[PPT] - Relativistic corrections to e + e cJ + in NRQCD Vladyslav PowerPoint Presentation

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Relativistic corrections to e+e− → χcJ + γ in NRQCD

Vladyslav Shtabovenko 1 in collaboration with N. Brambilla 1, W. Chen 2, Y. Jia 2 and A. Vairo 1

1Technische Universität München, Germany 2Institute of High Energy Physics Beijing, China

COLD QUANTUM COFFEE SEMINAR 16TH OF MAY, 2017, HEIDELBERG

Physik-Department T30f

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Outline

1

Theoretical framework and motivation Heavy quarkonia Nonrelativistic QCD

2

Relativistic corrections to the exclusive production e+e− → χcJ + γ Overview of the existing results Contributions from the higher Fock state |Q¯ Qg NRQCD-factorized production cross-sections Perturbative matching between QCD and NRQCD Final results

3

Automatic nonrelativistic calculation with FEYNCALC Nature of the problem FeynCalc FeynCalc 9.3 FeynOnium

4

Summary and Outlook

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Theoretical framework and motivation Heavy quarkonia b b c c

◮ Bound states of a heavy quark and a heavy

antiquark of the same flavor

◮ Heavy quarks: charm, bottom and top ◮ The top quark decays too fast to form a bound state ◮ Q¯

Q-bound states: charmonia (c¯ c) and bottomonia (b¯ b)

◮ Heavy quarkonia are an ideal laboratory to test our understanding of QCD

◮ Nonrelativistic system ◮ Rich phenomenology ◮ Creation/annihilation of Q¯

Q-pairs in short-distance processes

◮ Formation of Q¯

Q-bound states in long-distance (nonperturbative) processes

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Theoretical framework and motivation Heavy quarkonia

◮ Production of heavy quarkonia is governed by an interplay of perturbative and

non-perturbative effects:

◮ The formation of a heavy quark pair

from a collision process can be evaluated in perturbation theory.

◮ However, we should expand not only

in αs but also in the relative heavy quark velocity v!

◮ The evolution of a heavy quark pair

to a physical quarkonium requires non-perturbative input.

◮ We need a way to disentangle those effects from each other ⇒ Factorization. ◮ Even the perturbative part alone is not simple due to the presence of different

entangled scales ⇒ Multiscale problem in a non-relativistic system with a hierarchy

f scales.

◮ The nonperturbative part requires a clear field theoretical definition (e. g. for the

evaluation on the lattice)

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Theoretical framework and motivation Heavy quarkonia

◮ First heavy quarkonia (J/ψ and ψ(2s)) were discovered over 40 years ago

[Aubert et al., 1974], [Augustin et al., 1974]

◮ Crucial for the establishment of QCD as the correct theory of strong interactions

(November revolution of 1974, evidence for the existence of the charm quark, . . . )

◮ Early attempts to develop a theoretical description of heavy quarkonia:

phenomenological models

◮ Spectra from potential models ◮ Color singlet model [Einhorn & Ellis, 1975, Ellis et al., 1976, Carlson & Suaya, 1976] ◮ Color evaporation model

[Fritzsch, 1977, Halzen, 1977, Halzen & Matsuda, 1978, Gluck et al., 1978]

◮ Common short-comings of model-based approaches

◮ Require tuning to data ◮ Relation to the full QCD is unclear ◮ Predictions fail when the experimental precision increases ◮ No way to calculate higher order corrections systematically

◮ Effective Field Theory (EFT) methods [Weinberg, 1979, Wilson, 1974]: the modern way to

treat Q¯ Q-bound states

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Theoretical framework and motivation Heavy quarkonia

◮ The general idea is to construct a QFT that approximates the given high energy

theory at energies E ≪ Λ

◮ using the most appropriate degrees of freedom and ◮ providing the simplest description of the relevant physics

◮ The EFT approach is, in general, applicable to systems with several

well-sepearated dynamical scales Λ ≫ Λ1 ≫ Λ2 ≫ . . .

◮ The main steps to construct an EFT [Pich, 1998]

◮ Identify the relevant scales, symmetries and degrees of freedom ◮ The most general EFT Lagrangian: expansion in the small ratios of the relevant scales,

contains all operators Oi compatible with the symmetries. LEFT =

i

ci Λdi−4 Oi

◮ Introduce power-counting rules ⇒ Systematics, predictive power ◮ If possible, determine matching coefficients ci from comparing suitable quantities in the

high energy theory and in the EFT ⇒ Matching

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Theoretical framework and motivation Nonrelativistic QCD

◮ Relevant dynamical scales of a heavy quarkonium are

m

hard

≫ mv

soft

≫ mv2

ultrasoft

with v2

c ∼ 0.3, v2 b ∼ 0.1 ◮ ⇒ relativistic corrections are very important for

charmonia!

◮ The formation of a Q¯

Q-pair occurs within a distance 1/m (short distance process)

◮ The formation of a heavy quarkonium happens over distances of order 1/(mv) or

larger in the quarkonium rest frame (long distance process).

◮ A suitable EFT for studying quarkonium production is Non-Relativistic QCD

(NRQCD) [Caswell & Lepage, 1986, Bodwin et al., 1995]

◮ Starting from the full QCD, all scales above mv are integrated out. ◮ We can always do this perturbatively, since m ≫ ΛQCD ◮ The effects of the high-energy contributions are encoded in the matching coefficients

cn(αs(m), µ) multiplying NRQCD operators On(µ).

◮ LNRQCD =

n

cn(αs(m),µ) mdn−4

On(µ) is an expansion in αs and v.

◮ ∞-number of operators with increasing mass dimension. ◮ Contributions to a process at the given accuracy estimated by velocity scaling rules.

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Theoretical framework and motivation Nonrelativistic QCD

◮ NRQCD is not a model, it precisely reproduces the full QCD at energies E ≪ m

rder by order in 1/m.

◮ Non-perturbative contributions go inside long distance matrix elements (LDME)

On(µ).

◮ LDMEs must be extracted from experiment or calculated on the lattice, but they do

not depend on the short-distance process (universality).

◮ Predictive power of NRQCD: Extract LDMEs from one measurement and use them

for predictions in a different measurement.

◮ Matching condition on the level of cross-sections for production

σ(Q¯ Q)

pert. QCD

!

=

n

Fn(αs(m), µ) mdn−4 0|OQ¯

Q n (µ)|0 |pert. NRQCD. ◮ Once the matching coefficients Fn(αs(m), µ) are determined in perturbative

matching, we can write down the NRQCD factorized production cross-section σ(H) =

n

Fn(αs(m), µ) mdn−4 0|OH

n (µ)|0 .

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Theoretical framework and motivation Nonrelativistic QCD

◮ NRQCD tells us that a Q¯

Q-pair evolving into quarkonium does not necessarily has to be in the color singlet configuration.

◮ Fock-state expansion of a heavy quarkonium

|H ∼ a0 |Q¯ Q + a1 |Q¯ Qg + a2 |Q¯ Qgg + . . .

◮ Higher order Fock states with Q¯

Q-pairs in the color octet (CO) configuration are suppressed by power of v.

◮ Nevertheless, they must be taken into account when higher order relativistic or

radiative corrections are computed!

◮ The presence of the CO mechanism is an important feature that distinguishes

NRQCD from other approaches.

◮ Studying the importance of the CO contributions for phenomenology is an important

test for the validity of NRQCD.

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Relativistic corrections to the exclusive production e+e− → χcJ + γ Overview of the existing results

◮ Electromagnetic spin triplet P-wave quarkonium production in e+e−-annihilation:

virtual photon decays into a hard (kp ∼ m) on-shell photon and χcJ:

e− e+ γ χcJ

◮ No experimental data available, good perspectives for this measurement will exist at

Belle II in Japan

◮ An early study [Chung et al., 2008] based on O(α0 sv0) results predicted cross-sections

that might be measurable at B-factories (here for √s = 10.6 GeV):

◮ σ(e+e− → χc0 + γ) = 1.3 fb ◮ σ(e+e− → χc1 + γ) = 13.7 fb ◮ σ(e+e− → χc2 + γ) = 5.3 fb

◮ Subsequently, corrections of order O(αsv0) ( [Sang & Chen, 2010b], [Li et al., 2009]),

O(α0

sv2) ( [Li et al., 2013, Chao et al., 2013]) and finally O(αsv2) ( [Xu et al., 2014]) were

btained as well.

◮ C.f. also treatment in the light cone formalism [Braguta, 2010], [Wang & Yang, 2014]

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Relativistic corrections to the exclusive production e+e− → χcJ + γ Contributions from the higher Fock state |Q¯ Qg

◮ So far, all the previous NRQCD studies were concerned with the operators that

contribute through the dominant Fock state |Q¯ Q.

◮ However, at O(α0 sv2) operators that contribute through the subleading Fock state

|Q¯ Qg show up as well ⇒ CO mechanism of NRQCD.

e− e+ γ χcJ

◮ The external gluon is soft (|pg| ∼ mv) and thus must be treated in NRQCD. It is a

part of the perturbative quarkonium system Q(p1)¯ Q(p2)g(kg).

◮ The availability of the CS O(αsv2) corrections suggests that CO O(α0 sv2) corrections

should be determined as well.

◮ In this talk we will not discuss the numerical impact of these corrections, but rather

concentrate on explaining the analytical calculation and the methodology.

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Relativistic corrections to the exclusive production e+e− → χcJ + γ NRQCD-factorized production cross-sections

◮ Decay:, factorization of perturbative and nonperturbative contributions holds to all

rders in αs [Bodwin et al., 1995]

◮ Production: an all-order proof is still lacking ⇒ NRQCD factorization conjecture ◮ Nevertheless, NRQCD factorization in quarkonium production is widely used in the

phenomenology

◮ A general recipe for production calculations in NRQCD [Bodwin et al., 1995]

σ(Q¯ Q)

pert. QCD

!

=

n

Fn(αs(m), µ) mdn−4 0|OQ¯

Q n (µ)|0 |pert. NRQCD. 1

Write down the production cross-section σ up to the desired order in v

2

Evaluate σ in perturbative NRQCD by replacing |H with |Q¯ Q (or |Q¯ Qg, . . . )

3

Calculate σ(Q¯ Q) (σ(Q¯ Qg), . . . ) in perturbative QCD

4

Expand both sides in v

5

Read off the matching coefficients

6

Plug them back into σ

7

Numerics

◮ Common complications

◮ Nonrelativistic expansion in v breaks manifest Lorentz covariance of QCD ◮ Huge number of terms once we go beyond LO in v ◮ Analytic results for matching coefficients are highly desirable ⇒ numerical evaluation of

loop integrals is not really useful

◮ Many nonperturbative parameters (LDMEs) are still undetermined

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Relativistic corrections to the exclusive production e+e− → χcJ + γ NRQCD-factorized production cross-sections

Exclusive production of χcJ in NRQCD at O(v2) σ(e+e− → χc0 + γ) = F1(3P0) 3m2 0|χ†(− i 2

↔

D · σ)ψ|χc0 χc0|ψ†(− i 2

↔

D · σ)χ|0 + G1(3P0) 6m4

0|χ†(− i

2

↔

D · σ)ψ|χc0 χc0|ψ†(− i 2

↔

D · σ)(− i 2

↔

D)2χ|0 + h.c.

+ iT8(3P0)

3m3

0|χ†(− i

2 ↔

D · σ)ψ|χc0 χc0|ψ†(gE · σ)χ|0 + h.c.

≡ F1(3P0)

m2 0|O1(3P0)|0 + G1(3P0) m4 0|P1(3P0)|0 + T8(3P0) m3 0|T8(3P0)|0

◮ bold font denotes Cartesian 3-vectors ◮ ψ (χ) annihilates (creates) a heavy quark (antiquark) ◮ σ is the Pauli vector ◮ D ≡ ∇ − igA ◮ A is the gluon field ◮ E is the chromoelectric field ◮ ψ† ↔

D χ ≡ ψ†(Dχ) − (Dψ)†χ

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Relativistic corrections to the exclusive production e+e− → χcJ + γ NRQCD-factorized production cross-sections

σ(e+e− → χc1 + γ) = F1(3P1) 2m2 0|χ†(− i 2

↔

D × σ)ψ|χc1 · χc1|ψ†(− i 2

↔

D × σ)χ|0 + G1(3P1) 4m4

0|χ†(− i

2

↔

D × σ)ψ|χc1 · χc1|ψ†(− i 2

↔

D × σ)(− i 2

↔

D)2χ|0 + h.c.

+ iT8(3P1)

2m3

0|χ†(− i

2 ↔

D × σ)ψ|χc1 · χc1|ψ†(gE × σ)χ|0 + h.c.

≡ F1(3P1)

m2 0|O1(3P1)|0 + G1(3P1) m4 0|P1(3P1)|0 + T8(3P1) m3 0|T8(3P1)|0 σ(e+e− → χc2 + γ) = F1(3P2) m2 0|χ†(− i 2 ← → D (iσj))ψ|χc2 χc2|ψ†(− i 2 ← → D (iσj))χ|0 + G1(3P2) 2m4

0|χ†(− i

2 ← → D (iσj))ψ|χc2 χc2|ψ†(− i 2 ← → D (iσj))(− i 2

↔

D)2χ|0 + h.c.

+ iT8(3P2)

m3

0|χ†(− i

2

← → D (iσj))ψ|χc2 χc2|ψ†(gE(iσj))χ|0 + h.c.

≡ F1(3P2)

m2 0|O1(3P2)|0 + G1(3P2) m4 0|P1(3P2)|0 + T8(3P2) m3 0|T8(3P2)|0

◮ a(ibj) ≡ aibj+ajbi 2

− 1

3δij(a · b).

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Relativistic corrections to the exclusive production e+e− → χcJ + γ NRQCD-factorized production cross-sections

◮ Exclusive production (such as e+e− → χcJ + γ ): NRQCD factorization at the

amplitude level [Braaten & Chen, 1998, Braaten & Lee, 2003]

Apert. QCD

!

=

n

ci1...ik

n

Q¯ Q| ψ†Ki1...ik

n

χ |0 ≡ Apert. NRQCD

◮ Ki1...ik

n

are polynomials in D, E, B and contain a spin matrix and a color matrix

◮ ci1...ik

n

are short-distance coefficients

◮ The matching condition is an equality between the on-shell amplitude to produce a

heavy Q¯ Q-pair in pQCD and a sum of quarkonium-to-vacuum matrix elements multiplied or contracted with short-distance coefficients in pert. NRQCD.

◮ The cn are then substituted into NRQCD-factorized production amplitudes

ANRQCD =

n

ci1...ik

n

H| ψ†Ki1...ik

n

χ |0

◮ Squaring ANRQCD and integrating over the phase space of the physical quarkonium

we obtain previously shown NRQCD production cross-sections

◮ In general, not all matrix elements from Apert. NRQCD will also appear in ANRQCD ◮ QCD amplitudes do not have a definite angular momentum J ⇒ in the matching we

will determine more short-distance coefficients than required in

◮ Important crosscheck: Apert. QCD − Apert. NRQCD !

= 0 order by order in v

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Relativistic corrections to the exclusive production e+e− → χcJ + γ NRQCD-factorized production cross-sections ◮ The amplitude-level matching is very advantageous

◮ The matching calculation is simpler, no need to square QCD amplitudes ◮ Need to deal with a smaller number of terms on the QCD side ◮ Avoids the nonrelativistic expansion of the phase space measure

◮ Can be employed only in exclusive processes, i. e. not for hadronic production ◮ NRQCD-factorized amplitudes for e+e− → χcJ + γ at O(v2)

AJ=0

NRQCD = cJ=0 1

m2 χc0|ψ†

− i

2 ← → D · σ

χ|0

+ cJ=0

3

m4 χc0|ψ†

− i

2 ← → D · σ − i 2 ← → D 2 χ|0 + dJ=0

1

m3 χc0|ψ†gE · σχ|0 , AJ=1

NRQCD = (cJ=1 1

)i m2 χc1|ψ†

− i

2 ← → D × σ i χ|0 + (cJ=1

3

)i m4 χc1|ψ†

− i

2 ← → D × σ i − i 2 ← → D 2 χ|0 + (dJ=1

1

)i m3 χc1|ψ†(gE × σ)iχ|0 , AJ=2

NRQCD = (cJ=2 1

)ij m2 χc2|ψ†

− i

2 ← → D (iσj)

χ|0

+ (cJ=2

3

)ij m4 χc2|ψ†

− i

2 ← → D (iσj) − i 2 ← → D 2 χ|0 + (dJ=

1 )ij

m3 χc2|ψ†gE(iσj)χ|0

◮ Our goal: determine the short distance coefficients dJ=0 2

, (dJ=1

2

)i and (dJ=1

2

)ij.

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Relativistic corrections to the exclusive production e+e− → χcJ + γ Perturbative matching between QCD and NRQCD

◮ Operators that involve only D and σ contribute already through |Q¯

Q.

◮ Operators with one power of E or B require inclusion of |Q¯

Qg.

◮ What does this mean for the QCD side of the matching?

◮ Contributions from |Q¯

Q ⇒ 2 QCD diagrams two produce on-shell Q¯ Q

e− e+ γ Q ¯ Q e− e+ γ Q ¯ Q

◮ Contributions from |Q¯

Qg ⇒ 6 QCD diagrams two produce on-shell Q¯ Qg with an ultrasoft gluon (pg ∼ mv2)

e− e+ γ Q ¯ Qg

◮ Tree-level but NLO in v (complexity is the main challenge) ◮ First explicit calculation of matching coefficients multiplying chromoelectric LDMEs

in heavy quarkonium production

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Relativistic corrections to the exclusive production e+e− → χcJ + γ Perturbative matching between QCD and NRQCD

◮ Emission of an ultrasoft gluon from an external heavy quark line leads to IR

singularities

◮ More specifically, the amplitude contains terms singular in the limit |pg| → 0 ◮ Such terms cancel in the matching: QCD and NRQCD share the same IR behavior ◮ The cancellation requires inclusion of NRQCD operators with Lagrangian insertions ψ†(p1) χ(p2)

= i Q(p1,R)¯ Q(p2,R)|ψ†χ|0

ψ†(p1) χ(p2) A(pg) + ψ†(p1) χ(p2) A(pg) + ψ†(p1) χ(p2) A(pg)

= i Q(p1,R)¯ Q(p2,R)g(pg,R)|ψ†χ|0L2−f , L2−f = ψ† D2 2m + σ · gB 2m + [D·, gE] 8m2 + iσ · [D×, gE] 8m2 + D4 8m3 + {D2, σ · gB} 8m3

ψ

+ χ†

− D2

2m − σ · gB 2m + [D·, gE] 8m2 + iσ · [D×, gE] 8m2 − D4 8m3 + {D2, σ · gB} 8m3

χ
V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Relativistic corrections to the exclusive production e+e− → χcJ + γ Perturbative matching between QCD and NRQCD

Expansion of the QCD amplitudes

◮ Nowadays, matching calculations on the QCD side are usually carried out using the

covariant projector technique [Bodwin & Petrelli, 2002].

◮ Manifest Lorentz covariance of this approach greatly facilitates automation via

suitable software tools (e. g. FeynCalc [Mertig et al., 1991, Shtabovenko et al., 2016], FDC

[Wang, 2004], Redberry [Bolotin & Poslavsky, 2013])

◮ For this calculation we used the threshold expansion method of Braaten and Chen

[Braaten & Chen, 1996a], i. e. noncovariant matching. In particular, this involves following

manipulations of the QCD amplitude

◮ Using explicit representation of Dirac spinors ◮ Projecting out J = 0, J = 1 and J = 2 contributions via decomposition of Cartesian

tensors, e.g. aibj J=0 → 1 3δij(a · b) aibj J=1 → aibj − ajbi 2 aibj J=2 → aibj + ajbi 2 − 1 3 δij(a · b)

◮ Removing spurious terms via Schouten-like identity

εijkpl − εjklpi + εklipj − εlijpk = 0

◮ This way we can explicitly see how our QCD and NRQCD amplitudes exactly match

rder by order in v and how the IR singularities cancel on both sides.
V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

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Relativistic corrections to the exclusive production e+e− → χcJ + γ Perturbative matching between QCD and NRQCD

Choice of the frame

◮ The NRQCD side of the matching is usually evaluated in the frame where the heavy

quarkonium is at rest (rest frame).

◮ The QCD side can be evaluated in any convenient frame [Bodwin et al., 1995]. ◮ The most convenient frame for a production process is usually the CM frame of the

colliding particles (CM frame).

◮ Two possibilities to carry out the matching:

◮ Evaluate QCD amplitudes in the rest frame. Then boost the short distance coefficients in

the NRQCD production amplitude to the CM frame. Use Gremm-Kapustin relations

[Gremm & Kapustin, 1997] to eliminate heavy quarkonium mass from the matching

coefficients.

◮ Evaluate QCD amplitudes in the CM frame. For the nonrelativistic expansion, express

the momenta of the moving constituents in terms of the soft rest frame momenta.

◮ We applied both approaches and arrived to the same final results.

e+ e- H γ

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Relativistic corrections to the exclusive production e+e− → χcJ + γ Perturbative matching between QCD and NRQCD

Matching in the rest frame

◮ If we choose to work in the rest frame, our process exhibits strong similarities to the

decay χcJ → 2γ:

◮ Matching coefficients of the chromoelectric operators in decay are known

[Ma & Wang, 2002a], [Brambilla et al., 2006], talk of W.L. Sang at QWG 2013.

◮ We can recover them from our results in the limit s → 0 ◮ After we obtain the NRQCD production amplitudes with matching coefficients from

the rest frame, we boost them to the CM frame using heavy quarkonium kinematics.

◮ The appearance of the physical quarkonium mass MH is eliminated via following

Gremm-Kapustin relations [Ma & Wang, 2002b] 0|P1(3PJ)|0 = mEχcJ 0|O1(3PJ)|0 + m 0|T8(3PJ)|0 (2) with EB = MH − 2m.

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Relativistic corrections to the exclusive production e+e− → χcJ + γ Perturbative matching between QCD and NRQCD

P p2 q p1

◮ Of course, we can also work in the CM frame from the very beginning. ◮ Then the short distance coefficients do not require any further boosts. ◮ The calculation is slightly more involved as compared to to rest frame:

◮ In the CM frame, the energies and 3-momenta of Q, ¯

Q and g are not small.

◮ Using boost matrix formalism [Braaten & Chen, 1996b], we can rewrite all those energies

and 3-momenta in terms of soft/ultrasoft rest frame momenta.

◮ The generalization to the 3-body kinematics is straightforward.

◮ Braaten-Chen formalism for a 2-body system

◮ Jacobi momenta: p1 = 1

2 P + Q,

p2 = 1

2P − Q

◮ CM frame:

P = ( √ P2 + P2, P) Q = (

Q2 + Q2, Q)

◮ Rest frame (p1,R + p2,R = 0,

q ≡ p1,R = −p2,R): QR ≡ q = (0, q) PR = (2

q2 + M2, 0) ≡ (2Eq, 0)

◮ Boost matrix establishes connection between the two frames, e.g.

Qµ = Λµ

νQν R = Λµ iqi with

Λ0

i = Pi

2Eq , Λi

j = δij +

P0 2Eq − 1

ˆ

Piˆ Pj, P0 =

4E2

q + P2

◮ So we can expand the CM-frame amplitude in the soft rest frame momentum |q|.

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Relativistic corrections to the exclusive production e+e− → χcJ + γ Perturbative matching between QCD and NRQCD

◮ In a 3-body system the relations become slightly more complicated. ◮ Jacobi momenta

p1 = 1 3P + Q1 − Q2, p2 = 1 3P − Q1 − Q2, kg = 1 3P + 2Q2.

◮ In the rest frame, where p1,R + p2,R + pg,R = PR = 0, we have

p1,R = 1 3PR + q1 − q2, p2,R = 1 3PR − q1 − q2, kg,R = 1 3PR + 2q2,

r

q1 = 1 2 (p1,R − p2,R) ≡ (q0

1, q1)

q2 = 1 6 (2kg,R − p1,R − p2,R) ≡ (q0

2, q2)

q0

1 = 1

2

(q1 − q2)2 + m2 −
(q1 + q2)2 + m2
,

q0

2 = 1

6

4|q2| −
(q1 − q2)2 + m2 −
(q1 + q2)2 + m2
with |q1| and |q2| being our rest frame soft expansion parameters.
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Relativistic corrections to the exclusive production e+e− → χcJ + γ Perturbative matching between QCD and NRQCD ◮ CM frame vectors Q1 and Q2 in terms of q1 and q2

Qµ

1 = Λµ νqν 1

Qµ

2 = Λµ νqν 2 ◮ Boost matrix

Λ0

0 =

1 − P2

P2 , Λ0

i = Λi 0 =

Pi √ P2 , Λi

j = δij +

1 − P2

P2 − 1

ˆ

Piˆ Pj

◮ Boosted Dirac bilinears

¯ u(p1)γµv(p2) = Λµ

ν¯

uR(p1,R)γνvR(p2,R), ¯ u(p1)γµγ5v(p2) = Λµ

ν¯

uR(p1,R)γνγ5vR(p2,R)

◮ Rest frame bilinears (N1N2 is the normalization factor)

¯ uR(p1,R)γ0vR(p2,R) = N1N2ξ† p1,R · σ E1,R + m + p2,R · σ E2,R + m

η

¯ uR(p1,R)γivR(p2,R) = N1N2ξ†

σi +

1 E1,R + m 1 E2,R + m

pi

1,R(p2,R · σ) + pi 2,R(p1,R · σ)

+ (p1,R · p2,R)σi − i(p1,R × p2,R)i

η

¯ uR(p1,R)γ0γ5vR(p2,R) = N1N2ξ†

1 +

1 E1,R + m 1 E2,R + m (p1,R · p2,R + iσ · (p1,R × p2,R)

η

¯ uR(p1,R)γ0γ5vR(p2,R) = N1N2ξ†

pi

1,R − i(p1,R × σ)i

E1,R + m + pi

2,R + i(p2,R × σ)i

E2,R + m

η
V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

Electromagnetic χcJ production 24 / 40

SLIDE 25

Relativistic corrections to the exclusive production e+e− → χcJ + γ Final results

Final results (including the known O(αsv0) [Sang & Chen, 2010a, Li & Chao, 2009] correction to F1(3PJ)) σ(e+e− → χc0 + γ) = (4πα)3e4

Q(1 − 3r)2

18πm3s2(1 − r)

1 + C0

0(r)αs

π

0|O1(3P0)|0BBL

− (13 − 18r + 25r2) 10m2(1 − 4r + 3r2) 0|P1(3P0)|0BBL + 2r(2 − 3r) m(1 − 4r + 3r2) 0|T8(3P0)|0BBL

new correction

for χc0

◮ The subscript “BBL

” denotes the nonrelativistic normalization of the LDMEs H(P)|H(P′)BBL = (2π)3δ3(P − P′),

◮ Explicit values of Ci j(r) can be found the Appendix B of

[Sang & Chen, 2010a]

◮ eQ is the heavy quark charge ◮ m is the heavy quark mass ◮ s is the CM energy ◮ r = 4m2/s

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Electromagnetic χcJ production 25 / 40

SLIDE 26

Relativistic corrections to the exclusive production e+e− → χcJ + γ Final results

σ(e+e− → χc1 + γ) = (4πα)3Q4(1 + r) 3πm3s2(1 − r)

1 + C0

1(r) + rC1 1(r)

1 + r αs π

0|O1(3P1)|0BBL

− (11 − 20r − 11r2) 10m2(1 − r2) 0|P1(3P1)|0BBL − (3 − 3r − 4r2) 2m(1 − r2) 0|T8(3P1)|0BBL

new correction

for χc1

σ(e+e− → χc2 + γ) = (4πα)3Q4(1 + 3r + 6r2) 9πm3s2(1 − r) ×

1 + C0

2(r) + 3rC1 2(r) + 6r2C2 2(r)

1 + 3r + 6r2 αs π

0|O1(3P2)|0BBL

− (1 + 4r − 30r2) 10m2(1 + 3r + 6r2) 0|P1(3P2)|0BBL − (3 + r − 6r2 − 18r3) 2m(1 − r)(1 + 3r + 6r2) 0|T8(3P2)|0BBL

new correction

for χc2

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

Electromagnetic χcJ production 26 / 40

SLIDE 27

Relativistic corrections to the exclusive production e+e− → χcJ + γ Final results

◮ What is the impact of the new chromoelectric LDMEs for the phenomenology? ◮ In total, each cross-section depends on three LDMEs (nonperturbative quantities) ◮ However, the values of the matrix elements

0|O1(3PJ)|0BBL , 0|P1(3PJ)|0BBL , 0|T8(3PJ)|0BBL are poorly known

◮ no experimental cross-section measurements to fit to ◮ no lattice determinations

◮ The same matrix elements also enter NRQCD-factorized decay rates for χc0,2 → γγ ◮ The value of 0|O1(3PJ)|0BBL could be taken from potential models ◮ Gremm-Kapustin relations reduce the number of independent parameters

0|P1(3PJ)|0BBL = mEχcJ 0|O1(3PJ)|0BBL + m 0|T8(3PJ)|0BBL (3)

◮ However, the results are very sensitive to the value of EχcJ, so that we cannot make

a solid prediction . . .

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

Electromagnetic χcJ production 27 / 40

SLIDE 28

Automatic nonrelativistic calculation with FEYNCALC Nature of the problem

◮ Conventional packages for loop calculations are designed to work with manifestly

Lorentz covariant expressions

◮ However, for calculations in nonrelativistic EFTs we also need tools that can work

with objects like gµν, gµ0, gµi, g00, gij, δij ǫ0µνρ, ǫµνi, ǫµij, γ0, γi, / p.

◮ From the technical point of view, this is clearly feasible (albeit painful) ◮ Unfortunately, as of now there are no public packages designed for such purposes ◮ People either use private codes or computes everything by pen and paper ◮ With the upcoming FEYNCALC 9.3, this will change . . .

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Electromagnetic χcJ production 28 / 40

SLIDE 29

Automatic nonrelativistic calculation with FEYNCALC FeynCalc

◮ FEYNCALC is an open source (GPLv3)

MATHEMATICA package for symbolic semi-automatic evaluation of Feynman diagrams and algebraic expressions in QFT.

◮ Suitable for evaluating both single expressions and full Feynman diagrams. ◮ The calculation can be organized in many different ways (flexibility) ◮ Extensive typesetting for better readability ◮ Lorentz index contractions, SU(N) algebra, Dirac algebra, etc. ◮ Passarino-Veltman reduction of one-loop amplitudes to standard scalar integrals ◮ Basic support for manipulating multi-loop integrals ◮ General tools for non-commutative algebra ◮ BUT: Essentially only algebraic manipulations, everything else requires extra tools.

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Electromagnetic χcJ production 29 / 40

SLIDE 30

Automatic nonrelativistic calculation with FEYNCALC FeynCalc

◮ People behind FEYNCALC

◮ Rolf Mertig (GluonVision GmbH): original author of the package, first release 1991 ◮ Frederik Orellana (Technical University of Denmark): joined 1997 ◮ VS (TUM, soon Zhejiang University): joined 2014

◮ Recent developments (since 2014)

◮ Large parts of the code improved or rewritten from scratch. ◮ Public source code repository on GITHUB: https://github.com/FeynCalc ◮ Online documentation https://feyncalc.github.io/reference ◮ Ships with many sample calculations ◮ Extensive unit testing framework ◮ New and improved functions for loop calculations.

◮ I always put big emphasis on using FEYNCALC for EFT calculations

◮ Original motivation for FEYNHELPERS [Shtabovenko, 2016]: Matching calculations in

relativistic EFTs

◮ Upcoming FEYNCALC 9.3 and FEYNONIUM: Matching calculations in nonrelativistic EFTs

(in particular NRQED/NRQCD [Caswell & Lepage, 1986, Bodwin et al., 1995])

◮ Currently only tree-level NR calculations, extension to 1-loop in progress

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Electromagnetic χcJ production 30 / 40

SLIDE 31

Automatic nonrelativistic calculation with FEYNCALC FeynCalc 9.3

After having loaded the package

In[1]:= $LoadAddOns={"FeynOnium"};

<<FeynCalc‘

we can manipulate Cartesian vectors

In[2]:= CV[p,i]CV[q,j]KD[i,j]

Contract[%]

Out[2]=

_

pi _ qj _ δ

ij

Out[3]=

_

p·

_

q

r Levi-Civita tensors

In[4]:= CLC[i,j,k]CLC[i,l,m]

Contract[%]

Out[4]=

_

ǫijk _ ǫilm

Out[5]=

_

δ

jl _

δ

km− _

δ

jm _

δ

kl

in the same way as Lorentz objects.

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Electromagnetic χcJ production 31 / 40

SLIDE 32

Automatic nonrelativistic calculation with FEYNCALC FeynCalc 9.3

This includes undoing contractions of indices

In[6]:= CSP[p,q]

Uncontract[%,q,CartesianPair→ → →All]

Out[6]=

_

p·

_

q

Out[7]=

_

p$AL($24) _ q$AL($24)

and expanding Cartesian scalar products

In[8]:= CSP[p1+p2,q1+q2]

ExpandScalarProduct[%]

Out[8]= (

_

p1+

_

p2)·(

_

q1+

_

q2)

Out[9]=

_

p1·

_

q1+

_

p1·

_

q2+

_

p2·

_

q1+

_

p2·

_

q2

In[10]:= MomentumCombine[%] Out[10]= (

_

p1+

_

p2)·(

_

q1+

_

q2)

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Electromagnetic χcJ production 32 / 40

SLIDE 33

Automatic nonrelativistic calculation with FEYNCALC FeynCalc 9.3

It is also possible to rewrite Lorentz objects in terms of Cartesian objects

In[11]:= FV[p,µ

µ µ] LorentzToCartesian[%]

Out[11]=

_

pµ

Out[12]= p0 _

g0µ−

_

p$ _ g$µ

In[13]:= SP[p,q]

LorentzToCartesian[%]

Out[13]=

_

p·

_

q

Out[14]= p0 q0−

_

p·

_

q

In[15]:= GS[p]

LorentzToCartesian[%]

Out[15]=

_

γ·

_

p

Out[16]= p0 _

γ0−

_

γ·

_

p

r do the inverse

In[17]:= CSP[p,q]

CartesianToLorentz[%]

Out[17]=

_

p·

_

q

Out[18]= p0 q0−

_

p·

_

q

In[19]:= CGS[p]

CartesianToLorentz[%]

Out[19]=

_

γ·

_

p

Out[20]= p0 _

γ0−

_

γ·

_

p

V. Shtabovenko (TUM) @ CQC, Heidelberg, 16.05.2017

Electromagnetic χcJ production 33 / 40

SLIDE 34

Automatic nonrelativistic calculation with FEYNCALC FeynCalc 9.3

The dimensionality of Cartesian objects can be changed in the same way as for Lorentz

bjects

In[21]:= CV[p,i]

ChangeDimension[%,D]

Out[21]=

_

pi

Out[22]= pi

Furthermore, the new FEYNCALC can also perform summations over polarization vectors, that are contracted with Cartesian vectors

In[23]:= SP[kp,kp]=0;

PolarizationSum[i,j,kp,n,Heads→ → →{CartesianIndex,CartesianIndex}]

Out[23]=

_

δ

ij − _

n2

_

kp

i _

kp

j

(

_

kp·

_

n)

2

+

_

ni _ kp

j+ _

kp

i _

nj

_

kp·

_

n

DiracSimplify is capable of simplifying chains of Dirac matrices with temporal or spatial components

In[24]:= TGA[].TGA[]

DiracSimplify[%]

Out[24]=

_

γ0.

_

γ0

Out[25]= 1 In[26]:= CGA[i].CGS[p].CGA[j].CGA[i]

DiracSimplify[%]

Out[26]=

_

γi.(

_

γ·

_

p).

_

γj.

_

γi

Out[27]= (

_

γ·

_

p).

_

γj+4

_

pj

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Electromagnetic χcJ production 34 / 40

SLIDE 35

Automatic nonrelativistic calculation with FEYNCALC FeynCalc 9.3

The same goes also for Dirac traces

In[28]:= CGA[i,j,k,l]

DiracTrace[%,DiracTraceEvaluate→ → →True]

Out[28]=

_

γi.

_

γj.

_

γk.

_

γl

Out[29]= 4 (

_

δ

il _

δ

jk− _

δ

ik _

δ

jl+ _

δ

ij _

δ

kl)

A new function that simplifies chains of Pauli matrices is called PauliTrick

In[30]:= CSI[i,i]

PauliTrick[%]

Out[30]=

_

σi.

_

σi

Out[31]= 3 In[32]:= CSIS[p,p]

PauliTrick[%]

Out[32]= (

_

σ·

_

p).(

_

σ·

_

p)

Out[33]=

_

p2

In[34]:= CSI[i,j,i]

PauliTrick[%] Contract[%]

Out[34]=

_

σi.

_

σj.

_

σi

Out[35]=

_

σj+i

_

ǫij$MU($26) (

_

δ

i$MU($26)+i _

σ$MU($27) _ ǫ$MU($26)i$MU($27))

Out[36]= −

_

σj

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Electromagnetic χcJ production 35 / 40

SLIDE 36

Automatic nonrelativistic calculation with FEYNCALC FeynOnium

Let us also mention some functions provided by the FEYNONIUM extension. With FMSpinorChainExplicit2 we can rewrite arbitrary Dirac spinor chains in terms of Pauli matrices and Pauli spinors, which is very useful for matching calculations between QCD and NRQCD. The general results turn out to be quite large, which is why we do not display them here

In[37]:= SpinorUBar[p1,m].GA[mu].SpinorV[p2,m]

FMSpinorChainExplicit2[%,FMSpinorNormalization→ → →"nonrelativistic"]

Out[37]=

_

u(p1,m).

_

γmu.v(p2,m) ...

For practical purposes it is more useful to specify the kinematics explicitly, e. g. to consider ¯ u(p1, m)γµv(p2, m) at 0th order in the relative momentum between p1 and p2, which returns a very short result

In[38]:= FCClearScalarProducts[]

TC[p1]=m; TC[p2]=m; CartesianMomentum[p1|p2]=0; FMSpinorChainExplicit2[SpinorUBar[p1,m].GA[mu].SpinorV[p2,m],FMSpinorNormalization → → →"nonrelativistic"]

Out[38]=

_

g$MU($34)mu ξ†.

_

σ$MU($34).η

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Electromagnetic χcJ production 36 / 40

SLIDE 37

Automatic nonrelativistic calculation with FEYNCALC FeynOnium

Another important operation is the projection of different angular components from the given Cartesian tensor. This is again often needed in the matching, where we want to equate QCD and NRQCD amplitudes of the same total angular momentum J. The trivial example, is to consider the tensor ki

1kj 2

and to extract its components that correspond to J = 0, J = 1 and J = 2 using FMCartesianTensorDecomposition

In[39]:= CV[k1,i]CV[k2,j]

FMCartesianTensorDecomposition[%,{k1,k2},0] FMCartesianTensorDecomposition[%%,{k1,k2},1] FMCartesianTensorDecomposition[%%%,{k1,k2},2]

Out[39]=

_

k1

i _

k2

j

Out[40]=

1 3

_

δ

ij ( _

k1·

_

k2)

Out[41]=

1 2

_

k1

i _

k2

j− 1

2

_

k2

i _

k1

j

Out[42]= − 1

3

_

δ

ij ( _

k1·

_

k2)+ 1 2

_

k2

i _

k1

j+ 1

2

_

k1

i _

k2

j

Currently, FEYNONIUM can project out the J = 0, 1, 2 components of tensors up to rank 5. If needed, higher ranks or J-values can be easily added to the code.

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Electromagnetic χcJ production 37 / 40

SLIDE 38

Automatic nonrelativistic calculation with FEYNCALC FeynOnium

Finally, in complex nonrelativistic calculations one often encounters spurious terms that vanish by the virtue of the 3-dimensional Schouten identity εijkpl − εjklpi + εklipj − εlijpk = 0, where p is an arbitrary Cartesian vector. In general, it is very difficult to apply this identity in a systematic way, which is why FEYNONIUM features a tool that facilitates this task. FMCartesianSchoutenBruteForce tries out all possible combinations that can be formed out of the given list of Cartesian vectors and checks if this helps to reduce the number of terms in the expression. This is illustrated in the following simple example, where we have (ˆ k · ˆ q)

(ˆ

k · ˆ q)(ˆ k · (ε∗(k1) × ε∗(k2))) − (ˆ q · (ε∗(k1) × ε∗(k2)))

,

with (ˆ k · ˆ k) = (ˆ q · ˆ q) = 1, (ˆ k · ε∗(k1)) = (ˆ k · ε∗(k2)) = 0.

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Electromagnetic χcJ production 38 / 40

SLIDE 39

Automatic nonrelativistic calculation with FEYNCALC FeynOnium

Here FMCartesianSchoutenBruteForce can readily find the correct version of Schouten identity that reduces the given expression to zero.

In[43]:= FCClearScalarProducts[];

CSP[khat,Polarization[k1,−I,Transversality→ → →True]]=0; CSP[khat,Polarization[k2,−I,Transversality→ → →True]]=0; CSP[khat,khat]=1; CSP[qhat,qhat]=1; CSP[khat,qhat] (CSP[khat,qhat] CLC[][khat,Polarization[k1,−I,Transversality→ → →True], Polarization[k2,−I,Transversality→ → →True]]− CLC[][qhat,Polarization[k1,−I,Transversality→ → →True],Polarization[k2,−I,Transversality → → →True]]) FMCartesianSchoutenBruteForce[%,{khat,khat,qhat,Polarization[k1,−I,Transversality → → →True],Polarization[k2,−I,Transversality→ → →True]}]

Out[43]= (

_

khat·

_

qhat) ((

_

khat·

_

qhat)

_

ǫ

_ khat_ ε∗(k1)_ ε∗(k2)− _

ǫ

_ qhat_ ε∗(k1)_ ε∗(k2))

FMCartesianSchoutenBruteForce: 2 terms were removed using {

_

ǫ

_ qhat_ ε∗(k1)_ ε∗(k2)→( _

khat·

_

qhat)

_

ǫ

_ khat_ ε∗(k1)_ ε∗(k2)}

Of course, the function is also useful in more complicated cases.

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Electromagnetic χcJ production 39 / 40

SLIDE 40

Summary and Outlook

◮ We studied CO O(α0 sv2) corrections to the electromagnetic quarkonium production

process e+e− → χcJ + γ using NRQCD factorization at the amplitude level.

◮ Operators that appear at this order contain chromoelectric field E and thus

contribute only through subleading Fock states |Q¯ Qg.

◮ We determined short distance coefficients multiplying these operators by using

threshold expansion technique of Braaten and Chen

◮ The matching on the QCD side was performed both in the rest frame and in the CM

frame of the heavy quarkonium and the results agree.

◮ Cancellation of IR singularities in QCD and NRQCD was checked explicitly. ◮ This process could be, in principle, measured at B-factories, but so far no such

measurements exist.

◮ Similar investigations can be carried out also for other suitable processes in

NRQCD.

◮ Many ideas developed during this project will help us to make FEYNCALC more

useful for NR calculations.

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Electromagnetic χcJ production 40 / 40

SLIDE 41

Backup

Results of [Chung et al., 2008] I

Input parameters: √s = 10.58 GeV, mc = 1.4 ± 0.2 GeV, mχc0 = 3.41475 GeV, mχc1 = 3.51066 GeV, mχc2 = 3.55620 GeV

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Electromagnetic χcJ production 41 / 40

SLIDE 42

Backup

Results of [Chung et al., 2008] II

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Electromagnetic χcJ production 42 / 40

SLIDE 43

Backup

Results of [Xu et al., 2014] I

mc = 1.5 ± 0.1 GeV

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Electromagnetic χcJ production 43 / 40

SLIDE 44

Backup

Results of [Xu et al., 2014] II

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Electromagnetic χcJ production 44 / 40

SLIDE 45

Backup

Results of [Xu et al., 2014] III

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Electromagnetic χcJ production 45 / 40

SLIDE 46

Backup

Results of [Braguta, 2010]

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Electromagnetic χcJ production 46 / 40

SLIDE 47

Backup

Velocity scaling rules in NRQCD (BBL counting)

◮ αs(mv) ∼ v ◮ ψ ∼ (mv)3/2 ◮ χ ∼ (mv)3/2 ◮ Dt ∼ (mv)2 ◮ D ∼ mv ◮ gE ∼ m2v3 ◮ gB ∼ m2v4 ◮ gφ ∼ mv2 (in Coulomb gauge) ◮ gA ∼ mv3 (in Coulomb gauge) ◮ |H H| ∼ 1 (non-relativistic normalization) normalization)

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Electromagnetic χcJ production 47 / 40

SLIDE 48

Backup

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Phys. Rev. Lett., 33, 1404–1406.

Augustin, J. E. et al. (1974). Discovery of a Narrow Resonance in e+e− Annihilation.

Phys. Rev. Lett., 33, 1406–1408.

[Adv. Exp. Phys.5,141(1976)]. Bodwin, G. T., Braaten, E., & Lepage, G. P . (1995). Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium.

Phys. Rev., D51, 1125–1171.

[Erratum: Phys. Rev.D55,5853(1997)]. Bodwin, G. T. & Petrelli, A. (2002). Order-v4 corrections to S-wave quarkonium decay.

Phys. Rev. D, 66, 094011.

[Erratum: Phys. Rev.D87,no.3,039902(2013)]. Bolotin, D. A. & Poslavsky, S. V. (2013). Introduction to Redberry: a computer algebra system designed for tensor manipulation.

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Electromagnetic χcJ production 40 / 40

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Backup

Braaten, E. & Chen, Y.-Q. (1996a). Helicity decomposition for inclusive J / psi production.

Phys. Rev., D54, 3216–3227.

Braaten, E. & Chen, Y.-Q. (1996b). Helicity Decomposition for Inclusive J/ψ Production.

Phys. Rev. D, 54, 3216–3227.

Braaten, E. & Chen, Y.-Q. (1998). Renormalons in electromagnetic annihilation decays of quarkonium.

Phys. Rev., D57, 4236–4253.

[Erratum: Phys. Rev.D59,079901(1999)]. Braaten, E. & Lee, J. (2003). Exclusive double charmonium production from e+ e- annihilation into a virtual photon.

Phys. Rev., D67, 054007.

[Erratum: Phys. Rev.D72,099901(2005)]. Braguta, V. V. (2010). Exclusive C=+ charmonium production in e+e− → H + γ at B-factories within light cone formalism. Phys.Rev.D, 82:074009,2010.

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SLIDE 50

Backup

Brambilla, N., Mereghetti, E., & Vairo, A. (2006). Electromagnetic quarkonium decays at order v**7. JHEP, 08, 039. [Erratum: JHEP04,058(2011)]. Carlson, C. E. & Suaya, R. (1976). Hadronic Production of psi/J Mesons.

Phys. Rev., D14, 3115.

Caswell, W. E. & Lepage, G. P . (1986). Effective Lagrangians for Bound State Problems in QED, QCD, and Other Field Theories.

Phys. Lett., 167B, 437–442.

Chao, K.-T., He, Z.-G., Li, D., & Meng, C. (2013). Search for C = + charmonium states in e+e− → γ + X at BEPCII/BESIII. Chung, H. S., Lee, J., & Yu, C. (2008). Exclusive heavy quarkonium + gamma production from e+ e- annihilation into a virtual photon.

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Einhorn, M. B. & Ellis, S. D. (1975). Hadronic Production of the New Resonances: Probing Gluon Distributions.

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Backup

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