肖博文副教授2023年3月17日报告.pdf

Introduction NLO and Threshold Resummation Summary and Outlook Precision Study of Gluon Saturation: Experimental Analysis versus Theoretical Approach Bo-Wen Xiao School of Science and Engineering, CUHK-Shenzhen G. A. Chirilli, Bo-Wen Xiao, Feng Yuan, Y. Shi, L. Wang, S.Y. Wei, Bo-Wen Xiao, Phys. Rev. Lett. 108, 122301 (2012). Phys. Rev. Lett. 128, 202302 (2022). 1 / 28 Introduction NLO and Threshold Resummation Summary and Outlook QCD and Gauge Symmetry Trinity of Color: QCD Lagrangian : L = iψ̄γ µ Dµ ψ − mψ̄ψ − 1 µν [F Fµν ] 4 a = ∂ Aa − ∂ Aa − gf b c with Dµ ≡ ∂µ + igAaµ ta and Fµν µ ν ν µ abc Aµ Aν . = + Gauge symmetry is w.r.t. the local phase of charge particles. The same physics (phase) can be described using different orientations of the arrows (phases of the quark wavefunction) with a compensating gauge field. Non-Abelian gauge field theory. Invariant under SU(3) gauge transformation. 2 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Kinoshita-Lee-Nauenberg Theorem = = KLN theorem: In a theory with massless fields, transition rates are free of the infrared divergence (soft and collinear) if the summation over initial and final degenerate states is carried out. The KLN theorem: infrared divergences appear because some of states are physically “degenerate”, but we treat them as different. A state with a quark accompanied by a collinear gluon is degenerate with a state with a single quark. A state with a soft gluon is almost degenerate with a state with no gluon (virtual). 3 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Infrared Safety Two kinds of IR divergences: collinear and soft divergences. For a suitable defined inclusive observable (e.g., σe+ e− →hadrons ), there is a cancellation between the soft and collinear singularities occurring in the real and virtual contributions. Kinoshita-Lee-Nauenberg theorem Any new observables must have a definition which does not distinguish between parton ↔ parton + soft gluon parton ↔ two collinear partons Observables that respect the above constraint are called infrared safe observables. Infrared safety is a requirement that the observable is calculable in pQCD. 4 / 28 Introduction NLO and Threshold Resummation Summary and Outlook e+ e− annihilation Phase space for real emission Because ofp1 momentum conq q servation, q, q̄ and g lie in a γ ∗ plane. p3 γ∗ g p3 q q̄ _ q x2 q̄ = p2 singularities p1 p2 g Useful variables are the energy fractions and invariant masses x1  2  P= 2E Γ[2−] √2 , 4πµ =2 s Born diagram ( ) gives σ0 = αem sNc q eq xs + x + xΓ[2−2] 2p .p √ =1−x = y √ is NLO: real contribution (3 body final state). xi ≡ 2E s Q with Q = √ xi ij dσ3 dx1 dx2 = CF i i 1 j 2 3 k Jets in e+ e− Annihilation – p.6 x12 + x22 αs σ0 2π (1 − x1 )(1 − x2 ) 5 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Dimensional Regularization Dimensional regularization: Analytically continue in the number of dimensions from d = 4 to d = 4 − 2. Convert the soft and collinear divergence into poles in . To keep gs dimensionless, substitue gs → gs µ with renormalization scale µ. At the end of the day, one finds  2 −   Q Γ[1 − ] 2 αs (µ) 3 19 2π 2 σr = σ0 CF + + − 2π 4πµ2 Γ[1 − 2] 2  2 3  2 −   Q Γ[1 − ] αs (µ) 2 3 2π 2 σv = σ0 CF − 2 − −8+ 2π 4πµ2 Γ[1 − 2]   3   and the sum lim→0 σ = σ0 1 + 34 CF αsπ(µ) + O(αs2 ) . (Almost) Complete Cancellation between real and virtual. For more exclusive observables, the cancellation is not always complete. 6 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Factorization and NLO Calculation Factorization is about separation of short distant physics (perturbatively calculable hard factor) from large distant physics (Non perturbative). σ ∼ xf (x) ⊗ H ⊗ Dh (z) ⊗ F(k⊥ ) NLO (1-loop) calculation always contains various kinds of divergences. Some divergences can be absorbed into the corresponding evolution equations. Renormalization: cutting off infinities and hiding the ignorance. The rest of divergences should be canceled. Hard factor αs (1) H + ··· 2π NLO should always be finite and free of divergence of any kind. (0) H = HLO + 7 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Large Logarithms NLO vs NLL Naive αs expansion sometimes is not sufficient! LO NLO NNLO · · · LL 1 αs L (αs L)2 ··· NLL αs αs (αs L) · · · ··· ··· ··· Evolution → Resummation of large logs. LO evolution resums LL; NLO ⇒ NLL. 8 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Saturation Physics (Color Glass Condensate) 色玻璃凝聚理论: 描述高能极限下强子内部高密度胶子的涌现性质 Gluon density grows rapidly as x gets small. BFKL evolution! Resummation of the αs ln 1x ⇒ BFKL equation. Hard at NLO! (20 years) Many gluons with fixed size packed in a confined hadron, gluons overlap and recombine ⇒ Non-linear QCD dynamics (BK/JIMWLK) ⇒ ultra-dense gluonic matter Saturation = Multiple Scattering (MV model) + Small-x (high energy) evolution 9 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Ultimate Questions and Challenges in QCD To understand our physical world, we have to understand QCD! Three pillars of EIC Physics: How does the spin of proton arise? (Spin puzzle) What are the emergent properties of dense gluon system? How does proton mass arise? Mass gap: million dollar question. EICs: keys to unlocking these mysteries! Many opportunities will be in front of us! 10 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Dual Descriptions of Deep Inelastic Scattering 深度非弹性散射的双重描述: ... Bjorken frame Dipole frame Bjorken: partonic picture is manifest. Saturation shows up as limit of number density. Dipole: the partonic picture is no longer manifest. Saturation appears as the unitarity limit for scattering. Convenient to resum the multiple gluon interactions. F2 (x, Q2 ) = X f e2f Q2 S⊥ 4π 2 αem Z 1 0 dz Z 2 d2 r⊥ |ψ (z, r⊥ , Q)| i h 1 − S(2) (Qs r⊥ ) 11 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Wilson Lines in Color Glass Condensate Formalism The Wilson loop (color singlet dipole) in McLerran-Venugopalan (MV) model 1 Nc Q2 (x −y )2 − s ⊥4 ⊥ TrU(x⊥ )U † (y⊥ ) =e x⊥ y⊥ ··· ··· ··· Dipole amplitude S(2) then produces the quark kT spectrum via Fourier transform Z 2 E dN d x⊥ d2 y⊥ −ik⊥ ·(x⊥ −y⊥ ) 1 D † F(k⊥ ) ≡ 2 = e TrU(x )U (y ) ⊥ ⊥ . d k⊥ (2π)2 Nc 12 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Geometrical Scaling in DIS 深度非弹性散射总截面[Golec-Biernat, Stasto, Kwiecinski; 01, Munier, Peschanski, 03] All data (x ≤ 0.01, Q2 ≤ 450GeV 2 ) is function of a single variable τ = Q2 /Q2s . Define Q2s (x) = (x0 /x)λ GeV2 with x0 = 3.04 × 10−3 and λ = 0.288. 13 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Forward hadron production in pA collisions 利用相对稀疏的质子（氘核）作为探针来探测重核中的稠密胶子的性质 Inclusive hadron production in small-x formalism [Dumitru, Jalilian-Marian, 02] Dilute-dense factorization at forward rapidity Z 1 h pA→hX i dσLO dz = x q (x , µ)F (k )D (z, µ) + x g(x , µ) F̃ (k )D (z, µ) . x2 ⊥ x2 ⊥ 1 f 1 1 1 h/q h/g 2 d2 p⊥ dyh τ z √ ! At forward rapidity, the hadron is produced as follows (at LO) � � 1 dσ K dz = d2 b xfq/p (x)F (xA , q⊥ )Dh/q (z) 2 dyd2 p⊥ (2π)2 xF z � 2 �� d r⊥ iq⊥ ·r⊥ 1 � � F (xA , q⊥ ) = e Tr U (0)U † (r⊥ ) x A (2π)2 Nc ! Observation at high energy s ! projectile: x1 ∼ target: x2 ∼ p⊥ = z q⊥ q⊥ Dipole gluon distribution follows B-K evolution equation, which can be solved numerically Comparison with RHIC data p √⊥ e+y ∼ 1 s p √⊥ e−y � 1 s Jan 8, 2013 Albaete-Marquet, 2010 valence gluon Zhongbo Kang, LANL 7 Tuesday, January 8, 2013 ! The spin asymmetry becomes the largest at forward rapidity region, Proton: Collinear PDFs and FFs (Strongly depends on µ2 ).; Nucleus: Small-x gluon! corresponding to ! ! ! The partons in the projectile (the polarized proton) have very large momentum Early attempts: [Dumitru, Hayashigaki, Jalilian-Marian, 06; Altinoluk, Kovner 11] fraction x: dominated by the valence quarks (spin effects are valence effects) [Altinoluk, Armesto, Beuf, Kovner, Lublinsky, 14] The partons in the target (the unpolarized proton or nucleus) have very small momentum fraction x: dominated the small-x Full NLO: [Chirilli, BXbyand Yuan,gluons 12] Thus spin asymmetry in the forward region could probe both 14 / 28 Introduction NLO and Threshold Resummation Summary and Outlook d+Au collisions at RHIC 相对论重离子对撞机上的单强子产生: 氘+金核碰撞/质子+质子碰撞 1 d2 Nd+Au /d2 pT dη Rd+Au = hNcoll i d2 Npp /d2 pT dη BRAHMS Cronin effect at middle rapidity Rapidity evolution of the nuclear modification factors Rd+Au Promising evidence for gluon saturation effects 15 / 28 Introduction NLO and Threshold Resummation Summary and Outlook New LHCb Results [R. Aaet al. (LHCb Collaboration), Phys. Rev. Lett. 128 (2022) 142004] RpPb = 1 d2 Np+Pb /d2 pT dη hNcoll i d2 Npp /d2 pT dη Rapidity evolution of the nuclear modification factors RpPb similar to RHIC 16 / 28 Introduction NLO and Threshold Resummation Summary and Outlook NLO diagrams in the q → q channel G. A. Chirilli, Bo-Wen Xiao, Feng Yuan, Phys. Rev. Lett. 108, 122301 (2012). Take into account real (top) and virtual (bottom) diagrams together! Non-linear multiple interactions inside the grey blobs! Integrate over gluon phase space ⇒Divergences!. 17 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Factorization for single inclusive hadron productions Factorization for the p + A → H + X process [Chirilli, BX and Yuan, 12] [quark] (xp+ p , 0, 0) ξ kµ pµ , y [hadron] z (0, xa p− a , kg⊥ ) q µ [gluon] [nucleus] pµ a Pp− ≃ 0 k+ ≃ 0 P+ ≃ 0 A Rapidity Divergence Collinear Divergence (P) Collinear Divergence (F) Include all real and virtual graphs in all channels q → q, q → g, g → q(q̄) and g → g. 1. collinear to target nucleus; rapidity divergence ⇒ BK evolution for UGD F(k⊥ ). 2. collinear to the initial quark; ⇒ DGLAP evolution for PDFs 3. collinear to the final quark. ⇒ DGLAP evolution for FFs. 18 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Numerical implementation of the NLO result Single inclusive hadron production up to NLO Z Z αs xg xg (1) (0) dσ = xfa (x) ⊗ Da (z) ⊗ Fa (k⊥ ) ⊗ H + xfa (x) ⊗ Db (z) ⊗ F(N)ab ⊗ Hab . 2π Consistent implementation should include all the NLO αs corrections. NLO parton distributions. (MSTW or CTEQ) NLO fragmentation function. (DSS or others.) Use NLO hard factors. [Chirilli, BX and Yuan, 12] Use the one-loop approximation for the running coupling rcBK evolution equation for the dipole gluon distribution [Balitsky, Chirilli, 08; Kovchegov, Weigert, 07]. Full NLO BK evolution not available. Saturation physics at One Loop Order (SOLO). [Stasto, Xiao, Zaslavsky, 13] 19 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Numerical implementation of the NLO result Saturation physics at One Loop Order (SOLO). [Stasto, Xiao, Zaslavsky, 13] BRAHMS η = 2.2, 3.2 101 10−7  10−3 0 1 2 p⊥ [GeV] 3 BK d3 N dηd2 p⊥ rcBK   GeV−2 10−3 h 2 i
λ  (2) 1 Sxg = exp − r4 Q20 xxg0 ln e + Λr
λ   2 (2) Sxg = exp − r4 Q20 xxg0  η = 3.2 (×0.1) GeV−2 10−5 LO NLO data 10−2 10−3 10−4  η = 2.2 STAR η = 4 MV GBW 10−71 10 −3 d3 N dηd2 p⊥ d3 N dηd2 p⊥  GeV−2  10−1 10 GeV−2 d3 N dηd2 p⊥ LO NLO data  101 10−7 10−5 0 1 2 p⊥ [GeV] 3 0 1 2 p⊥ [GeV] 3 10−6 1 1.2 1.4 1.6 p⊥ [GeV] 1.8 2 Reduced factorization scale dependence! Catastrophe: Negative NLO cross-sections at high pT . Fixed order calculation in field theories is not guaranteed to be positive. Rapidity sub with kinematic constraints. [Watanabe, Xiao, Yuan, Zaslavsky, 15] 20 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Extending the applicability of CGC calculation Goal: find a solution within our current factorization (exactly resum αs ln 1/xg ) to extend the applicability of CGC. Other scheme choices certainly is possible. A lot of logs arise in pQCD loop-calculations: DGLAP, small-x, threshold, Sudakov. Breakdown of αs expansion occurs due to the appearance of logs in certain PS. Demonstrate onset of saturation and visualize smooth transition to dilute regime. Add’l consideration: numerically challenging due to limited computing resources. Towards a more complete framework. [Altinoluk, Armesto, Beuf, Kovner, Lublinsky, 14; Kang, Vitev, Xing, 14; Ducloue, Lappi and Zhu, 16, 17; Iancu, Mueller, Triantafyllopoulos, 16; Liu, Ma, Chao, 19; Kang, Liu, 19; Kang, Liu, Liu, 20;] 21 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Some thoughts Spiritual Retreat under the Auspices of the (Color) Trinity Sabbatical leave is important! Learn new technique and put it aside sometimes. Keep trying and take as long as it takes! 22 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Gluon Radiation at the Threshold y ⊥e Near threshold: radiated gluon has to be soft! τ = p√ density (τ = xp ξz ≤ 1) s ((1 − ξ)xpP + , q −, q⊥) (xpP + , 0, 0⊥) (ξxpP + , k −, k⊥) xg P − Gluon momentum: q+ = (1 − ξ)p+ q →0 Introduce an additional semi-hard scale Λ2 . 23 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Threshold Logarithms Y. Shi, L. Wang, S.Y. Wei, Bo-Wen Xiao, Phys. Rev. Lett. 128, 202302 (2022). Numerical integration (8-d in total) is notoriously hard in r⊥ space. Go to k⊥ space. In the coordinate space, we can identify two types of logarithms single log: ln 2 2 k⊥ k⊥ → ln , µ2r Λ2 ln µ2 µ2 → ln ; µ2r Λ2 double log: ln2 2 2 k⊥ 2 k⊥ → ln , µ2r Λ2 with µr ≡ c0 /r⊥ with c0 = 2e−γE . Introduce a semi-hard auxiliary scale Λ2 ∼ µ2r  Λ2QCD . Identify dominant r⊥ ! Dependences on µ2 , Λ2 cancel order by order. Choose “natural" values at fixed order.  CR /[CR +β1 ] 2 (1−ξ)k⊥ 2 2 For running coupling, Λ = ΛQCD Λ2 . Akin to CSS & Catani et al. QCD 24 / 28 Introduction NLO and Threshold Resummation Summary and Outlook BRAHMS dAu, y = 2.2 LO One-loop Resummed, Λ2 = 3 GeV2   dN /dyd2 pT GeV−2 100 10−3 10−12 BRAHMS dAu, y = 3.2 LO One-loop Resummed, Λ2 = 5 GeV2 10−2 −4 10 10−6 10−9   d3 N /dyd2 pT GeV−2 Numerical Results for pA spectra RHIC: Λ2 ∼ Q2s ; LHC, larger Λ2 . −6 10 √ sNN = 200 GeV µ2 = α2 (µ2min + p2⊥ ), α ∈ [2, 4] 2 4 6 8 10 −8 10 12 √ sNN = 200 GeV µ2 = α2 (µ2min + p2⊥ ), α ∈ [2, 4] 1 14 2 101 10−5 5 ATLAS pPb, 1.5 < y < 1.8 LO One-loop Resummed, Λ2 = 20 GeV2 10−3 −4 10 10−5 10−1 10−3 4 10−2   d3 N /dyd2 pT GeV−2   d3 σ/dyd2 pT mb/GeV2 103 3 pT [GeV] pT [GeV] STAR dAu, y = 4 LO One-loop Resummed, Λ2 = 7 GeV2 10−7 sNN = 200 GeV µ2 = α2 (µ2min + p2⊥ ), α ∈ [2, 4] 1 1.5 2 pT [GeV] 2.5 10−8 √ 10−9 µ2 = α2 (µ2min + p2⊥ ), α ∈ [2, 4] sNN = 5.02 TeV 5 10 15 20 25 pT [GeV] µ ∼ Q ≥ 2k⊥ (α > 2) at high pT . 2 → 2 hard scattering. Estimate higher order correction by varying µ and Λ. Threshold enhancement for σ. Nice agreement with data across many orders of magnitudes for different energies and pT ranges! 10−6 √ µ2 = α2 (µ2min + p2T ) & α ∈ [2, 4]; 30 35 25 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Comparison with the new LHCb data   d3σ/dyd2pT mb/GeV2 104 pPb, 2.5 < y < 3.0 LO pPb, 2.0 < y < 2.5 LO 103 pPb, 3.0 < y < 3.5 LO pPb, 3.5 < y < 4.0 LO pPb, 4.0 < y < 4.3 LO LHCb data: 2108.13115 102 101 Data Link 100 10−1 α ∈ [2, 4] µ 101 √ 2 = α (µ2min + p2⊥) pp, 2.0 < y < 2.5 One-loop   d3σ/dyd2pT mb/GeV2 10−22 10 2 sNN = 5.02 TeV pp, 2.5 < y < 3.0 One-loop 2 2 Λ ∈ [10, 40] GeV pp, 3.0 < y < 3.5 One-loop LHCb pp, 3.5 < y < 4.0 One-loop pp, 4.0 < y < 4.5 One-loop 100 10−1 10−2 10−3 10−4 RpPb, 2.0 < y < 2.5 Resummed 1.2 RpPb, 2.5 < y < 3.0 Resummed RpPb, 3.0 < y < 3.5 Resummed RpPb, 3.5 < y < 4.0 Resummed RpPb, 4.0 < y < 4.3 Resummed RpPb 1 0.8 DIS2021 µ ∼ (2 ∼ 4)pT with proper choice of Λ Threshold effect is not important at low pT for LHCb data. Saturation effects are still dominant. Predictions are improved from LO to NLO. 0.6 0.4 1 2 3 4 5 pT [GeV] 6 7 1 2 3 4 5 pT [GeV] 6 7 1 2 3 4 5 pT [GeV] 6 7 1 2 3 4 5 pT [GeV] 6 7 1 2 3 4 5 6 7 pT [GeV] 26 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Summary 10−3 𝑦<0 109 𝑦=0 (LHC)   d3 σ/dydp2T mb/GeV2   dN /dyd2 pT GeV−2 𝑦=0 (RHIC) BRAHMS dAu, y = 2.2 LO One-loop Resummed, Λ2 = 3 GeV2 100 𝑦>0 10−1 10−6 10−9 10−12 LHCb pPb, y = 4.0 − 4.3 LO One-loop Resummed, Λ2 = 20 GeV2 104 √ 10−6 Proton sNN = 200 GeV µ2 = α2 (µ2min + p2⊥ ), α ∈ [2, 4] 2 4 6 8 10 pT [GeV] 12 14 Forward Hadron Production Nucleus √ sNN = 5.02 TeV µ2 = α2 (µ2min + p2⊥ ), α ∈ [2, 4] 10−11 0 10 101 pT [GeV] Ten-Year Odyssey in NLO hadron productions in pA collisions in CGC. Towards the precision test of saturation physics (CGC) at RHIC and LHC. Next Goal:Global analysis for CGC combining data from pA and DIS. Exciting time of NLO CGC phenomenology with the upcoming EIC. 27 / 28 Introduction NLO and Threshold Resummation Summary and Outlook Threshold resummation in the CGC formalism Threshold logarithms: Sudakov soft gluon part and Collinear (plus-distribution) part. 2 /Λ2 , ln2 k2 /Λ2 ) are resummed via Sudakov factor. Soft single and double logs (ln k⊥ ⊥ Performing Fouier transformations  Z 2  µ2 −ik⊥ ·r⊥ d l⊥ c0 d2 r⊥ S(r⊥ )ln 2 e =− F(k⊥ + l⊥ ) − J0 ( l⊥ )F(k⊥ ) 2 (2π)2 µr µ πl⊥   Z 2 2 1 d l⊥ Λ µ2 F(l ) − F(k ) + F(k )ln . = − ⊥ ⊥ ⊥ π (l⊥ − k⊥ )2 Λ2 + (l⊥ − k⊥ )2 Λ2 Z Two equivalent methods to resum the collinear part (Pab (ξ) ln Λ2 /µ2 ): 1. Reverse DGLAP evolution; 2. RGE method (threshold limit ξ → 1). Introduce forward threshold quark jet function ∆q (Λ2 , µ2 , ω), which satisfies   Z d∆q (ω) d∆q (ω) αs C F 3 αs CF ω 0 ∆q (ω) − ∆q (ω 0 ) q =− =− dω . ln ω + ∆ (ω) + d ln µ2 d ln Λ2 π 4 π ω − ω0 0 Consistent with the threshold resummation in SCET[Becher, Neubert, 06]! 28 / 28