Determination of nucleon sigma terms I Lukas Varnhorst for the BMW - PowerPoint PPT Presentation

Determination of nucleon sigma terms I Lukas Varnhorst for the BMW collaboration University of Wuppertal Faculty 4 - Department of Physics BMW collaboration Lukas Varnhorst for BMW collaboration Sigma terms I 1 von 17

Introduction 1 Lattice setup 2 Analysis strategy 3 Crosscheck 4 Results 5 Lukas Varnhorst for BMW collaboration Sigma terms I 2 von 17

Introduction Nucleon sigma terms are defined as σ qN = m q � N | ¯ qq | N � − m q � 0 | ¯ qq | 0 � . They can be related — via the Feynmann Hellmann theorem — to the quark mass derivative of the nucleon mass: ∂ M N σ qN = m q . ∂ m q They are closely related to the contribution of individual quark flavors to the nucleon masses. The are relevant for dark matter detection experiments. Lukas Varnhorst for BMW collaboration Sigma terms I 3 von 17

Introduction � ∂ M N � uu + ¯ uu + ¯ σ udN = m ud � N | ¯ dd | N � − m ud � 0 | ¯ dd | 0 � = m ud � ∂ m ud � m s , a � ∂ M N � σ sN = m s � N | ¯ ss | N � − m s � 0 | ¯ ss | 0 � = m s � ∂ m s � m ud , a Light and strange sigma terms can be related to ”mesonic” sigma terms via a transformation matrix: �  �  ∂ M 2 ∂ M 2 � m ud � m ud K χ � π � � � � � M 2 ∂ m ud M 2 ∂ m ud σ udN � σ π N  m s , a  π K χ m s , a =  �  � σ sN ∂ M 2 σ K χ N ∂ M 2 �   m s � m s K χ � π � M 2 ∂ m s M 2 ∂ m s � m ud , a K χ π m ud , a with � � � ∂ M N ∂ M N � σ π N = M 2 σ K χ N = M 2 � and . � π π � ∂ M 2 ∂ M 2 � � π M 2 K χ , a K χ M 2 π , a M 2 K χ = 2 M 2 K − M 2 π Lukas Varnhorst for BMW collaboration Sigma terms I 4 von 17

Introduction =: J � �� � �   � ∂ M 2 ∂ M 2 � m ud m ud � K χ � σ π N π � � σ udN � � M 2 � M 2 ∂ m ud ∂ m ud �  m s , a  K χ π m s , a =  �  � ∂ M 2 σ sN σ K χ N ∂ M 2 �   m s � m s K χ � π � M 2 ∂ m s M 2 ∂ m s � �� � � m ud , a π K χ m ud , a next talk � �� � this talk Staggered fermions are well suited to determine J : Quark masses are easy to define Only pseudoscalar masses are required Available configurations bracket the physical point Lukas Varnhorst for BMW collaboration Sigma terms I 5 von 17

Lattice setup We used staggered N f = 2 + 1 + 1 configurations with tree-level improved Symmanik gauge action and a 2-stout smeared fermion action. 1.06 1.04 2 M 2( φ ) 2 M 2 π π 1.02 K − 1 − 1 M 2 M 2( φ ) K 1.00 β = 3 . 8400 (7) 0.98 β = 3 . 9200 (2) β = 4 . 0126 (3) 0.98 1.00 1.02 1.04 M 2 π M 2( φ ) π Lukas Varnhorst for BMW collaboration Sigma terms I 6 von 17

Analysis strategy We expand the pion and reduced kaon mass around the physical point like c 0 + c 1 , ud ( m ud − m ( φ ) ud ) + c 1 , s ( m s − m ( φ ) ) + . . . s But how to define m ( φ ) ud and m ( φ ) ? s Lukas Varnhorst for BMW collaboration Sigma terms I 7 von 17

Analysis strategy Physical values of the M ( φ ) depend on the gauge coupling: q c 0 + c 1 , ud ( m ud − m ( φ ) ud [ β ]) + c 1 , s ( m s − m ( φ ) [ β ]) + . . . s Lukas Varnhorst for BMW collaboration Sigma terms I 8 von 17

Analysis strategy The ratio r = m s / m ud of strange and light quark masses is a physical observable. We use it to rewrite the expansion as � � � � m ud r m s c 0 + c ′ + c ′ − 1 − 1 + . . . 1 , ud 1 , s m ( φ ) m ( φ ) [ β ] [ β ] s s We treat m ( φ ) [ β ] as a fit parameter per gauge coupling and assume s r = r 0 + r 1 a 2 + O ( a 4 ) Up to higher order correction c 1 , ud and c 1 , s are the matrix elements of J . Lukas Varnhorst for BMW collaboration Sigma terms I 9 von 17

Analysis strategy Our ensambles feature a constant m c / m s = 11 . 85. Using the expansion of the form � � � � m ud r m s c 0 + c ′ + c ′ − 1 − 1 + . . . 1 , ud 1 , s m ( φ ) m ( φ ) [ β ] [ β ] s s allows to extract derivative like e.g. � ∂ M 2 � π m s � ∂ m s � m ud , m c / m s , a Hence we had to introuce a term proportional to m c / m s to our fit function and use the relation � � � ∂ M 2 ∂ M 2 ∂ M 2 − m c � � � π π π m s = m s � � � ∂ m s ∂ m s m s ∂ ( m c / m s ) � � � m ud , m c , a m ud , m c / m s , a m ud , m s , a We generated a dedicated ensamble with m c / m s = 11 . 45 so that we are sensitive on the m c / m s direction. Lukas Varnhorst for BMW collaboration Sigma terms I 10 von 17

Analysis strategy We used the expansion up to quadratic order and included a 2 corrections on the leading terms: 1 , ud + d 1 , ud a 2 )∆ ud + ( c ′ 1 , s + d 1 , s a 2 )∆ s + c 2 , ud , s ∆ ud ∆ s c 0 + ( c ′ + c 2 , ud ∆ 2 ud + c 2 , s ∆ 2 s + c c / s ∆ c / s with ∆ ud = m ud ( r 0 + r 1 a 2 ) − 1 , m ( φ ) [ β ] s m s ∆ s = − 1 , m ( φ ) [ β ] s � m c � ( φ ) ∆ c / s = m c − . m s m s We use fit function of this for to simultaneous fit M 2 π , M 2 K χ and for scale setting f π . Lukas Varnhorst for BMW collaboration Sigma terms I 11 von 17

Analysis strategy Dependence of M 2 π on the light (left) and strange (right) quark mass: 19000 β = 4 . 0126 β = 4 . 0126 β = 3 . 84 18350 β = 3 . 84 18800 β = 3 . 92 β = 3 . 92 18300 18600 18250 π /MeV 2 π /MeV 2 18400 18200 M 2 M 2 18200 18150 18000 18100 17800 18050 0.98 0.99 1.00 1.01 1.02 1.03 1.04 0.98 1.00 1.02 1.04 1.06 m ud / ( m φ m s /m φ s /r ) s Lukas Varnhorst for BMW collaboration Sigma terms I 12 von 17

Analysis strategy Dependence of M 2 K χ on the light (left) and strange (right) quark mass: 236000 255000 β = 4 . 0126 β = 4 . 0126 β = 3 . 84 β = 3 . 84 235800 β = 3 . 92 β = 3 . 92 250000 235600 245000 235400 K χ /MeV 2 K χ /MeV 2 240000 235200 M 2 M 2 235000 235000 234800 230000 234600 225000 0.98 0.99 1.00 1.01 1.02 1.03 1.04 0.98 1.00 1.02 1.04 1.06 m ud / ( m φ s /r ) m s /m φ s Lukas Varnhorst for BMW collaboration Sigma terms I 13 von 17

Analysis strategy For the systematic error we used the histogram method and varied the fit function in the following ways: A early and a late plateou for the extraction of M 2 π , M 2 K χ , and f π . Quadratic or no quadratic terms in m ud and m s . All possible combinations of a 2 terms switched on and of. We weight individual fits with their respective AIC weight. We estimated the statistical error with the bootstrap procedured Lukas Varnhorst for BMW collaboration Sigma terms I 14 von 17

Crosscheck A independent analysis was carried out where the m ud ( M 2 π , M 2 K χ ) and m s ( M 2 π , M 2 K χ ) instead of M 2 π ( m ud , m s ) and M 2 π ( m ud , m s ) was fitted. The fits of this type allow for a direct determination of the inverse matrix J − 1 . The two analysis methods are physically closely related but are technically quite different. Both analysis procedures where implemented fully independently and show an excelent agreement. Lukas Varnhorst for BMW collaboration Sigma terms I 15 von 17

Crosscheck Example: Treatment of x and y errors. In one analysis there are only x -errors, in the other cases there are only y errors. For x errors we use the following procedure: Calculate χ 2 via � � f ( x + δ x ) − y � δ = δ x � χ 2 = � δ T C − 1 � δ i Generalizes to the case of several channels. Lukas Varnhorst for BMW collaboration Sigma terms I 16 von 17

Results preliminary The results for the mixing matrix are: LO χ PT M 2 m ud m ud = 0 . 99(3)(4) π statistical error M 2 π systematic error M 2 m ud K χ m ud = − 0 . 08(4)(6) M 2 K χ M 2 m s m s = 0 . 04(4)(2) π M 2 π M 2 m s K χ m s = 1 . 03(3)(2) M 2 K χ 0.2 0.0 0.2 0.4 0.6 0.8 1.0 The result for the strange to light quark mass ratio are m s = 27 . 293(33)(08) m ud m s (FLAG result: m ud = 27 . 30(34)). Lukas Varnhorst for BMW collaboration Sigma terms I 17 von 17