# Valence quark transversity at small $x$

## Abstract

In our previous work we established a formalism which allows one to determine the small-x asymptotics of any transverse momentum-dependent parton distribution function (TMD PDF) of the proton at small values of strong coupling. In this paper we apply this formalism to the valence quark transversity TMD. We relate the valence quark transversity to the transversely polarized dipole scattering amplitude, written in terms of the fundamental transversely polarized “Wilson line” operator, an expression for which we derive explicitly as well. We then write down the evolution equation for the transversely polarized dipole amplitude. Solving that equation we arrive at the following small-x asymptotics of the valence quark transversity in the large-N_{c} limit: $${h}_{1T}^{v}(x,{k}_{T}^{2}){\sim}{h}_{1T}^{{\perp}v}(x,{k}_{T}^{2}){\sim}{(\frac{1}{x})}^{{-}1+2\sqrt{\frac{{{\alpha}}_{s}{N}_{c}}{2{\pi}}}}$$. This result is in agreement with one of the two possible small-$x$ asymptotics for the transverse structure function found previously by Kirschner, Mankiewicz, Schäfer, and Szymanowski [Z. Phys. C 74, 501 (1997)].

- Authors:

- Publication Date:

- Research Org.:
- The Ohio State Univ., Columbus, OH (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP)

- OSTI Identifier:
- 1504280

- Alternate Identifier(s):
- OSTI ID: 1604160

- Grant/Contract Number:
- SC0004286; AC52-06NA25396

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Name: Physical Review D Journal Volume: 99 Journal Issue: 5; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 73 NUCLEAR PHYSICS AND RADIATION PHYSICS

### Citation Formats

```
Kovchegov, Yuri V., and Sievert, Matthew D. Valence quark transversity at small x. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.99.054033.
```

```
Kovchegov, Yuri V., & Sievert, Matthew D. Valence quark transversity at small x. United States. doi:10.1103/PhysRevD.99.054033.
```

```
Kovchegov, Yuri V., and Sievert, Matthew D. Fri .
"Valence quark transversity at small x". United States. doi:10.1103/PhysRevD.99.054033.
```

```
@article{osti_1504280,
```

title = {Valence quark transversity at small x},

author = {Kovchegov, Yuri V. and Sievert, Matthew D.},

abstractNote = {In our previous work we established a formalism which allows one to determine the small-x asymptotics of any transverse momentum-dependent parton distribution function (TMD PDF) of the proton at small values of strong coupling. In this paper we apply this formalism to the valence quark transversity TMD. We relate the valence quark transversity to the transversely polarized dipole scattering amplitude, written in terms of the fundamental transversely polarized “Wilson line” operator, an expression for which we derive explicitly as well. We then write down the evolution equation for the transversely polarized dipole amplitude. Solving that equation we arrive at the following small-x asymptotics of the valence quark transversity in the large-Nc limit: ${h}_{1T}^{v}(x,{k}_{T}^{2}){\sim}{h}_{1T}^{{\perp}v}(x,{k}_{T}^{2}){\sim}{(\frac{1}{x})}^{{-}1+2\sqrt{\frac{{{\alpha}}_{s}{N}_{c}}{2{\pi}}}}$. This result is in agreement with one of the two possible small-$x$ asymptotics for the transverse structure function found previously by Kirschner, Mankiewicz, Schäfer, and Szymanowski [Z. Phys. C 74, 501 (1997)].},

doi = {10.1103/PhysRevD.99.054033},

journal = {Physical Review D},

number = 5,

volume = 99,

place = {United States},

year = {2019},

month = {3}

}

DOI: 10.1103/PhysRevD.99.054033

*Citation information provided by*

Web of Science

Web of Science

Works referenced in this record:

##
Saturation physics and deuteron–gold collisions at RHIC

journal, January 2006

- Jalilianmarian, J.; Kovchegov, Y.
- Progress in Particle and Nuclear Physics, Vol. 56, Issue 1

##
Calculating TMDs of a large nucleus: Quasi-classical approximation and quantum evolution

journal, February 2016

- Kovchegov, Yuri V.; Sievert, Matthew D.
- Nuclear Physics B, Vol. 903

##
First Monte Carlo Global Analysis of Nucleon Transversity with Lattice QCD Constraints

journal, April 2018

- Lin, H. -W.; Melnitchouk, W.; Prokudin, A.
- Physical Review Letters, Vol. 120, Issue 15

##
Baryon stopping and valence quark distribution at small x

journal, January 2004

- Itakura, Kazunori; Kovchegov, Yuri V.; McLerran, Larry
- Nuclear Physics A, Vol. 730, Issue 1-2

##
Exclusive processes in perturbative quantum chromodynamics

journal, November 1980

- Lepage, G. Peter; Brodsky, Stanley J.
- Physical Review D, Vol. 22, Issue 9

##
Chiral-odd parton distributions and Drell-Yan processes

journal, May 1992

- Jaffe, R. L.; Ji, Xiangdong
- Nuclear Physics B, Vol. 375, Issue 3

##
Small- $x$ Asymptotics of the Quark Helicity Distribution

journal, January 2017

- Kovchegov, Yuri V.; Pitonyak, Daniel; Sievert, Matthew D.
- Physical Review Letters, Vol. 118, Issue 5

##
Double logarithmic asymptotics and regge singularities of quark amplitudes with flavour exchange

journal, March 1983

- Kirschner, R.; Lipatov, L. N.
- Nuclear Physics B, Vol. 213, Issue 1

##
First Extraction of Transversity from a Global Analysis of Electron-Proton and Proton-Proton Data

journal, May 2018

- Radici, Marco; Bacchetta, Alessandro
- Physical Review Letters, Vol. 120, Issue 19

##
Solution of the NLO BFKL equation and a strategy for solving the all-order BFKL equation

journal, June 2013

- Chirilli, Giovanni A.; Kovchegov, Yuri V.
- Journal of High Energy Physics, Vol. 2013, Issue 6

##
Quantum chromodynamics and other field theories on the light cone

journal, August 1998

- Brodsky, S.
- Physics Reports, Vol. 301, Issue 4-6

##
Studying the perturbative Reggeon

journal, January 2000

- Griffiths, S.; Ross, D. A.
- The European Physical Journal C, Vol. 12, Issue 2

##
Small-x asymptotics of the gluon helicity distribution

journal, October 2017

- Kovchegov, Yuri V.; Pitonyak, Daniel; Sievert, Matthew D.
- Journal of High Energy Physics, Vol. 2017, Issue 10

##
The Color Glass Condensate

journal, November 2010

- Gelis, Francois; Iancu, Edmond; Jalilian-Marian, Jamal
- Annual Review of Nuclear and Particle Science, Vol. 60, Issue 1

##
Gluon saturation and initial conditions for relativistic heavy ion collisions

journal, May 2014

- Albacete, J. L.; Marquet, C.
- Progress in Particle and Nuclear Physics, Vol. 76

##
Evolution at small : The color glass condensate

journal, October 2005

- Weigert, H.
- Progress in Particle and Nuclear Physics, Vol. 55, Issue 2

##
Small- x asymptotics of the quark helicity distribution: Analytic results

journal, September 2017

- Kovchegov, Yuri V.; Pitonyak, Daniel; Sievert, Matthew D.
- Physics Letters B, Vol. 772

##
Reggeon interactions in perturbative QCD

journal, September 1995

- Kirschner, R.
- Zeitschrift für Physik C Particles and Fields, Vol. 65, Issue 3

##
Transversity and Collins functions from SIDIS and ${e}^{+}{e}^{-}$ data

journal, March 2007

- Anselmino, M.; Boglione, M.; D’Alesio, U.
- Physical Review D, Vol. 75, Issue 5

##
Regge asymptotics of scattering with flavour exchange in QCD

journal, September 1995

- Kirschner, R.
- Zeitschrift für Physik C Particles and Fields, Vol. 67, Issue 3

##
Positivity Constraints for Spin-Dependent Parton Distributions

journal, February 1995

- Soffer, Jacques
- Physical Review Letters, Vol. 74, Issue 8

##
Regge asymptotics of scattering amplitudes in the logarithmic approximation of QCD

journal, March 1986

- Kirschner, R.
- Zeitschrift für Physik C Particles and Fields, Vol. 31, Issue 1

##
Relations between generalized and transverse momentum dependent parton distributions

journal, August 2007

- Meißner, S.; Metz, A.; Goeke, K.
- Physical Review D, Vol. 76, Issue 3

##
New mechanism for generating a single transverse spin asymmetry

journal, August 2012

- Kovchegov, Yuri V.; Sievert, Matthew D.
- Physical Review D, Vol. 86, Issue 3

##
Transverse polarisation of quarks in hadrons

journal, March 2002

- Barone, Vincenzo; Drago, Alessandro; Ratcliffe, Philip G.
- Physics Reports, Vol. 359, Issue 1-2

##
Quark-antiquark exchange in γ* γ* scattering

journal, September 2003

- Bartels, Jochen; Lublinsky, Michael
- Journal of High Energy Physics, Vol. 2003, Issue 09

##
Small- $x$ helicity evolution: An operator treatment

journal, March 2019

- Kovchegov, Yuri V.; Sievert, Matthew D.
- Physical Review D, Vol. 99, Issue 5

##
-Broadening and production processes versus dipole/quadrupole amplitudes at next-to-leading order

journal, November 2012

- Mueller, A. H.; Munier, S.
- Nuclear Physics A, Vol. 893

##
Helicity evolution at small $x$ : Flavor singlet and nonsinglet observables

journal, January 2017

- Kovchegov, Yuri V.; Pitonyak, Daniel; Sievert, Matthew D.
- Physical Review D, Vol. 95, Issue 1

##
First extraction of valence transversities in a collinear framework

journal, March 2013

- Bacchetta, Alessandro; Courtoy, A.; Radici, Marco
- Journal of High Energy Physics, Vol. 2013, Issue 3

##
Measuring the transverse polarization of quarks in the proton

journal, September 1992

- Cortes, J. L.; Pire, B.; Ralston, J. P.
- Zeitschrift für Physik C Particles and Fields, Vol. 55, Issue 3

##
Helicity evolution at small x

journal, January 2016

- Kovchegov, Yuri V.; Pitonyak, Daniel; Sievert, Matthew D.
- Journal of High Energy Physics, Vol. 2016, Issue 1

##
Production of dimuons from high-energy polarized proton-proton collisions

journal, May 1979

- Ralston, John P.; Soper, Davidson E.
- Nuclear Physics B, Vol. 152, Issue 1

##
Small x behaviour of the chirally-odd parton distribution h 1 ( x , Q 2 )

journal, May 1997

- Kirschner, R.; Mankiewicz, L.; Schäfer, A.
- Zeitschrift f�r Physik C Particles and Fields, Vol. 74, Issue 3