d . . cos(4φ) = dP.S. P S (9) ⟨ ⟩ dσ d . . ! dP.S. P S We compute the asymmetry! for two deferent centrality classes as well as for the UPC and the tagged UPC cases. The corresponding impact parameter range for a given centrality class is determined using the Glauber model(see the review article [47] and references therein). For the UPC, the asymmetry is averaged over the impact parameter range [2RA, ]. However, STAR experiments at RHIC measure pair production cross section together with the double electromagnetic∞ excitation in both ions. Neutrons emitted at forward angles by the fragmenting nuclei are measured, and used as a UPC trigger. Requiring lepton pair to be produced in coincidence with Coulomb breakup of the beam nuclei alters the impact parameter distribution compared with exclusive production. In order to incorporate the experimental conditions in the theoretical calculations, one can define a ”tagged” UPC cross section, ∞ 2 2π b⊥db⊥P (b⊥)dσ(b⊥, ...) (10) "2RA where the probability P (b⊥) of emitting a neutron from the scattered nucleus is often parameterized as [48],

3 3 −5 Z (A Z) −5 Z (A Z) P (b⊥)=5.45 10 − exp 5.45 10 − (11) ∗ A2/3b2 − ∗ A2/3b2 ⊥ # ⊥ $ As a matter of fact, the mean impact parameter is dramatically reduced in interactions with Coulomb dissociation. We plot the cos 4φ asymmetry for electron pair production at mid-rapidity as the function of the total transverse momentum q⊥ at the center mass energy √s = 200 GeV in Fig.2. The general trend is that the asymmetry increases when the impact parameter decreases. The overall q⊥ and b⊥ dependent behavior of the asymmetry for the diﬀerent collision species(Au and Ru) are similar, except for that the curves are slightly more ﬂat for the smaller nucleus. The asymmetry reaches a maximal value of 17%–22% percent around q⊥ 30 MeV for the centrality classes [60%-80%], [80%-99.9%], and the tagged UPC. For the unrestricted UPC, the as≈ymmetry is roughly twice smaller than that in the tagged UPC. The results obtained for di-muon production in Pb-Pb collisions at LHC energy shown in Fig.3 are Mapping the Initial Magneticrather close to these Field at RHIC Strength energy. • The cos 4Δ� modulation caused by vacuum birefringence depends on: • Field Strength • Field geometry (photon polarization orientations)

• Larger modulation for small impact parameters • Smaller modulation for large impact parameters

⇒ STAR’s neutron “tagged” UPCFIG. Measurement 2: Estimates ofagrees the cos with 4φ asymmetry theory prediction. as the function of q⊥ for the diﬀerent centralities at √s =200GeV.Theelectron To map the field, need more measurementsand positron rapidities at various and transverse � etc. momenta are integrated over the regions [-1,1], and [0.2 GeV, 0.4 GeV]. The asymmetries in Au-Au collisions and Ru-Ru collisions are shown in the leftplotandtherightplotrespectively. 3/3/20 Daniel Brandenburg 24

III. CONCLUSIONS

We study the impact parameter dependence of the cos 4φ azimuthal asymmetry for purely electromagnetic lepton pair production in heavy ion collisions at low q⊥. This asymmetry arises from the correlation between the polarization vector of the electric field coherently generated by a fast moving heavy ion and the associated equivalent photon’s transverse momentum. Such correlation reflects the nature of the boosted Coulomb potential. We found that the azimuthal asymmetry has a strong b⊥ dependence. To be more specific, the asymmetry decreases with increasing What can we learn about final188 state, mediumL. McLerran, V. Skokov effects? / Nuclear Physics A 929 (2014) 184–190

• Idea: Extremely small � → easily deflected by relatively small perturbations vacuum • Two proposals from different groups: 1. Lorentz-Force bending due to long- lived magnetic field [1] 2. Coulomb scattering through QGP

medium [2,3] Fig. 1. Magnetic field for static mediumL. with McLerran, Ohmic V. conductivity, Skokov, σ . Nuclear Physics A 929 (2014)Ohm 184–190 [1] STAR, Phys. Rev. Lett. 121 (2018) 132301 [2] S. R. Klein, et. al, Phys. Rev. Lett. 122, (2019), 132301 The decay of the conductivity owing to expansion of the medium can only decrease the life- [3] ATLAS Phys. Rev. Lett. 121 (2018) , 212301 time of the magnetic field and thus will not be considered here. Our simulations are done for Au–Au collisions at energy √s 200 GeV and fixed impact parameter b 6fm.InFig. 1 we February 5, 2020 Daniel Brandenburg 25 show time evolution of the magnetic= field in the origin x 0asafunctionoftheelectriccon-= ⃗ = 2 ductivity σOhm.Theresultsshowthatthelifetimeofthestrongmagneticfield(eB > mπ ) is not affected by the conductivity, if one uses realistic values obtained in Ref. [5].

4. Energy dependence

In the previous section, we established that for realistic values of the conductivities the electromagnetic fields in heavy-ion collisions are almost unmodified by the presence of the medium. Thus one can safely use the magnetic field generated by the original protons only. This magnetic field can be approximated as follows 1 cZ eB(t,x 0) , (18) ⃗ = = γ t2 (2R/γ )2 + where Z is the number of protons, R is the radius of the nuclei, γ is the Lorentz factor and, finally, c is some non-important numerical coefficient. We are interested on the effect of the magnetic field on the matter, otherwise the magnetic field does not contribute to photon production. Thus we need to compute the magnetic field at the time tm,characterizingmatterformationtime. 1 On the basis of a very general argument, one would expect that tm aQs− .Hereweassumed that the Color Glass Condensate (CGC) provides an appropriate description= of the early stage of heavy ion collisions, namely Qs ΛQCD;intheCGCframework,owingtothepresenceof only one dimensional scale, the matter≪ formation time is inversely proportional to the saturation scale. We also note that if the formation time for a particle is much less than this, the magnetic field has a correspondingly larger effect, as the magnetic field is biggest at early times. The phenomenological constraints from photon azimuthal anisotropy at the top RHIC energy demand RHIC tm 2R/γRHIC,i.e.a 2RQ /γRHIC.Usingthisrelation,wecanestimatethemagnitudeof ≈ = s 10-1 Centrality: 60-80% gg®ee (Zha et al.) Long-lived Magnetic Field? gg®ee (Zha et al.) with EM pe >0.2 GeV/c T gg®ee (STARlight) e ee ⃗ ) 10-2 |h |<1, |y |<1 � = � � + �⃗ ×� -2 STAR � -3 10 (a) (b) 0.4-0.76 GeV/c2 0.76-1.2 GeV/c2

dy) ((GeV/c) 0 0.002 0.004 0.006 0.008

2 T 2 p2 ((GeV/c)Au+Au )200 GeV p2 ((GeV/c)2) T U+U 193 GeV T 70 -3 N/(dp 10 2 60 d 50 (MeV/c) ñ

Assumptions: 2 T

(c) p 40 á (d) 1. Used STARLight � Spectra 1.2-2.6 GeV/c2 10-4 30 ± 0 0.002 0.004 0.006 0.5 1 1.5 2 2.5 2. All � travers 1fm through � ≈ 2 p2 ((GeV/c)2) M (GeV/c ) T ee 10 T (�� ≈ 30 MeV/�) STAR, Phys. Rev. Lett. 121 (2018) 132301 February 5, 2020 Daniel Brandenburg 26 [1] STAR, Phys. Rev. Lett. 121 (2018) 132301 [2] S. R. Klein, et. al, Phys. Rev. Lett. 122, (2019), 132301 �� → � � : UPC vs. Peripheral [3] ATLAS Phys. Rev. Lett. 121 (2018) , 212301 Characterize difference in spectra via ⟨� ⟩ ) ) 2 -2 arXiv : 1910.12400 2 � UPC Au+Au 60-80% Au+Au 1 STAR 10 � (MeV/c) Au+Au UPC Measured 38.1 ± 0.9 50.9 ± 2.5 Au+Au 60-80% (mb/(GeV/c) dy) ((GeV/c) 2 QED 37.6 48.5 2

10-1 + - 10

)/dP � range (fm) ≈ 20 ≈ 11.5 − 13.5

gg® e e (QED)´0.88 - N/(dP e 2 Au+Au UPC + e o Leading order QED calculation of ®

Au+Au 60-80%

g �� → � � describes both spectra (±1�) g (

-2 s o Best fit for spectra in 60-80% central collisions found (60-80%) d 1 10 for QED shape plus 14 ± 4 (stat.)±4 (syst.) MeV/c broadening (UPC) d 2 0.4 < Mee < 0.76 GeV/c o Due to additional final-state effects? 10-3 10-1 o Better statistics + higher precision is needed for a 0 0.001 0.002 0.003 0.004 0.005 0.006 definitive conclusion P2 (GeV/c)2 We have not yet compared QED calculation to the new, high precision data from ATLAS ( from Quark Matter 2019) – should provide more insight

February 5, 2020 Daniel Brandenburg 27 Summary & Conclusions 1. Measurements of exclusive �� → �� process 2. Experimental demonstration that the � spectra from �� → �� depends on impact parameter (4.8� observation) 3. First earth-based observation of Vacuum Birefringence : Observed (6.7�) via angular modulations in linear polarized �� → �� process o Breit-Wheeler process in HICs (UPC and in hadronic collisions) provides a new precision tool for studying the magnetic field in Heavy-Ion Collisions o More data will help mapping the initial field strength in detail o Higher precision data needed to conclusively determine if there are medium effects

11/05/19 Daniel Brandenburg 28 Thank you

Vacuum Breit-Wheeler Birefringence pair production Predicted 1936, Predicted 1934, STAR 2019 STAR 2019

3/3/20 Daniel Brandenburg 29 Extra Slides

3/3/20 Daniel Brandenburg 30 8 8

WhyWhy isis the Breit-WheelerBreit-Wheeler process process so elusive? so elusive? Breit-Wheeler Process, why so elusive? Breit-WheelerBreit-Wheeler and and Klein-Nishina Klein-Nishina cross-sections cross-sections • 28 28 The• Breit-WheelerThe Breit-Wheeler cross-section cross-section! ! 10 10 O. Pike Breit-Wheeler Pair Production Cross Section �: ! 2 2 4 e !2 m 2 m m 2 1 ! 4 e =⇡r0 2 2m 1+ m cosh m 1 ! ! =!⇡r !2 1+ ! m cosh ! ⇣ ⌘ 0⇢ !✓ ⇣ ⌘ ◆ ⇣ !⌘ ! m ! m⇣ 2 ⌘ ⇢ m✓ 2 ⇣ ⌘ ◆ ⇣ ⌘ 1+ 1m 2 m 2 ! ! r ! e e ! ✓ ⇣ ⌘1+◆ ⇣ ⌘ 1 ! ! r ! e e ! ✓ ⇣ ⌘ ◆ ⇣ ⌘

) is, at its peak, of the same order as that of

2 ! 29 (m 10 oComptonSame scatteringpeak andcross Dirac annihilation.section! as

) is, at its peak, of the same order as that of 2 29 • However,Compton to create scattering matter from a massless and Dirac (m 10 + Compton scattering and Dirac annihilation.! e e ! state,annihilation the centre-of-mass energy must be at least• 2However,m.! to create matter from a massless + e e 2 ! •oItCross has state,previously section,the notcentre-of-mass been �possible peaks toenergy promote mustat 10 be at m oenoughCreatingleast photons 2m.! abovematter threshold from for the massless process• to be observable and Breit-Wheeler 30 It has previously not been possible to promote 10 pairstate, production remember has eluded any : direct� = �� 0 2 4 6 8 10 enough photons above threshold for the !/m detection.o center of mass energy must be � ≥ process2� to be observable and Breit-Wheeler Breit30 and Wheeler, Phys Rev 46, 1087 (1934)! 10 pair production has eluded any direct Jauch and0 Rohrlich, The2 Theory of Photons4 and Electrons6 (1959)8 10 detection. February 5, 2020 !/m Daniel Brandenburg 31 Breit and Wheeler, Phys Rev 46, 1087 (1934)! Jauch and Rohrlich, The Theory of Photons and Electrons (1959) ATLAS Measurement of �� → � � arXiv:1806.08708 Phys. Rev. Lett. 121, 212301 (2018)

s 600 α N d

d 0 - 10 % 10 - 20 % 20 - 40 % 40 - 80 %

1 ATLAS Pb+Pb data N 400 sNN = 5.02 TeV > 80% data Pb+Pb, 0.49 nb-1 STARlight + 200 data overlay

0 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015 α α α α

s 80 A N

d 0 - 10 % 10 - 20 % 20 - 40 % 40 - 80 %

• ATLAS recently measured forward � � pairs s 1

N 60 Pb+Pb data • Poor momentum resolution, better angular resolution ATLASATLAS Measurements: > 80% data sNN = 5.02 TeV 40 � − � � > 4 GeV/c � = 1 − Pb+Pb, 0.49 nb-1 � 4 < � < 45 GeVSTARlight/� + 20 data overlay • Significant broadening observed in central collisions w.r.t > 80 % data 0

0 0.05 0.1 0 0.05 0.1 0 0.05 0.1 0 0.05 0.1 A A A A February 5, 2020 Daniel Brandenburg 32 PHYSICAL REVIEW LETTERS 122, 132301 (2019)

−Su Q;mμ;r PT-broadening effects are sensitive to the electromagnetic W b ; r N γγ b ; r e ð ⊥Þ; 2 property of the quark-gluon plasma, whereas the jet ð ⊥ ⊥Þ¼ ð ⊥ ⊥Þ ðÞ

PT-broadening effects depend on the strong interaction where Su is the Sudakov factor and will be presented below. property. The experimental and theoretical investigations of By setting S 0, one gets back to the results in previous u ¼ both phenomena will deepen our understanding of the hot studies [38–41]. The factor N γγ represents the incoming medium created in these collisions. The clear indication of photon flux overlap, lepton PT-broadening effects from ATLAS and STAR [29,30] should stimulate further study on dijet azimuthal 2 2 i k1 k2 ·r correlations in heavy ion collisions. N γγ b ; r xaxb d k1 d k2 e ð ⊥þ ⊥Þ ⊥ ð ⊥ ⊥Þ¼ ⊥ ⊥ The rest of the Letter is organized as follows. We first γZ γ × fA xa;k1 fB xb;k2 b; 3 study the azimuthal angular correlation for dileptons in ½ ð ⊥Þ ð ⊥Þ ðÞ UPCs in Sec. II. Then, we investigate the medium effects, including the QED multiple scattering effects and the where xa k1=PA and xb k2=PB. To simplify the above expression,¼ we have¼ introduced an impact magnetic effects in Secs. III and IV, respectively. Finally, γ γ Sec. V summarizes the paper. parameter b -dependent photon flux: fAfB b 2 2 ⊥ ½ ¼ Lepton pair production in ultraperipheral heavy ion d b1 d b2 Θ b Nγ b1 ;k1 Nγ b2 ;k2 , where ⊥ ⊥ ð ⊥Þ ð ⊥ ⊥Þ ð ⊥ ⊥Þ collisions.—The leading order production of lepton pairs Θ b denotes the impact parameter constraints for a R ð Þ comes from photon-photon scattering, see, Fig. 1(a). The particular centrality with b⃗ b⃗ 1 − b⃗ 2 , and individual ⊥ ¼ ⊥ ⊥ outgoing leptons have momenta p1 and p2, individual photon flux Nγ b1 ;k1 can be computed separately ð ⊥ ⊥Þ transverse momenta p1 and p2 , and rapidities y1 and y2, [38–42]. Here, the interdependence between the impact respectively. The leptons⊥ are produced⊥ dominantly back to parameter bi and the photon’s transverse momentum back in the transverse plane, i.e., p⃗ p⃗ 1 p⃗ 2 ≪ contribution k⊥ is ignored, which could introduce addi- j ⊥j ¼ j ⊥ þ ⊥j i p1 ∼ p2 . The incoming photons have the tional theoretical⊥ uncertainties. j ⊥j j ⊥j following momenta: k P = s ey1 ey2 P and k 1 p A 2 The Sudakov factor Su starts to appear at one-loop order, −y1 −y2 ¼ ⊥ ð þ Þ ¼ P =ps e e PB, where P represents p1 ∼ where soft photon radiations contribute to the dominant ⊥ ð þ Þ ffiffiffi ⊥ j ⊥j p2 , and the incoming nuclei have per-nucleon momenta logarithms in the kinematics of our interest. The typical j ⊥j ffiffiffi PA and PB. In the Sudakov resummation formalism, the Feynman diagrams for the real photon radiation are shown differential cross section can be written as [37] in Figs. 1(b),1(c). Applying the Eikonal approximation, see, e.g., Ref. [43], we obtain − AB γγ →μþμ 2 dσ ½ d r ip ·r σ ⊥ e W b ; r ; 1 2 2 0 2 ⊥ ⊥ 2p1 · p2 dy1dy2d p1 d p2 ¼ 2π ð ⊥ ⊥Þ ðÞ M 1 r 2 e2 M 0 2; 4 ⊥ ⊥ Z ð Þ ðÞ soft ðÞ j ¼ p1 · ksp2 · ks j j ðÞ where b denotes the centrality at a particular impact ⊥ ¯ 0 2 2 4 where M 0 is the leading order Born amplitude and parameter of AA collisions, σ0 MðÞ =16π Q with ðÞ ¯ 2 2 2 2 2 ¼ j j ks is the soft photon momentum. In the small total trans- M0 4π αe2 t u =tu, Q is the invariant massPHYSICAL for REVIEW LETTERS 122, 132301 (2019) thej leptonj ¼ð pair,Þ t andð uþareÞ the usual Mandelstam variables verse momentum region l p ≪ P , we have the 2 2 following behavior from⊥ ¼ the⊥ above⊥ contribution: for the 2 → 2 process.Qse Inr the∼ 1. coordinate Therefore, we space need which to take into account the ⊥ 2 2 2 2 2 allows one to convenientlymultiple take scattering care of effects. the transverse α=π 1=l ln Q =l mμ , where mμ is the lepton If we compare[1] S. R. Klein, the aboveet. al, Phys. dipole Rev. Lett. to 122, theð (2019), QCDÞð 132301 dipole⊥ Þ ð ⊥ þ Þ momentum conservation, W b ; r is the combination mass and l is related to ks . In order to derive the one- [49,50], we[2] will ATLAS find Phys. the Rev. Lett. following 121 (2018) differences. , 212301 First,⊥ ⊥ of incoming photon fluxes consideredð ⊥ ⊥Þ in previous studies loop result for Su, we need to Fourier transform the above Coulomb Scattering through QGPbecause the couplings in QED and QCD are dramatically [38 42] and all order Sudakov resummation (see, e.g., expression to the conjugate r space, and add the virtual – different, this introduces a major difference for the medium ⊥ Refs. [21,22,43]), photon contributions. Because of the lepton mass mμ, the PT-broadening effects, in addition to the difference in the • Charged particles may scatter off charge Sudakovcenters effects in mentioned QGP, above. Second,cancellation the saturation between the real and virtual diagrams will depend on the relative size of μ c =r and m , where modifying primordial pair �? scales depend on the charge density. Since only quarks r 0 μ −γE ¼ ⊥ carry electric charge, the QED saturation scalec0 will2e dependwith γEFIG.the 3. Euler Medium’s constant. modifications In the to end, the acoplanarity we find distribution, on the quark density, whereas the QCD saturationat one-loop¼ scale order with[37] different, values of the effective qLˆ . depends on both quark and gluon density. Their densities are proportional to the respective degree of freedoms if we αis too2 Q2 small (few percent of the P -broadening value) to (a) (b) (c) − ln 2 ; μr >mμT; PHYSICAL REVIEW LETTERS 122, 132301 (2019)assume the ratio of the thermal distributions of quarks and 2π μr S have any observational effects. 5 21 u 2 2 m2 FIG. 1. The leading ordergluons: and2 next-to-leadingNf∶16 [51]. Here orderNf QEDis the number of active¼ α MediumQ Q effects:μ Magnetic fields.—ThereðÞ has been a ( − ln 2 ln 2 ln 2 ; μr

132301-4 Motivation From STAR and ATLAS arXiv:1806.08708 Phys. Rev. Lett. 121, 212301 (2018)

600

N a d d

ATLAS 0 - 10 % 10 - 20 % 20 - 40 % 40 - 80 % > 80 % 1 N Gaussian + 400 Data background sNN = 5.02 TeV -1 Pb+Pb, 0.49 nb Background Convolution + background 200

0 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015 a a a a a 100

§ Describe the broadening in terms of UPC curve + 90 ATLAS Gaussian + [MeV] background kick from Coulombic multiple scattering (in QGP) 80 s = 5.02 TeV RMS T NN k 70 -1 Convolution + § Fits to data: � ≈ 40 − 50 MeV Pb+Pb, 0.49 nb background 60 § No significant centrality dependence, maybe a hint in 50 last bin 40 § Very different kinematics range than from STAR 30 dielectron measurement, �-field / Coulomb multiple 20 scattering may not be mutually exclusive 10 descriptions 0 50 100 150 200 250 300 350 400 á N ñ February 5, 2020 Daniel Brandenburg part 34 Motivation From STAR and ATLAS arXiv:1806.08708 Phys. Rev. Lett. 121, 212301 (2018)

600

N a d d

ATLAS 0 - 10 % 10 - 20 % 20 - 40 % 40 - 80 % > 80 % 1 N Gaussian + 400 Data background sNN = 5.02 TeV -1 Pb+Pb, 0.49 nb Background Convolution + background 200

0 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015 0 0.005 0.01 0.015 a a a a a 100

§ Describe the broadening in terms of UPC curve + 90 ATLAS Gaussian + [MeV] background kick from Coulombic multiple scattering (in QGP) 80 s = 5.02 TeV RMS T NN k 70 -1 Convolution + § Fits to data: � ≈ 40 − 50 MeV Pb+Pb, 0.49 nb background 60 § No significant centrality dependence, maybe a hint in 50 last bin 40 New QM19 § Very different kinematics range than from STAR 30 dielectron measurement, �-field / Coulomb multiple 20 scattering may not be mutually exclusive 10 descriptions 0 50 100 150 200 250 300 350 400 á N ñ February 5, 2020 Daniel Brandenburg part 35 3 Peripheral Data 5

400 400 -2 α Pb + Pb 5.02 TeV 0 - 10%α Pb + Pb 5.02 TeV 10 - 20% α Pb + Pb 5.02 TeV40 - 80% α Pb + Pb 5.02 TeV > 80% d d d d dN dN dN 600 dN 600

350 2 350 2

1 4 < M < 45 GeV/c 1 1 4 < M < 45 GeV/c 1 −1 N µµ data N N µµ data N 10 Au + Au 200 GeV p Au > 4 +GeV/c Au | η200| < 2.4 GeV Au + Au500 200p >GeV 4 GeV/c |η | < 2.4 500 data 300 T,µ µ 300data T,µ µ data 10−2 Centrality: 60 - 80% STARLight 250 Centrality: 60 - 80%QED 250STARLight Centrality:400 60 - 80% STARLightQED 400 gEPA 1 200 200gEPA 1 gEPA 1

dy (GeV/c) gEPA 2 300 gEPA300 2 T 2 gEPA 2 150 150gEPA 2 gEPA 2 QED 100 100QED 200 QED 200 N/dP

2 50 50 100 100

d −2 −3 10 0 0 10 0 0 0 0.005 0.01 0.005 0.01 0.015 0 0.005 0.01 0.005 0.01 0.015 p > 0.2 GeV/c p > 0.2 GeV/c p > 0.2 GeV/c T,e T,e α α T,e α α |y | < 1; |η | < 1 |y | < 1; |η | < 1 |y | < 1; |η | < 1 ee e 10−3 ee e ee e 2 FIG. 3. The distributions of the2 broadening variable, ↵, from the generalized2 EPA approach (gEPA2, dash blue lines) and 0.4 < Mee < 0.76 (GeV/c ) • 0.76Peripheral < Mee < 1.2 (GeV/c data) from both STAR1.2 < Mee < 2.6 (GeV/c ) QED (solid red line) for muon pairs in Pb+Pb collisions at psNN = 5.02 TeV for di↵erent centrality classes. The results are

10−3 and ATLAS are well described

0 0.002 0.004 0.006 ﬁltered0 with the0.002 ﬁducial cuts0.004 described in the0.006 text and0 normalized0.002 to unity to facilitate0.004 a direct0.006 comparison with experimental

T T 2T 2 data. Theby measurements QED calculation from ATLAS [25] are also plotted for comparison. P (GeV/c) P2 (GeV/c)2 P2 (GeV/c)2 • → No need for final state 2 ⌫ FIG. 1. The P distributions of electron-positronlution.effects? It pair should production be noted within that ⇡↵ the STARP /M acceptancell in a de- for theliding mass nucleiregions maintain 0.4 0.76 their velocities (a (ki ui⌫ ) ? 2 ' ? (left panel), 0.76 1.2(middlepanel),and1tector.2 setup2.6GeV where/c the(right sagitta panel) of a in particle 60 80% trajectory Au+Au is collisionsfunction) at psNN to=200GeV. simplify the calculation. In central TheFebruary STAR 5, measurements2020 [24] and calculationsDanielmuch Brandenburg from larger gEPA1, than the gEPA2, e↵ect of QED multiple and STARLight scattering in [15] the36 are alsocollisions, plotted for where comparison. the photon flux are generated See text for details of the models. detector material and from resolution of the experimen- predominantly by the participant nucleons, charge tal measurements, as is the case for the STAR Detector stopping may be an important correction to the within the measured kinematic range. The measured ↵ initial electromagnetic fields. distributions show broadening in hadronic Pb+Pb colli- and the subsequent result reads: sions with respect to UPCs.lowest Figure order 3 shows is the ↵ dis- omission of higher order contribution and multiple • pair production: tributions from our calculations in Pb+Pb6 collisions at d P (~q) 4 4 We1 have ignored2 higher-order corrections in both psNN = 5.02 TeV for di↵erent centrality classes. The re- 4 4 2 2 2 2 3 3 =(Z↵) 2 6the initial electromagneticd q1 field [10] and Sudakov Z e dw1 dw2 d k1 d k2 sultsF ( arek1 filtered) with the fiduciald cuts:p+dpTµp > 4 GeV/c, (2⇡) 2✏+2✏ = 16 ? ? e↵ect [33], Z which should be quite small in the low 2 2 2 and ⌘µ2 < 2.4, and normalized to facilitate a direct com- 1 (4⇡) w1 w2 (2⇡) (2⇡) |k1| F (N0)F (N1)F (N3)F (N4)[NP0Nand1N3 smallN4]↵ range. It has been pointed out that Z parison with experimental(6) data from ATLAS. The mea- ? 2 2 1 there may be significant multiple pair production F ( k2) 2 2 surements from ATLAS [25] can be wellTr described(p + m by)[N the u/ (p q + m)u/ + k k (w ,w ) / 2D 1 / in/ the1 same event2 [36],(8) which may complicate the 2 1 2 gEPA2 1 and2 QED calculations within⇥ uncertainties.{ ⇥ k2 ? ? 1 calculation1 and measurement. There have been proposals in theN literatureu/ (q regardingp + m)u/ ](p m)[N u/ 2X 2 /1 /+ 1 /+ 5D 2 possible final-state e↵ects to explain the P broadening. final-state e↵ects of magnetic field deflection and ? •1 Two such proposals are that the(p/ broadening/q /q is+ duem) tou/ + N5X umultiple/ (/q + Coulomb/q p/ scattering: where (w1,w2) is the cross-section averaged over the 1 1 1 1 + deflection by the residual magnetic field trapped in an The STAR and ATLAS collaborations have demon- scalar and pseudoscalar polarization. This is exactly the + m)u/ ] , electrically conducting QGP [24, 37] or due2 } to multiple strated that it is possible to identify and measure EPA expression commonly used in the literatureCoulomb and scattering used in the hot and dense medium [25, the Breit-Wheeler process accompanying the cre- in comparison to recent experiments [6].33]. The All the spectral proposed mechanismswith including this study ation of QGP. This opens new opportunity using shape [15, 33], which is insensitive to therequire collision extraordinarily cen- strong electromagnetic2 fields, an this2 process as a probe of emerging QCD phenom- interdisciplinary subject of intenseN0 = interestq1,N1 across= many[q1 (p+ + p )] , trality, is the result of integrating over the whole impact ena [8]. scientific communities. There are a few assumptions2 and 2 parameter space as shown in Eq. 31 to Eq. 32 [9] and N3 = (q1 + q) ,N4 = [qIn+( summary,q1 p+ wep study)] , the impact-parameter depen- caveats in our calculation which deserve further studies: 2 2 dence of the Breit-Wheeler process in heavy-ion collisions subsequently inserting an impact-parameter dependent N2D = (q1 p ) + m , continuous charge distribution without point-like within the framework of the(9) external QED field and the photon flux (w1,w2,b), as shown in Eq.• 36 to 43 in 2 2 structure: N2X = (q1 p+) + m , approximations used to arrive at the Equivalent Photon Ref. [9]. Approximation, and with a full QED calculation based on It has been shown [38, 39] that the substructures of2 2 N5D = (q1 + q p ) + mtwo, lowest-order Feynman diagrams. We further demon- We have also performed a QED calculationprotons at leading- and quarks in nuclei and their fluctuations can significantly alter the electromagnetic field in-2 strate2 that the P spectrum from the STARLight model order based on Ref [30, 31] and extended its original cal- N = (q + q p ) + m , ? 5X 1 + calculation used by the recent comparisons as a baseline culation to all impact parameters as a functionside the of nucleus the at any given instant. This should result in an observable e↵ect deserving further the- results from averaging over the whole impact parameter transverse momentum of the produced pair. The lowest- where p+ and p are the momenta of the created lep- oretical and experimental investigation. The e↵ect space and is therefore by definition independent of impact order two-photon interaction is a second-order process tons, the longitudinal componentsparameter. of Ourq1 are model given results by can qualitatively describe is most prominent in central1 collisions where the with two Feynman diagrams contributing, asATLAS shown results in haveq10 large= uncertainties[(✏+ + ✏ )+ and where(p+z + pbothz)], theq1Pz =broadeningq10/, ✏+ observedand at RHIC as well as the 2 ? Fig. 2 of Ref. [30, 31]. Similarly, the straight-lineSTAR approx- currently lacks✏ theare necessary the energies statistics of the for a producedacoplanarity leptons, broadening and m observedis the at the LHC. It provides measurement. mass of lepton. In the calculationa practical of P (~q procedure), the traces for studying and the Breit-Wheeler pro- imation for the incoming projectile and target nuclei is cess with ultra-strong electromagnetic fields in a control- applied as in the case of all EPA calculations.projectile Otherwise, and targetmatrices nuclei maintain have been the same handled ve- bylable the fashion. Mathematica This outcome package indicates that the broaden- a full QED calculation of the di↵erential• cross-sectionlocity vector beforeFeynCalc and after collision: [34]. The multi-dimensionaling originates integration predominantly is per- from the initial electromag- with two photons colliding to create two leptonsThe has very been first assumptionformed in with Eq. 1 is the that Monte both col- Carlonetic (MC) field integration strength that routine varies significantly with impact calculated. Following the derivation of Ref. [30, 31], the VEGAS [35]. cross section for pair production of leptons is given by The gEPA1 and QED calculations are shown in Fig. 1 as dash-dotted and solid lines, respectively, together with experimental data points and the STARLight calcula- 6 ~ 6 2 d P (b) 2 d P (~q) 2 iq~ ~b tions. It is clear that there is a di↵erence between the = d b 3 3 = d q 3 3 d be · , (7) d p+d p d p+d p gEPA1 and the QED calculations. The most striking Z Z Z di↵erence is in the P spectral shape. The QED curves describe the spectra quite? well with a smooth distribution d6P (q~) and the di↵erential probability 3 3 in QED at the d p+d p of the cross-section increasing from high to low P ,but ? Photon virtuality and differential cross section Impact Parameter Dependence B gEPA 1 gEPA 2 QED Data Mass Range 5 STAR 80 2 Au+Au UPC 0.4 < Mee < 0.76 GeV/c -

(MeV/c) +

ñ 2 gg® e e (QED) 0.76 < Mee < 1.2 GeV/c 4 2 70 + -

P 2 gg® e e (STARLight) á 1.2 < Mee < 2.6 GeV/c 3 (mb/(GeV/c))

60 ) Au+Au 200 GeV - e p > 0.2 GeV/c + T,e 2 e

50 |y | < 1; |h | < 1 ee e ®

dP g 1

g 2 0.4 < Mee < 0.76 GeV/c 40 ( s

d 0 0 0.02 0.04 0.06 0.08 0.1 30 0 10 20 30 40 P (GeV/c) b (fm) QED (and gEPA parameterization) describe data Note: gEPA1 vs. gEPA2 : gEPA2 includes Larger ⟨�⟩ from impact parameter dependence no phase term to approximate full QED result evidence for significant photon virtuality

oStill only models, can we experimentally investigate impact parameter dependence : →Compare UPC vs. same process in peripheral collisions

February 5, 2020 Daniel Brandenburg 37 Application : Mapping the Magnetic Field EPA Two Photon Distribution The colliding photons in the �� → �� process originate from the Lorentz- Two-photon density (arb. contracted electromagnetic fields norm.) photon density is related to energy Uniformly Charged flux of the electromagnetic fields Sphere �� ion, � = 6.38 fm � ∝ �⃗ = �×� � = 19 fm

For highly Lorentz-contracted fields � ≈ � with � ⊥ � and �⃗ ∝ � ≈ �

Equivalent Photon Approximation, photon density (single ion): 1 1 4� � � � � � + � ⁄� ⋅ � �; � = � �, � = � �, � = � � �� 2� � + � ⁄�

[1] M. Vidovic ,́ et al., Phys. Rev. C 47, 2308 (1993). [2] C. F. v. Weizsa cker̈ , Z. Phys. 88, 612 (1934).

11/05/19 Daniel Brandenburg 38 Example : Light-by-Light Scattering

50 ATLAS Observed Light-by-Light ATLAS 45 Scattering in UPCs: Pb+Pb s = 5.02 TeV 40 NN • Purely quantum

Events / 0.005 35 Data 2018, 1.7 nb-1 mechanical process 30 Signal (γγ → γγ) CEP gg → γγ (�) 25 γγ → ee • Light-by-Light 20 Sys. unc. scattering involves real 15 photons by definition 10 ATLAS, Nature Physics 13 (2017), 852 5 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Aφ

Figure11/05/19 2: The diphoton A distributionDaniel Brandenburg 39 for events satisfying the signal selection, but before the A < 0.01 requirement. Data are shown as points, while the histograms represent the expected signal and background levels.

18 ATLAS 25 ATLAS 16 Pb+Pb sNN = 5.02 TeV Pb+Pb sNN = 5.02 TeV 14 20 Events / GeV 12 Data 2018, 1.7 nb-1 Data 2018, 1.7 nb-1 Signal (γγ → γγ) Events / 0.2 GeV Signal (γγ → γγ) 10 CEP gg → γγ 15 CEP gg → γγ γγ → ee γγ → ee 8 Sys. unc. Sys. unc. 10 6

4 5 2

0 0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 m [GeV] pγγ [GeV] γγ T (a) (b)

Figure 3: Kinematic distributions for event candidates: (a) ! diphoton invariant mass, (b) diphoton transverse momentum. Data (points) are compared with the sum of signal and background expectations (histograms). Systematic uncertainties of the signal and background processes, excluding that of the luminosity, are shown as shaded bands. and resolution eects. The C factor is defined as the ratio of the number of selected MC signal events passing the selection and after applying data/MC correction factors to the number of generated MC signal events satisfying the fiducial requirements. It is found to be C = 0.350 0.024. The uncertainty in C ± is estimated by varying the data/MC correction factors within their uncertainties, as well as using an alternative signal MC sample based on calculations from Ref. [29]. The overall uncertainty is dominated by uncertainties in the photon reconstruction eciency (4%) and the trigger eciency (2%). The measured fiducial cross section is 78 13 (stat.) 7 (syst.) 3 (lumi.) nb, which can be compared ± ± ± with the predicted values of 51 5 nb from Ref. [29] and 50 5 nb from SuperChic3 MC simulation [28]. ± ± The experiment-to-prediction ratios are 1.53 0.33 and 1.56 0.33, respectively. ± ± In summary, this Letter reports the observation of light-by-light scattering in quasi-real photon interactions from ultraperipheral Pb+Pb collisions at psNN = 5.02 TeV recorded in 2018 by the ATLAS experiment. After applying all selection criteria, 59 data events are observed in the signal region, while 12 3 background ±

7 COLLI SION OF TWOBreitLI GHT-WheelerQUANTA Process,i09i why so elusive? for the transformation of momentum and energy. oThisAlreadyis Dirac's in 1934recombination Breit andformula Wheelerwith knew it was hard, maybe impossible? Dirac's The collision cross section for pair production DECEMn =coshHER 20.15, One1934could also deriveP H (21.YS I3)CAI REVI EW VOI-U M E over all angles depends only on t'o'devi and from Dirac's recombination formula and the one may thus use Eqs. (21) for the calculation relations (17'), (18'). The other Collisionformulas (20),of Two Light Quanta of these cross sections for any angle between (21) require, however, the more detailed calcu- ** light quanta by applying lations, the resultsG. BREITof whichAND JQHNwereA. WHEELER,reported Department of Physics, ¹m York University above. (Received October 23, 1934) o' & = sin' o v (rp/2) ((vi'vq') sin (y/2)) (23') been As Thehasrecombinationreportedof free electronsat the andWashingtonfree positrons polarizations, and formulas are given for the angular dis- production due to collisions of The polarizations of the light quanta are changed meeting,and its pairconnection with the Compton effect have been tribution of photons due to recombination. The results are cosmictreatedraysby Diracwith beforethe thetemperatureexperimental radiationdiscovery of ofthe applied to the collision of high energy photons of cosmic by the Lorentz transformation and thus only Gregory Breit interstellarpositron. Inspacethe presentis muchnotetooare smallgiven analogousto be ofcalcula-any radiation with the temperature radiation of interstellar (21.3) has in general a simple meaning. ' tions for the production of positron electron pairs as a space. The effect on the absorption of such. quanta is found interest. We do not give the explicit calculations, Using Eq. (17'), comparing it with Eq. (18') result of the collision of two light quanta. The angular to be negligibly small. sincedistributionthe resultof isthedueejectedto thepairsordersis calculatedof magnitudefor diff'erent and Eq. (21.3), one obtains the probability of rather than exact relations. It is also hopeless to recombination per unit volume per second as try to observe the pair formation in laboratory c~.ov(o/2) (~P) experiments~WO simultaneouslywith two beamsactingof x-rayslight wavesor p-rayswith meeting eachvectorotherpotentialson account of the smallness —exp ( 2tbw—/h) I2/(Bw)'. (2) in the frame of zero momentum in terms of the of cr and the insufficiently large available densities A;=a;* exp —(~;l —k;r) I John Wheeler electron and, positron densities p., p„. Trans- of quanta. In {the considerations of Williams, Here 8 is a dimensionless number depending on forming frame in which the electrons are to a however, the large nuclear+a;electricexp {2(cv;tfields—k,leadr) } to(1) initial momenta and spin and the polarizations at rest one has large densities of quanta in moving frames of of the quanta. 6$' is the difference in energy of February 5, 2020 Daniel Brandenburg 40 reference.are consideredThis, togetheras actingwithon thean electron.large numberUnder the initial and the final states. Thus if S' of nucleiithe inliuenceavailableof thein wavesunit volumea single electronof ordinarywave = —IWI is the energy of the electron in its materials,function increasesP'2) is changed,the effectand totheobservableperturbed initial state and if hv1, kv2 are the energies of the amounts.functionAnalyzingmay be expandedthe field ofaccordingthe nucleusto powersinto quanta, then As for pair productions the number of recom- of a, u*. The phenomena of pair production and quanta by a procedure similar to that of v. 8W =c(P22+m2c2) 2+ W1 —h1 1 —hv2, of recombination have to do with the terms in binations per unit volume per unit time is Weizsacker, 4 he finds that if one quantum hv a&*a2* and. u1c&, respectively, as is obvious frorr1 W1 = — Lorentz invariant and thus in E' (system where per cm' is incident on a nucleus of charge Ze then W, (3) the theory of quantization of light waves. We electron is at rest) this number per unit electron the number of pairs produced is' where P2 Pl+ Pl+ P2 and positron density is consider first the pair production. We let the function $(0) represent an electron in a negative is the final momentum of the electron and energy state. It is convenient for practical P1, P& are the momenta of the quanta. The total applications to normalize P"& so as to have the electron density due to the two light quanta is ' Two light waves polarized parallel or perpendicular to whereelectrona(C) isdensitygiven equalby Eq.to(21.unity.3), hv=mc'$It is alsoandun- obtained by summing expression (2) over all each other retain their relative polarization when viewed =2m.necessary to use quantized light waves in the possible states of negative energy. The equal and from another frame of reference if they travel in the same a e'/hc. The evaluation of the integral shows production since the results with opposite spin directions for every p& contribute direction. If, however, they travel in opposite directions tha, tpairo(P) is in asymptoticproblem, agreement with the the relative polarization is in general changed. On the quantized waves are known to be identical with to the density. One is thus only interested in other an unpolarized beam remains corresponding formula of Heitler and Sauter' hand, unpolarized those obtained means of ordinary waves. the average for 8 over the different directions 0. when viewed from any frame of reference. Thus Eq. (21.3) for high P. by in conjunction with (23') always applies to the collision of 'As a result of the calculation one finds that at of the positron spin. This average will be called a quantum with quanta having random polarizations. time t after the of waves 8'. There are 2P1'dP1d~1 U/h' electronic states For colliding head-on relative polarizations 4 a application the the quanta the C. F. v. Weizsacker, Zeits. f. Physik 88, 612 (1934). of negative energy in the fundamental volume V are the same as in the frame of zero momentum, and for ' Wewaveare veryfunctionmuch containsindebted toa Dr.termE.whichJ, Williamsmay forbe such quanta Eq. (23') with cr as given by Eqs. (21.1), (21.2) permissioninterpretedto quote ashis referringresults. to an electron in a for which the momentum is p1 and the direction may be applied directly to. the calculation of collisions 'W. Heitler and F. Sauter, Nature 132, 892 (1933). is within the solid. angle dco1. Each of (hese has between quanta polarized parallel or perpendicular to Cf. alsopositiveJ. R. Oppenheimerenergy stateandwithM. S.a Plesset,momentumPhys. andRev.a a density U. The number of positron electron each other whether the total momentum is zero or not. 44, 53spin(1933).coordinate which are functions of the 1/ pairs produced per cm' corresponding to the original momentum and spin and of the momenta absolute value of positron momentum being and polarizations of the light quanta. The between and in the direction — and density of electrons corresponding to this wave p1 p1+dp1 P1 in solid angle ds&1 is thus obtained from (2) by function may be put into the form multiplying it by 2P1'dp1dco1/h'. Integrating over * Now at Department of Physics, University of Wis- dpi, and making use of consin. ** ~ National Research Fellow now at Copenhagen. d(5W) I Pl/Wl+P1P2/P1W2jdp1 (3) 1087 COLLI SION OF TWO LI GHT QUANTA i09i

for the transformation of momentum and energy. This is Dirac's recombination formula with The collision cross section for pair production Dirac's n =cosh 20. One could also derive (21.3) over all angles depends only on t'o'devi and from Dirac's recombination formula and the one may thus use Eqs. (21) for the calculation relations (17'), (18'). The other formulas (20), of these cross sections for any angle between (21) require, however, the more detailed calcu- COLLI SION OF TWO LIlightGHTquantaQUANTAby applying i09i lations, the results of which were reported Breit-Wheeler Process, whyabove. so elusive? o' & = sin' o v (rp/2) ((vi'vq') sin (y/2)) (23') As has been reported at the Washington for the transformation of momentum and energy. oThisAlreadyis Dirac's in 1934recombination Breit andformula Wheelerwith knew it was hard, maybe impossible? meeting, pair production due to collisions of The collision cross section for pair production Dirac'sThenpolarizations=cosh 20. Oneof thecouldlight quantaalso deriveare changed DECEM HER 15, 1934 P H (21.YS I3)CAI cosmicREVI raysEW with the temperature radiationVOI-U MofE t'o'devi by the Lorentz transformation and thus only over all angles depends only on and from Dirac's recombination formula and the interstellar space is much too small to be of any (21.3) has in general a simple meaning. ' one may thus use Eqs. (21) for the calculation relations (17'), (18').V. WeizsäckerThe other formulas (20), interest. We do not give the explicit calculations, Using Eq. (17'), comparing itCollisionwith Eq. of(18')Two Light Quanta of these cross sections for any angle between (21) require, however, the more detailed calcu- since the result is due to the orders of magnitude and Eq. (21.3), one obtains the probability **of G. BREIT AND JQHN A. WHEELER, Departmentrather thanof Physics,exact ¹relations.m York UniversityIt is also hopeless to light quanta by applying lations,recombinationthe resultsper unitof whichvolume perweresecondreportedas above. (Received Octobertry to23,observe1934) the pair formation in laboratory & o' = sin' o ((vi'vq') sin v (23') (rp/2) (y/2)) As has been reportedc~.ov(o/2) at(~P)the Washington experimentsOr maybewith two notbeams impossible!of x-rays or p-rays The recombination of free electrons and free positrons meetingopolarizations,each andotherformulason accountare givenof thefor thesmallnessangular dis- meeting,and its pairconnectionproductionwith the dueComptonto effectcollisionshave beenof tribution of photons due to recombination. The results are The polarizations of the light quanta are changed in the frame of zero momentum in terms of the of cr and the insufficiently large available densities cosmictreatedraysby Diracwith beforethe thetemperatureexperimental radiationdiscovery of ofthe applied to the collision of high energy photons of cosmic electron and, positron densities p., Trans- by the Lorentz transformation and thus only positron. In the present note are given analogousp„. calcula- ofradiationquanta. withIn thethe temperatureconsiderationsradiationof Williams,of interstellar Gregory Breit interstellarforming spacea frameis muchin toowhichsmallthe toelectronsbe of anyare (21.3) has in general a simple meaning. ' tions for theto production of positron electron pairs as a however,space. Thetheeffectlargeon thenuclearabsorptionelectricof such.fieldsquantalead istofound interest.at restWeonedohasnot give the explicit calculations, Using Eq. (17'), comparing it with Eq. (18') result of the collision of two light quanta. The angular largeto bedensitiesnegligibly ofsmall.quanta in moving frames of sincedistributionthe resultof isthedueejectedto thepairsordersis calculatedof magnitudefor diff'erent with number and Eq. (21.3), one obtains the probability of reference. This, together the large rather than exact relations. It is also hopeless to of nucleii available in unit volume of ordinary recombination per unit volume per second as try to observe the pair formation in laboratory materials, increases the effect to observable field of the nucleus into c~.ov(o/2) experiments~WO simultaneouslywith two beamsactingof x-rayslight wavesor p-rayswith amounts. Analyzing the (~P) As for pair productions the number of recom- procedure similar of v. meeting eachvectorotherpotentialson account of the smallness quanta by a to that binations per unit volumeE. J. Williamsper unit time is Weizsacker, 4 he finds—expthat( 2tbw—if one/h)quantumI2/(Bw)'.hv (2) in the frame of zero momentum in terms of the of cr and the insufficiently large available densities Lorentz invariant— and— thus in E' (system where per cm' is incident on a nucleus of charge Ze then electron and, positron densities Trans- A;=a;* exp { (~;l k;r) I E. J. Williams Phys. Rev. 45, 729 (1934) John Wheeler p., p„. of quanta.electron Inis attherest)considerationsthis number perofunitWilliams,electron theHerenumber8 is ofa dimensionlesspairs produced numberis' depending on forming frame in which the electrons are K. F. Weizsacker, Z. Physik , 612 (1934) to a however,and positronthe largedensitynuclearis+a;electricexp {2(cv;tfields—k,leadr) } to(1) initial momenta and spin and the polarizations at rest one has large densities of quanta in moving frames of of the quanta. 6$' is the difference in energy of February 5, 2020 Daniel Brandenburg 41 reference.are consideredThis, togetheras actingwithon thean electron.large numberUnder the initial and the final states. Thus if S' the inliuence of the waves a single electron wave = — the of nucleii' available in unit volume of ordinary IWI is energy of the electron in its Two lightP'2)waves polarized parallel or perpendicular to where a(C) is given by Eq. (21.3), hv=mc'$ and functioneach other retain istheirchanged,relative polarizationand thewhenperturbedviewed initial state and if hv1, kv2 are the energies of the materials, increases the effect to observable =2m.e'/hc. The evaluation of the shows functionfrom anothermayframebe ofexpandedreference if accordingthey travel intothepowerssame a quanta, then integral amounts.direction.AnalyzingIf, however,thetheyfieldtravelof intheoppositenucleusdirectionsinto tha, t o(P) is in asymptotic agreement with the As for pair productions the number of recom- ofthea,relativeu*. Thepolarizationphenomenais in ofgeneralpair productionchanged. On theand quanta by a procedure similar to that of v. corresponding8W =c(P22+m2c2)formula2+ W1of —Heitlerh1 1 —hv2,and Sauter' ofotherrecombinationhand, an unpolarizedhave tobeamdo withremainstheunpolarizedterms in binations per unit volume per unit time is Weizsacker,when viewed4 hefromfindsany framethatof reference.if one quantumThus Eq. (21.hv3) for high P. E' a&*a2*in conjunctionand. u1c&,with respectively,(23') always appliesas isto obviousthe collisionfrorr1of W1 = —W, (3) Lorentz invariant and thus in (system where per cm'a quantumis incidentwith quantaon a havingnucleusrandomof chargepolarizations.Ze then theFortheoryquanta ofcollidingquantizationhead-on theofrelativelight polarizationswaves. We 4 C. F. v. Weizsacker, Zeits. f. Physik 88, 612 (1934). electron is at rest) this number per unit electron the number of pairs produced is' where P2 Pl+ Pl+ P2 considerare the samefirstas inthethe pairframe production.of zero momentum,We andlet forthe ' We are very much indebted to Dr. E. J, Williams for and positron density is such quanta Eq. (23') with cr as given by Eqs. (21.1), (21.2) permission to quote his results. functionmay be applied$(0) representdirectly to.antheelectroncalculationin ofa collisionsnegative is'W.theHeitlerfinaland momentumF. Sauter, Natureof the132, electron892 (1933).and between quanta polarized parallel or perpendicular to Cf. also J. R. Oppenheimer and M. S. Plesset, Phys. Rev. energyeach otherstate.whether Itthe istotalconvenientmomentum is forzero orpracticalnot. 44,P1,53 P&(1933).are the momenta of the quanta. The total applications to normalize P"& so as to have the electron density due to the two light quanta is ' Two light waves polarized parallel or perpendicular to whereelectrona(C) isdensitygiven equalby Eq.to(21.unity.3), hv=mc'$It is alsoandun- obtained by summing expression (2) over all each other retain their relative polarization when viewed =2m.necessary to use quantized light waves in the possible states of negative energy. The equal and from another frame of reference if they travel in the same a e'/hc. The evaluation of the integral shows production since the results with opposite spin directions for every p& contribute direction. If, however, they travel in opposite directions tha, tpairo(P) is in asymptoticproblem, agreement with the the relative polarization is in general changed. On the quantized waves are known to be identical with to the density. One is thus only interested in other an unpolarized beam remains corresponding formula of Heitler and Sauter' hand, unpolarized those obtained means of ordinary waves. the average for 8 over the different directions 0. when viewed from any frame of reference. Thus Eq. (21.3) for high P. by in conjunction with (23') always applies to the collision of 'As a result of the calculation one finds that at of the positron spin. This average will be called a quantum with quanta having random polarizations. time t after the of waves 8'. There are 2P1'dP1d~1 U/h' electronic states For colliding head-on relative polarizations 4 a application the the quanta the C. F. v. Weizsacker, Zeits. f. Physik 88, 612 (1934). of negative energy in the fundamental volume V are the same as in the frame of zero momentum, and for ' Wewaveare veryfunctionmuch containsindebted toa Dr.termE.whichJ, Williamsmay forbe such quanta Eq. (23') with cr as given by Eqs. (21.1), (21.2) permissioninterpretedto quote ashis referringresults. to an electron in a for which the momentum is p1 and the direction may be applied directly to. the calculation of collisions 'W. Heitler and F. Sauter, Nature 132, 892 (1933). is within the solid. angle dco1. Each of (hese has between quanta polarized parallel or perpendicular to Cf. alsopositiveJ. R. Oppenheimerenergy stateandwithM. S.a Plesset,momentumPhys. andRev.a a density U. The number of positron electron each other whether the total momentum is zero or not. 44, 53spin(1933).coordinate which are functions of the 1/ pairs produced per cm' corresponding to the original momentum and spin and of the momenta absolute value of positron momentum being and polarizations of the light quanta. The between and in the direction — and density of electrons corresponding to this wave p1 p1+dp1 P1 in solid angle ds&1 is thus obtained from (2) by function may be put into the form multiplying it by 2P1'dp1dco1/h'. Integrating over * Now at Department of Physics, University of Wis- dpi, and making use of consin. ** ~ National Research Fellow now at Copenhagen. d(5W) I Pl/Wl+P1P2/P1W2jdp1 (3) 1087