TF InclusiveDeltaDoverD - Revision history

Foretony at 19:03, 22 September 2018

2018-09-22T19:03:49Z

Foretony at 19:03, 22 September 2018

2018-09-22T19:03:09Z

Foretony at 18:44, 22 September 2018

2018-09-22T18:44:42Z

Foretony at 18:44, 22 September 2018

2018-09-22T18:44:07Z

Foretony at 18:25, 22 September 2018

2018-09-22T18:25:36Z

Foretony at 18:25, 22 September 2018

2018-09-22T18:25:08Z

Foretony at 18:22, 22 September 2018

2018-09-22T18:22:34Z

Foretony at 18:21, 22 September 2018

2018-09-22T18:21:45Z

Foretony at 18:21, 22 September 2018

2018-09-22T18:21:32Z

Foretony at 16:44, 22 September 2018

2018-09-22T16:44:45Z

@@ Line 37: / Line 37: @@
-<math> A_1(x,Q^2) \equiv \frac{\sigma_{1/2}^T - \sigma_{3/2}^T}{\sigma_{1/2}^T - \sigma_{3/2}^T} = \frac{g_1(x,Q^2) - \frac{Q^2}{\nu^2} g_2(x,Q^2)}{F_1(x,Q^2)} \approx \frac{q_1(x,Q^2)}{F_1(x,Q^2)}</math>
+<math> A_1(x,Q^2) \equiv \frac{\sigma_{1/2}^T - \sigma_{3/2}^T}{\sigma_{1/2}^T - \sigma_{3/2}^T} = \frac{g_1(x,Q^2) - \frac{Q^2}{\nu^2} g_2(x,Q^2)}{F_1(x,Q^2)} \approx \frac{g_1(x,Q^2)}{F_1(x,Q^2)}</math>
@@ Line 50: / Line 50: @@
 Then using the above approximation for A_1
-<math> \frac{\Delta d}{d} = \frac{4}{15} \left ( 4 + \frac{u}{d}\right ) \frac{q_1^n}{F_1^n} + \frac{1}{15} \left ( 1 + 4\frac{u}{d}\right ) \frac{q_1^p}{F_1^p}</math>
+<math> \frac{\Delta d}{d} = \frac{4}{15} \left ( 4 + \frac{u}{d}\right ) \frac{g_1^n}{F_1^n} + \frac{1}{15} \left ( 1 + 4\frac{u}{d}\right ) \frac{g_1^p}{F_1^p}</math>

← Older revision		Revision as of 19:03, 22 September 2018
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	<math> A_1^p = \frac{4\Delta u + \Delta d}{4u+d} \;\;\;\;\ A_1^n = \frac{\Delta u + 4\Delta d}{u+4d}</math> <ref>https://arxiv.org/abs/hep-ph/9809255 PHYSICAL REVIEW D, VOLUME 59, 034013 Valence quark spin distribution functions Nathan Isgur , https://arxiv.org/abs/hep-ph/0411005 The Spin Structure of the Proton Steven D. Bass Rev.Mod.Phys.77:1257-1302,2005</ref>		<math> A_1^p = \frac{4\Delta u + \Delta d}{4u+d} \;\;\;\;\ A_1^n = \frac{\Delta u + 4\Delta d}{u+4d}</math> <ref>https://arxiv.org/abs/hep-ph/9809255 PHYSICAL REVIEW D, VOLUME 59, 034013 Valence quark spin distribution functions Nathan Isgur , https://arxiv.org/abs/hep-ph/0411005 The Spin Structure of the Proton Steven D. Bass Rev.Mod.Phys.77:1257-1302,2005</ref>
		+
		+	One can use the two equations above and solve for the polarized quark distributions assuming the unpolarized are known and get , for example
		+
		+	<math> \frac{\Delta d}{d} = \frac{4}{15} \left ( 4 + \frac{u}{d}\right ) A_1^n + \frac{1}{15} \left ( 1 + 4\frac{u}{d}\right ) A_1^p</math>
		+
		+	Then using the above approximation for A_1
		+
		+	<math> \frac{\Delta d}{d} = \frac{4}{15} \left ( 4 + \frac{u}{d}\right ) \frac{q_1^n}{F_1^n} + \frac{1}{15} \left ( 1 + 4\frac{u}{d}\right ) \frac{q_1^p}{F_1^p}</math>
		+

	<math>g_1^d \approx \left ( 1 - 1.5 \omega_D \right ) \left ( g_1^n + g_1^p \right )</math><ref> Eq. 28 from https://arxiv.org/abs/1505.07877 which is based on https://arxiv.org/abs/0809.4308</ref>		<math>g_1^d \approx \left ( 1 - 1.5 \omega_D \right ) \left ( g_1^n + g_1^p \right )</math><ref> Eq. 28 from https://arxiv.org/abs/1505.07877 which is based on https://arxiv.org/abs/0809.4308</ref>

@@ Line 42: / Line 42: @@
 In the non-relativistic constituent quark model
-<math> A_1^p = \frac{4\Delta u + \Delta d}{4u+d} \;\;\;\;\ A_1^n = \frac{\Delta u + 4\Delta d}{u+4d}</math> <ref>https://arxiv.org/abs/hep-ph/9809255 PHYSICAL REVIEW D, VOLUME 59, 034013 Valence quark spin distribution functions Nathan Isgur  , https://arxiv.org/abs/hep-ph/0411005 The Spin Structure of the Proton
+<math> A_1^p = \frac{4\Delta u + \Delta d}{4u+d} \;\;\;\;\ A_1^n = \frac{\Delta u + 4\Delta d}{u+4d}</math> <ref>https://arxiv.org/abs/hep-ph/9809255 PHYSICAL REVIEW D, VOLUME 59, 034013 Valence quark spin distribution functions Nathan Isgur  , https://arxiv.org/abs/hep-ph/0411005 The Spin Structure of the Proton Steven D. Bass Rev.Mod.Phys.77:1257-1302,2005</ref>
 <math>g_1^d \approx \left ( 1 - 1.5 \omega_D \right ) \left ( g_1^n + g_1^p \right )</math><ref> Eq. 28 from https://arxiv.org/abs/1505.07877 which is based on  https://arxiv.org/abs/0809.4308</ref>

@@ Line 42: / Line 42: @@
 In the non-relativistic constituent quark model
-<math> A_1^p = \frac{4\Delta u + \Delta d}{4u+d} \;\;\;\;\ A_1^n = \frac{\Delta u + 4\Delta d}{u+4d}</math>
+<math> A_1^p = \frac{4\Delta u + \Delta d}{4u+d} \;\;\;\;\ A_1^n = \frac{\Delta u + 4\Delta d}{u+4d}</math> <ref>https://arxiv.org/abs/hep-ph/9809255 PHYSICAL REVIEW D, VOLUME 59, 034013 Valence quark spin distribution functions Nathan Isgur  , https://arxiv.org/abs/hep-ph/0411005 The Spin Structure of the Proton
 <math>g_1^d \approx \left ( 1 - 1.5 \omega_D \right ) \left ( g_1^n + g_1^p \right )</math><ref> Eq. 28 from https://arxiv.org/abs/1505.07877 which is based on  https://arxiv.org/abs/0809.4308</ref>

@@ Line 42: / Line 42: @@
 In the non-relativistic constituent quark model
-<math> A_1^p = \frac{4\Delta u + \Delta d}{4u+d} \;\;\;\;\ A_1^n = \frac{4\Delta u + 4\Delta d}{u+4d}</math>
+<math> A_1^p = \frac{4\Delta u + \Delta d}{4u+d} \;\;\;\;\ A_1^n = \frac{\Delta u + 4\Delta d}{u+4d}</math>
 <math>g_1^d \approx \left ( 1 - 1.5 \omega_D \right ) \left ( g_1^n + g_1^p \right )</math><ref> Eq. 28 from https://arxiv.org/abs/1505.07877 which is based on  https://arxiv.org/abs/0809.4308</ref>

← Older revision		Revision as of 18:25, 22 September 2018
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	<math> A_1(x,Q^2) \equiv \frac{\sigma_{1/2}^T - \sigma_{3/2}^T}{\sigma_{1/2}^T - \sigma_{3/2}^T} = \frac{g_1(x,Q^2) - \frac{Q^2}{\nu^2} g_2(x,Q^2)}{F_1(x,Q^2)} \approx \frac{q_1(x,Q^2)}{F_1(x,Q^2)}</math>		<math> A_1(x,Q^2) \equiv \frac{\sigma_{1/2}^T - \sigma_{3/2}^T}{\sigma_{1/2}^T - \sigma_{3/2}^T} = \frac{g_1(x,Q^2) - \frac{Q^2}{\nu^2} g_2(x,Q^2)}{F_1(x,Q^2)} \approx \frac{q_1(x,Q^2)}{F_1(x,Q^2)}</math>

		+
		+	In the non-relativistic constituent quark model
		+
		+	<math> A_1^p = \frac{4\Delta u + \Delta d}{4u+d} \;\;\;\;\ A_1^n = \frac{4\Delta u + 4\Delta d}{u+4d}</math>

	<math>g_1^d \approx \left ( 1 - 1.5 \omega_D \right ) \left ( g_1^n + g_1^p \right )</math><ref> Eq. 28 from https://arxiv.org/abs/1505.07877 which is based on https://arxiv.org/abs/0809.4308</ref>		<math>g_1^d \approx \left ( 1 - 1.5 \omega_D \right ) \left ( g_1^n + g_1^p \right )</math><ref> Eq. 28 from https://arxiv.org/abs/1505.07877 which is based on https://arxiv.org/abs/0809.4308</ref>

← Older revision		Revision as of 18:22, 22 September 2018
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−	<math> A_1(x,Q^2) \equiv \frac{\sigma_{1/2}^T - \sigma_{3/2}^T}{\sigma_{1/2}^T - \sigma_{3/2}^T} = \frac{g_1(x,Q^2) - \frac{Q^2}{\nu^2} g_2(x,Q^2)}{F_1(x,Q^2)}</math>	+	<math> A_1(x,Q^2) \equiv \frac{\sigma_{1/2}^T - \sigma_{3/2}^T}{\sigma_{1/2}^T - \sigma_{3/2}^T} = \frac{g_1(x,Q^2) - \frac{Q^2}{\nu^2} g_2(x,Q^2)}{F_1(x,Q^2)} \approx \frac{q_1(x,Q^2)}{F_1(x,Q^2)}</math>

← Older revision		Revision as of 18:21, 22 September 2018
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−	<math> A_1(x,Q^2) equiv \frac{\sigma_{1/2}^T - \sigma_{3/2}^T}{\sigma_{1/2}^T - \sigma_{3/2}^T} = \frac{g_1(x,Q^2) - \frac{Q^2}{\nu^2} g_2(x,Q^2)}{F_1(x,Q^2)}</math>	+	<math> A_1(x,Q^2) \equiv \frac{\sigma_{1/2}^T - \sigma_{3/2}^T}{\sigma_{1/2}^T - \sigma_{3/2}^T} = \frac{g_1(x,Q^2) - \frac{Q^2}{\nu^2} g_2(x,Q^2)}{F_1(x,Q^2)}</math>

← Older revision		Revision as of 18:21, 22 September 2018
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	<math> g_1^p(x) =\frac{1}{2} \left [\frac{4}{9} \Delta u(x) + \frac{1}{9} \Delta d(x)\right ] \;\;\;\;\;</math> <math> g_1^n(x)=\frac{1}{2} \left [\frac{4}{9} \Delta d(x) + \frac{1}{9} \Delta u(x)\right ]</math>		<math> g_1^p(x) =\frac{1}{2} \left [\frac{4}{9} \Delta u(x) + \frac{1}{9} \Delta d(x)\right ] \;\;\;\;\;</math> <math> g_1^n(x)=\frac{1}{2} \left [\frac{4}{9} \Delta d(x) + \frac{1}{9} \Delta u(x)\right ]</math>
		+
		+
		+
		+	<math> A_1(x,Q^2) equiv \frac{\sigma_{1/2}^T - \sigma_{3/2}^T}{\sigma_{1/2}^T - \sigma_{3/2}^T} = \frac{g_1(x,Q^2) - \frac{Q^2}{\nu^2} g_2(x,Q^2)}{F_1(x,Q^2)}</math>

@@ Line 13: / Line 13: @@
 using the above definition to define the proton and neutron unpolarized structure function :
-<math> F_1^p(x) \equiv \frac{1}{2} \sum_q e_i^2 q_i^p(x) = \frac{1}{2}\left [  \left( \frac{2}{3} \right)^2 u^p(x)+ \left( \frac{-1}{3} \right)^2 d^p(x)\right ] =\frac{1}{2} \left [\frac{4}{9}u(x) + \frac{1}{9}d(x)\right ]</math>
+<math> F_1^p(x) \equiv \frac{1}{2} \sum_q e_i^2 q_i^p(x) = \frac{1}{2}\left [  \left( \frac{2}{3} \right)^2 u^p(x)+ \left( \frac{-1}{3} \right)^2 d^p(x)\right ] =\frac{1}{2} \left [\frac{4}{9}u^p(x) + \frac{1}{9}d^p(x)\right ]</math>
-<math> F_1^n(x) \equiv \frac{1}{2} \sum_q e_i^2 q_i^p(x) = \frac{1}{2}\left [  \left( \frac{2}{3} \right)^2 u^n(x)+ \left( \frac{-1}{3} \right)^2 d^n(x)\right ] =\frac{1}{2} \left [\frac{4}{9}d(x) + \frac{1}{9}u(x)\right ]</math>
+<math> F_1^n(x) \equiv \frac{1}{2} \sum_q e_i^2 q_i^p(x) = \frac{1}{2}\left [  \left( \frac{2}{3} \right)^2 u^n(x)+ \left( \frac{-1}{3} \right)^2 d^n(x)\right ] =\frac{1}{2} \left [\frac{4}{9}u^n(x) + \frac{1}{9}d^n(x)\right ]</math>